diff --git a/SRC/CMakeLists.txt b/SRC/CMakeLists.txt index a84f784182..7c767b1693 100644 --- a/SRC/CMakeLists.txt +++ b/SRC/CMakeLists.txt @@ -87,7 +87,7 @@ set(SLASRC sgebrd.f sgecon.f sgeequ.f sgees.f sgeesx.f sgeev.f sgeevx.f sgehd2.f sgehrd.f sgelq2.f sgelqf.f sgels.f sgelst.f sgelsd.f sgelss.f sgelsy.f sgeql2.f sgeqlf.f - sgeqp3.f sgeqr2.f sgeqr2p.f sgeqrf.f sgeqrfp.f sgerfs.f sgerq2.f sgerqf.f + sgeqp3.f sgeqp3rk.f sgeqr2.f sgeqr2p.f sgeqrf.f sgeqrfp.f sgerfs.f sgerq2.f sgerqf.f sgesc2.f sgesdd.f sgesv.f sgesvd.f sgesvdx.f sgesvx.f sgetc2.f sgetf2.f sgetri.f sggbak.f sggbal.f @@ -102,7 +102,7 @@ set(SLASRC slangb.f slange.f slangt.f slanhs.f slansb.f slansp.f slansy.f slantb.f slantp.f slantr.f slanv2.f slapll.f slapmt.f - slaqgb.f slaqge.f slaqp2.f slaqps.f slaqsb.f slaqsp.f slaqsy.f + slaqgb.f slaqge.f slaqp2.f slaqps.f slaqp2rk.f slaqp3rk.f slaqsb.f slaqsp.f slaqsy.f slaqr0.f slaqr1.f slaqr2.f slaqr3.f slaqr4.f slaqr5.f slaqtr.f slar1v.f slar2v.f ilaslr.f ilaslc.f slarf.f slarfb.f slarfb_gett.f slarfg.f slarfgp.f slarft.f slarfx.f slarfy.f @@ -177,7 +177,8 @@ set(CLASRC cgbtf2.f cgbtrf.f cgbtrs.f cgebak.f cgebal.f cgebd2.f cgebrd.f cgecon.f cgeequ.f cgees.f cgeesx.f cgeev.f cgeevx.f cgehd2.f cgehrd.f cgelq2.f cgelqf.f - cgels.f cgelst.f cgelsd.f cgelss.f cgelsy.f cgeql2.f cgeqlf.f cgeqp3.f + cgels.f cgelst.f cgelsd.f cgelss.f cgelsy.f cgeql2.f cgeqlf.f + cgeqp3.f cgeqp3rk.f cgeqr2.f cgeqr2p.f cgeqrf.f cgeqrfp.f cgerfs.f cgerq2.f cgerqf.f cgesc2.f cgesdd.f cgesv.f cgesvd.f cgesvdx.f cgesvj.f cgejsv.f cgsvj0.f cgsvj1.f @@ -213,7 +214,7 @@ set(CLASRC clanhb.f clanhe.f clanhp.f clanhs.f clanht.f clansb.f clansp.f clansy.f clantb.f clantp.f clantr.f clapll.f clapmt.f clarcm.f claqgb.f claqge.f - claqhb.f claqhe.f claqhp.f claqp2.f claqps.f claqsb.f + claqhb.f claqhe.f claqhp.f claqp2.f claqps.f claqp2rk.f claqp3rk.f claqsb.f claqr0.f claqr1.f claqr2.f claqr3.f claqr4.f claqr5.f claqsp.f claqsy.f clar1v.f clar2v.f ilaclr.f ilaclc.f clarf.f clarfb.f clarfb_gett.f clarfg.f clarfgp.f clarft.f @@ -287,7 +288,7 @@ set(DLASRC dgebrd.f dgecon.f dgeequ.f dgees.f dgeesx.f dgeev.f dgeevx.f dgehd2.f dgehrd.f dgelq2.f dgelqf.f dgels.f dgelst.f dgelsd.f dgelss.f dgelsy.f dgeql2.f dgeqlf.f - dgeqp3.f dgeqr2.f dgeqr2p.f dgeqrf.f dgeqrfp.f dgerfs.f dgerq2.f dgerqf.f + dgeqp3.f dgeqp3rk.f dgeqr2.f dgeqr2p.f dgeqrf.f dgeqrfp.f dgerfs.f dgerq2.f dgerqf.f dgesc2.f dgesdd.f dgesv.f dgesvd.f dgesvdx.f dgesvx.f dgetc2.f dgetf2.f dgetrf.f dgetrf2.f dgetri.f dgetrs.f dggbak.f dggbal.f @@ -302,7 +303,7 @@ set(DLASRC dlangb.f dlange.f dlangt.f dlanhs.f dlansb.f dlansp.f dlansy.f dlantb.f dlantp.f dlantr.f dlanv2.f dlapll.f dlapmt.f - dlaqgb.f dlaqge.f dlaqp2.f dlaqps.f dlaqsb.f dlaqsp.f dlaqsy.f + dlaqgb.f dlaqge.f dlaqp2.f dlaqps.f dlaqp2rk.f dlaqp3rk.f dlaqsb.f dlaqsp.f dlaqsy.f dlaqr0.f dlaqr1.f dlaqr2.f dlaqr3.f dlaqr4.f dlaqr5.f dlaqtr.f dlar1v.f dlar2v.f iladlr.f iladlc.f dlarf.f dlarfb.f dlarfb_gett.f dlarfg.f dlarfgp.f dlarft.f dlarfx.f dlarfy.f @@ -375,7 +376,8 @@ set(ZLASRC zgbtf2.f zgbtrf.f zgbtrs.f zgebak.f zgebal.f zgebd2.f zgebrd.f zgecon.f zgeequ.f zgees.f zgeesx.f zgeev.f zgeevx.f zgehd2.f zgehrd.f zgelq2.f zgelqf.f - zgels.f zgelst.f zgelsd.f zgelss.f zgelsy.f zgeql2.f zgeqlf.f zgeqp3.f + zgels.f zgelst.f zgelsd.f zgelss.f zgelsy.f zgeql2.f zgeqlf.f + zgeqp3.f zgeqp3rk.f zgeqr2.f zgeqr2p.f zgeqrf.f zgeqrfp.f zgerfs.f zgerq2.f zgerqf.f zgesc2.f zgesdd.f zgesv.f zgesvd.f zgesvdx.f zgesvx.f zgesvj.f zgejsv.f zgsvj0.f zgsvj1.f @@ -412,7 +414,7 @@ set(ZLASRC zlanhe.f zlanhp.f zlanhs.f zlanht.f zlansb.f zlansp.f zlansy.f zlantb.f zlantp.f zlantr.f zlapll.f zlapmt.f zlaqgb.f zlaqge.f - zlaqhb.f zlaqhe.f zlaqhp.f zlaqp2.f zlaqps.f zlaqsb.f + zlaqhb.f zlaqhe.f zlaqhp.f zlaqp2.f zlaqps.f zlaqp2rk.f zlaqp3rk.f zlaqsb.f zlaqr0.f zlaqr1.f zlaqr2.f zlaqr3.f zlaqr4.f zlaqr5.f zlaqsp.f zlaqsy.f zlar1v.f zlar2v.f ilazlr.f ilazlc.f zlarcm.f zlarf.f zlarfb.f zlarfb_gett.f diff --git a/SRC/Makefile b/SRC/Makefile index 40041b8991..bfa4d0fdca 100644 --- a/SRC/Makefile +++ b/SRC/Makefile @@ -119,7 +119,7 @@ SLASRC = \ sgebrd.o sgecon.o sgeequ.o sgees.o sgeesx.o sgeev.o sgeevx.o \ sgehd2.o sgehrd.o sgelq2.o sgelqf.o \ sgels.o sgelst.o sgelsd.o sgelss.o sgelsy.o sgeql2.o sgeqlf.o \ - sgeqp3.o sgeqr2.o sgeqr2p.o sgeqrf.o sgeqrfp.o sgerfs.o \ + sgeqp3.o sgeqp3rk.o sgeqr2.o sgeqr2p.o sgeqrf.o sgeqrfp.o sgerfs.o \ sgerq2.o sgerqf.o sgesc2.o sgesdd.o sgesv.o sgesvd.o sgesvdx.o sgesvx.o \ sgetc2.o sgetf2.o sgetri.o \ sggbak.o sggbal.o sgges.o sgges3.o sggesx.o \ @@ -134,7 +134,7 @@ SLASRC = \ slangb.o slange.o slangt.o slanhs.o slansb.o slansp.o \ slansy.o slantb.o slantp.o slantr.o slanv2.o \ slapll.o slapmt.o \ - slaqgb.o slaqge.o slaqp2.o slaqps.o slaqsb.o slaqsp.o slaqsy.o \ + slaqgb.o slaqge.o slaqp2.o slaqps.o slaqp2rk.o slaqp3rk.o slaqsb.o slaqsp.o slaqsy.o \ slaqr0.o slaqr1.o slaqr2.o slaqr3.o slaqr4.o slaqr5.o \ slaqtr.o slar1v.o slar2v.o ilaslr.o ilaslc.o \ slarf.o slarfb.o slarfb_gett.o slarfg.o slarfgp.o slarft.o slarfx.o slarfy.o \ @@ -211,8 +211,8 @@ CLASRC = \ cgbtf2.o cgbtrf.o cgbtrs.o cgebak.o cgebal.o cgebd2.o cgebrd.o \ cgecon.o cgeequ.o cgees.o cgeesx.o cgeev.o cgeevx.o \ cgehd2.o cgehrd.o cgelq2.o cgelqf.o \ - cgels.o cgelst.o cgelsd.o cgelss.o cgelsy.o cgeql2.o cgeqlf.o cgeqp3.o \ - cgeqr2.o cgeqr2p.o cgeqrf.o cgeqrfp.o cgerfs.o \ + cgels.o cgelst.o cgelsd.o cgelss.o cgelsy.o cgeql2.o cgeqlf.o \ + cgeqp3.o cgeqp3rk.o cgeqr2.o cgeqr2p.o cgeqrf.o cgeqrfp.o cgerfs.o \ cgerq2.o cgerqf.o cgesc2.o cgesdd.o cgesv.o cgesvd.o cgesvdx.o \ cgesvj.o cgejsv.o cgsvj0.o cgsvj1.o \ cgesvx.o cgetc2.o cgetf2.o cgetri.o \ @@ -246,7 +246,7 @@ CLASRC = \ clanhb.o clanhe.o \ clanhp.o clanhs.o clanht.o clansb.o clansp.o clansy.o clantb.o \ clantp.o clantr.o clapll.o clapmt.o clarcm.o claqgb.o claqge.o \ - claqhb.o claqhe.o claqhp.o claqp2.o claqps.o claqsb.o \ + claqhb.o claqhe.o claqhp.o claqp2.o claqps.o claqp2rk.o claqp3rk.o claqsb.o \ claqr0.o claqr1.o claqr2.o claqr3.o claqr4.o claqr5.o \ claqsp.o claqsy.o clar1v.o clar2v.o ilaclr.o ilaclc.o \ clarf.o clarfb.o clarfb_gett.o clarfg.o clarft.o clarfgp.o \ @@ -321,7 +321,7 @@ DLASRC = \ dgebrd.o dgecon.o dgeequ.o dgees.o dgeesx.o dgeev.o dgeevx.o \ dgehd2.o dgehrd.o dgelq2.o dgelqf.o \ dgels.o dgelst.o dgelsd.o dgelss.o dgelsy.o dgeql2.o dgeqlf.o \ - dgeqp3.o dgeqr2.o dgeqr2p.o dgeqrf.o dgeqrfp.o dgerfs.o \ + dgeqp3.o dgeqp3rk.o dgeqr2.o dgeqr2p.o dgeqrf.o dgeqrfp.o dgerfs.o \ dgerq2.o dgerqf.o dgesc2.o dgesdd.o dgesv.o dgesvd.o dgesvdx.o dgesvx.o \ dgetc2.o dgetf2.o dgetrf.o dgetri.o \ dgetrs.o dggbak.o dggbal.o dgges.o dgges3.o dggesx.o \ @@ -336,7 +336,7 @@ DLASRC = \ dlangb.o dlange.o dlangt.o dlanhs.o dlansb.o dlansp.o \ dlansy.o dlantb.o dlantp.o dlantr.o dlanv2.o \ dlapll.o dlapmt.o \ - dlaqgb.o dlaqge.o dlaqp2.o dlaqps.o dlaqsb.o dlaqsp.o dlaqsy.o \ + dlaqgb.o dlaqge.o dlaqp2.o dlaqps.o dlaqp2rk.o dlaqp3rk.o dlaqsb.o dlaqsp.o dlaqsy.o \ dlaqr0.o dlaqr1.o dlaqr2.o dlaqr3.o dlaqr4.o dlaqr5.o \ dlaqtr.o dlar1v.o dlar2v.o iladlr.o iladlc.o \ dlarf.o dlarfb.o dlarfb_gett.o dlarfg.o dlarfgp.o dlarft.o dlarfx.o dlarfy.o \ @@ -412,7 +412,8 @@ ZLASRC = \ zgbtf2.o zgbtrf.o zgbtrs.o zgebak.o zgebal.o zgebd2.o zgebrd.o \ zgecon.o zgeequ.o zgees.o zgeesx.o zgeev.o zgeevx.o \ zgehd2.o zgehrd.o zgelq2.o zgelqf.o \ - zgels.o zgelst.o zgelsd.o zgelss.o zgelsy.o zgeql2.o zgeqlf.o zgeqp3.o \ + zgels.o zgelst.o zgelsd.o zgelss.o zgelsy.o zgeql2.o zgeqlf.o \ + zgeqp3.o zgeqp3rk.o \ zgeqr2.o zgeqr2p.o zgeqrf.o zgeqrfp.o zgerfs.o zgerq2.o zgerqf.o \ zgesc2.o zgesdd.o zgesv.o zgesvd.o zgesvdx.o \ zgesvj.o zgejsv.o zgsvj0.o zgsvj1.o \ @@ -449,7 +450,7 @@ ZLASRC = \ zlanhe.o \ zlanhp.o zlanhs.o zlanht.o zlansb.o zlansp.o zlansy.o zlantb.o \ zlantp.o zlantr.o zlapll.o zlapmt.o zlaqgb.o zlaqge.o \ - zlaqhb.o zlaqhe.o zlaqhp.o zlaqp2.o zlaqps.o zlaqsb.o \ + zlaqhb.o zlaqhe.o zlaqhp.o zlaqp2.o zlaqps.o zlaqp2rk.o zlaqp3rk.o zlaqsb.o \ zlaqr0.o zlaqr1.o zlaqr2.o zlaqr3.o zlaqr4.o zlaqr5.o \ zlaqsp.o zlaqsy.o zlar1v.o zlar2v.o ilazlr.o ilazlc.o \ zlarcm.o zlarf.o zlarfb.o zlarfb_gett.o \ diff --git a/SRC/cgeqp3rk.f b/SRC/cgeqp3rk.f new file mode 100755 index 0000000000..70789e64fb --- /dev/null +++ b/SRC/cgeqp3rk.f @@ -0,0 +1,1091 @@ +*> \brief \b CGEQP3RK computes a truncated Householder QR factorization with column pivoting of a complex m-by-n matrix A by using Level 3 BLAS and overwrites m-by-nrhs matrix B with Q**H * B. +* +* =========== DOCUMENTATION =========== +* +* Online html documentation available at +* http://www.netlib.org/lapack/explore-html/ +* +*> \htmlonly +*> Download CGEQP3RK + dependencies +*> +*> [TGZ] +*> +*> [ZIP] +*> +*> [TXT] +*> \endhtmlonly +* +* Definition: +* =========== +* +* SUBROUTINE CGEQP3RK( M, N, NRHS, KMAX, ABSTOL, RELTOL, A, LDA, +* $ K, MAXC2NRMK, RELMAXC2NRMK, JPIV, TAU, +* $ WORK, LWORK, RWORK, IWORK, INFO ) +* IMPLICIT NONE +* +* .. Scalar Arguments .. +* INTEGER INFO, K, KMAX, LDA, LWORK, M, N, NRHS +* REAL ABSTOL, MAXC2NRMK, RELMAXC2NRMK, RELTOL +* .. +* .. Array Arguments .. +* INTEGER IWORK( * ), JPIV( * ) +* REAL RWORK( * ) +* COMPLEX A( LDA, * ), TAU( * ), WORK( * ) +* .. +* +* +*> \par Purpose: +* ============= +*> +*> \verbatim +*> +*> CGEQP3RK performs two tasks simultaneously: +*> +*> Task 1: The routine computes a truncated (rank K) or full rank +*> Householder QR factorization with column pivoting of a complex +*> M-by-N matrix A using Level 3 BLAS. K is the number of columns +*> that were factorized, i.e. factorization rank of the +*> factor R, K <= min(M,N). +*> +*> A * P(K) = Q(K) * R(K) = +*> +*> = Q(K) * ( R11(K) R12(K) ) = Q(K) * ( R(K)_approx ) +*> ( 0 R22(K) ) ( 0 R(K)_residual ), +*> +*> where: +*> +*> P(K) is an N-by-N permutation matrix; +*> Q(K) is an M-by-M orthogonal matrix; +*> R(K)_approx = ( R11(K), R12(K) ) is a rank K approximation of the +*> full rank factor R with K-by-K upper-triangular +*> R11(K) and K-by-N rectangular R12(K). The diagonal +*> entries of R11(K) appear in non-increasing order +*> of absolute value, and absolute values of all of +*> them exceed the maximum column 2-norm of R22(K) +*> up to roundoff error. +*> R(K)_residual = R22(K) is the residual of a rank K approximation +*> of the full rank factor R. It is a +*> an (M-K)-by-(N-K) rectangular matrix; +*> 0 is a an (M-K)-by-K zero matrix. +*> +*> Task 2: At the same time, the routine overwrites a complex M-by-NRHS +*> matrix B with Q(K)**H * B using Level 3 BLAS. +*> +*> ===================================================================== +*> +*> The matrices A and B are stored on input in the array A as +*> the left and right blocks A(1:M,1:N) and A(1:M, N+1:N+NRHS) +*> respectively. +*> +*> N NRHS +*> array_A = M [ mat_A, mat_B ] +*> +*> The truncation criteria (i.e. when to stop the factorization) +*> can be any of the following: +*> +*> 1) The input parameter KMAX, the maximum number of columns +*> KMAX to factorize, i.e. the factorization rank is limited +*> to KMAX. If KMAX >= min(M,N), the criterion is not used. +*> +*> 2) The input parameter ABSTOL, the absolute tolerance for +*> the maximum column 2-norm of the residual matrix R22(K). This +*> means that the factorization stops if this norm is less or +*> equal to ABSTOL. If ABSTOL < 0.0, the criterion is not used. +*> +*> 3) The input parameter RELTOL, the tolerance for the maximum +*> column 2-norm matrix of the residual matrix R22(K) divided +*> by the maximum column 2-norm of the original matrix A, which +*> is equal to abs(R(1,1)). This means that the factorization stops +*> when the ratio of the maximum column 2-norm of R22(K) to +*> the maximum column 2-norm of A is less than or equal to RELTOL. +*> If RELTOL < 0.0, the criterion is not used. +*> +*> 4) In case both stopping criteria ABSTOL or RELTOL are not used, +*> and when the residual matrix R22(K) is a zero matrix in some +*> factorization step K. ( This stopping criterion is implicit. ) +*> +*> The algorithm stops when any of these conditions is first +*> satisfied, otherwise the whole matrix A is factorized. +*> +*> To factorize the whole matrix A, use the values +*> KMAX >= min(M,N), ABSTOL < 0.0 and RELTOL < 0.0. +*> +*> The routine returns: +*> a) Q(K), R(K)_approx = ( R11(K), R12(K) ), +*> R(K)_residual = R22(K), P(K), i.e. the resulting matrices +*> of the factorization; P(K) is represented by JPIV, +*> ( if K = min(M,N), R(K)_approx is the full factor R, +*> and there is no residual matrix R(K)_residual); +*> b) K, the number of columns that were factorized, +*> i.e. factorization rank; +*> c) MAXC2NRMK, the maximum column 2-norm of the residual +*> matrix R(K)_residual = R22(K), +*> ( if K = min(M,N), MAXC2NRMK = 0.0 ); +*> d) RELMAXC2NRMK equals MAXC2NRMK divided by MAXC2NRM, the maximum +*> column 2-norm of the original matrix A, which is equal +*> to abs(R(1,1)), ( if K = min(M,N), RELMAXC2NRMK = 0.0 ); +*> e) Q(K)**H * B, the matrix B with the orthogonal +*> transformation Q(K)**H applied on the left. +*> +*> The N-by-N permutation matrix P(K) is stored in a compact form in +*> the integer array JPIV. For 1 <= j <= N, column j +*> of the matrix A was interchanged with column JPIV(j). +*> +*> The M-by-M orthogonal matrix Q is represented as a product +*> of elementary Householder reflectors +*> +*> Q(K) = H(1) * H(2) * . . . * H(K), +*> +*> where K is the number of columns that were factorized. +*> +*> Each H(j) has the form +*> +*> H(j) = I - tau * v * v**H, +*> +*> where 1 <= j <= K and +*> I is an M-by-M identity matrix, +*> tau is a complex scalar, +*> v is a complex vector with v(1:j-1) = 0 and v(j) = 1. +*> +*> v(j+1:M) is stored on exit in A(j+1:M,j) and tau in TAU(j). +*> +*> See the Further Details section for more information. +*> \endverbatim +* +* Arguments: +* ========== +* +*> \param[in] M +*> \verbatim +*> M is INTEGER +*> The number of rows of the matrix A. M >= 0. +*> \endverbatim +*> +*> \param[in] N +*> \verbatim +*> N is INTEGER +*> The number of columns of the matrix A. N >= 0. +*> \endverbatim +*> +*> \param[in] NRHS +*> \verbatim +*> NRHS is INTEGER +*> The number of right hand sides, i.e. the number of +*> columns of the matrix B. NRHS >= 0. +*> \endverbatim +*> +*> \param[in] KMAX +*> \verbatim +*> KMAX is INTEGER +*> +*> The first factorization stopping criterion. KMAX >= 0. +*> +*> The maximum number of columns of the matrix A to factorize, +*> i.e. the maximum factorization rank. +*> +*> a) If KMAX >= min(M,N), then this stopping criterion +*> is not used, the routine factorizes columns +*> depending on ABSTOL and RELTOL. +*> +*> b) If KMAX = 0, then this stopping criterion is +*> satisfied on input and the routine exits immediately. +*> This means that the factorization is not performed, +*> the matrices A and B are not modified, and +*> the matrix A is itself the residual. +*> \endverbatim +*> +*> \param[in] ABSTOL +*> \verbatim +*> ABSTOL is REAL +*> +*> The second factorization stopping criterion, cannot be NaN. +*> +*> The absolute tolerance (stopping threshold) for +*> maximum column 2-norm of the residual matrix R22(K). +*> The algorithm converges (stops the factorization) when +*> the maximum column 2-norm of the residual matrix R22(K) +*> is less than or equal to ABSTOL. Let SAFMIN = DLAMCH('S'). +*> +*> a) If ABSTOL is NaN, then no computation is performed +*> and an error message ( INFO = -5 ) is issued +*> by XERBLA. +*> +*> b) If ABSTOL < 0.0, then this stopping criterion is not +*> used, the routine factorizes columns depending +*> on KMAX and RELTOL. +*> This includes the case ABSTOL = -Inf. +*> +*> c) If 0.0 <= ABSTOL < 2*SAFMIN, then ABSTOL = 2*SAFMIN +*> is used. This includes the case ABSTOL = -0.0. +*> +*> d) If 2*SAFMIN <= ABSTOL then the input value +*> of ABSTOL is used. +*> +*> Let MAXC2NRM be the maximum column 2-norm of the +*> whole original matrix A. +*> If ABSTOL chosen above is >= MAXC2NRM, then this +*> stopping criterion is satisfied on input and routine exits +*> immediately after MAXC2NRM is computed. The routine +*> returns MAXC2NRM in MAXC2NORMK, +*> and 1.0 in RELMAXC2NORMK. +*> This includes the case ABSTOL = +Inf. This means that the +*> factorization is not performed, the matrices A and B are not +*> modified, and the matrix A is itself the residual. +*> \endverbatim +*> +*> \param[in] RELTOL +*> \verbatim +*> RELTOL is REAL +*> +*> The third factorization stopping criterion, cannot be NaN. +*> +*> The tolerance (stopping threshold) for the ratio +*> abs(R(K+1,K+1))/abs(R(1,1)) of the maximum column 2-norm of +*> the residual matrix R22(K) to the maximum column 2-norm of +*> the original matrix A. The algorithm converges (stops the +*> factorization), when abs(R(K+1,K+1))/abs(R(1,1)) A is less +*> than or equal to RELTOL. Let EPS = DLAMCH('E'). +*> +*> a) If RELTOL is NaN, then no computation is performed +*> and an error message ( INFO = -6 ) is issued +*> by XERBLA. +*> +*> b) If RELTOL < 0.0, then this stopping criterion is not +*> used, the routine factorizes columns depending +*> on KMAX and ABSTOL. +*> This includes the case RELTOL = -Inf. +*> +*> c) If 0.0 <= RELTOL < EPS, then RELTOL = EPS is used. +*> This includes the case RELTOL = -0.0. +*> +*> d) If EPS <= RELTOL then the input value of RELTOL +*> is used. +*> +*> Let MAXC2NRM be the maximum column 2-norm of the +*> whole original matrix A. +*> If RELTOL chosen above is >= 1.0, then this stopping +*> criterion is satisfied on input and routine exits +*> immediately after MAXC2NRM is computed. +*> The routine returns MAXC2NRM in MAXC2NORMK, +*> and 1.0 in RELMAXC2NORMK. +*> This includes the case RELTOL = +Inf. This means that the +*> factorization is not performed, the matrices A and B are not +*> modified, and the matrix A is itself the residual. +*> +*> NOTE: We recommend that RELTOL satisfy +*> min( 10*max(M,N)*EPS, sqrt(EPS) ) <= RELTOL +*> \endverbatim +*> +*> \param[in,out] A +*> \verbatim +*> A is COMPLEX array, dimension (LDA,N+NRHS) +*> +*> On entry: +*> +*> a) The subarray A(1:M,1:N) contains the M-by-N matrix A. +*> b) The subarray A(1:M,N+1:N+NRHS) contains the M-by-NRHS +*> matrix B. +*> +*> N NRHS +*> array_A = M [ mat_A, mat_B ] +*> +*> On exit: +*> +*> a) The subarray A(1:M,1:N) contains parts of the factors +*> of the matrix A: +*> +*> 1) If K = 0, A(1:M,1:N) contains the original matrix A. +*> 2) If K > 0, A(1:M,1:N) contains parts of the +*> factors: +*> +*> 1. The elements below the diagonal of the subarray +*> A(1:M,1:K) together with TAU(1:K) represent the +*> orthogonal matrix Q(K) as a product of K Householder +*> elementary reflectors. +*> +*> 2. The elements on and above the diagonal of +*> the subarray A(1:K,1:N) contain K-by-N +*> upper-trapezoidal matrix +*> R(K)_approx = ( R11(K), R12(K) ). +*> NOTE: If K=min(M,N), i.e. full rank factorization, +*> then R_approx(K) is the full factor R which +*> is upper-trapezoidal. If, in addition, M>=N, +*> then R is upper-triangular. +*> +*> 3. The subarray A(K+1:M,K+1:N) contains (M-K)-by-(N-K) +*> rectangular matrix R(K)_residual = R22(K). +*> +*> b) If NRHS > 0, the subarray A(1:M,N+1:N+NRHS) contains +*> the M-by-NRHS product Q(K)**H * B. +*> \endverbatim +*> +*> \param[in] LDA +*> \verbatim +*> LDA is INTEGER +*> The leading dimension of the array A. LDA >= max(1,M). +*> This is the leading dimension for both matrices, A and B. +*> \endverbatim +*> +*> \param[out] K +*> \verbatim +*> K is INTEGER +*> Factorization rank of the matrix A, i.e. the rank of +*> the factor R, which is the same as the number of non-zero +*> rows of the factor R. 0 <= K <= min(M,KMAX,N). +*> +*> K also represents the number of non-zero Householder +*> vectors. +*> +*> NOTE: If K = 0, a) the arrays A and B are not modified; +*> b) the array TAU(1:min(M,N)) is set to ZERO, +*> if the matrix A does not contain NaN, +*> otherwise the elements TAU(1:min(M,N)) +*> are undefined; +*> c) the elements of the array JPIV are set +*> as follows: for j = 1:N, JPIV(j) = j. +*> \endverbatim +*> +*> \param[out] MAXC2NRMK +*> \verbatim +*> MAXC2NRMK is REAL +*> The maximum column 2-norm of the residual matrix R22(K), +*> when the factorization stopped at rank K. MAXC2NRMK >= 0. +*> +*> a) If K = 0, i.e. the factorization was not performed, +*> the matrix A was not modified and is itself a residual +*> matrix, then MAXC2NRMK equals the maximum column 2-norm +*> of the original matrix A. +*> +*> b) If 0 < K < min(M,N), then MAXC2NRMK is returned. +*> +*> c) If K = min(M,N), i.e. the whole matrix A was +*> factorized and there is no residual matrix, +*> then MAXC2NRMK = 0.0. +*> +*> NOTE: MAXC2NRMK in the factorization step K would equal +*> R(K+1,K+1) in the next factorization step K+1. +*> \endverbatim +*> +*> \param[out] RELMAXC2NRMK +*> \verbatim +*> RELMAXC2NRMK is REAL +*> The ratio MAXC2NRMK / MAXC2NRM of the maximum column +*> 2-norm of the residual matrix R22(K) (when the factorization +*> stopped at rank K) to the maximum column 2-norm of the +*> whole original matrix A. RELMAXC2NRMK >= 0. +*> +*> a) If K = 0, i.e. the factorization was not performed, +*> the matrix A was not modified and is itself a residual +*> matrix, then RELMAXC2NRMK = 1.0. +*> +*> b) If 0 < K < min(M,N), then +*> RELMAXC2NRMK = MAXC2NRMK / MAXC2NRM is returned. +*> +*> c) If K = min(M,N), i.e. the whole matrix A was +*> factorized and there is no residual matrix, +*> then RELMAXC2NRMK = 0.0. +*> +*> NOTE: RELMAXC2NRMK in the factorization step K would equal +*> abs(R(K+1,K+1))/abs(R(1,1)) in the next factorization +*> step K+1. +*> \endverbatim +*> +*> \param[out] JPIV +*> \verbatim +*> JPIV is INTEGER array, dimension (N) +*> Column pivot indices. For 1 <= j <= N, column j +*> of the matrix A was interchanged with column JPIV(j). +*> +*> The elements of the array JPIV(1:N) are always set +*> by the routine, for example, even when no columns +*> were factorized, i.e. when K = 0, the elements are +*> set as JPIV(j) = j for j = 1:N. +*> \endverbatim +*> +*> \param[out] TAU +*> \verbatim +*> TAU is COMPLEX array, dimension (min(M,N)) +*> The scalar factors of the elementary reflectors. +*> +*> If 0 < K <= min(M,N), only the elements TAU(1:K) of +*> the array TAU are modified by the factorization. +*> After the factorization computed, if no NaN was found +*> during the factorization, the remaining elements +*> TAU(K+1:min(M,N)) are set to zero, otherwise the +*> elements TAU(K+1:min(M,N)) are not set and therefore +*> undefined. +*> ( If K = 0, all elements of TAU are set to zero, if +*> the matrix A does not contain NaN. ) +*> \endverbatim +*> +*> \param[out] WORK +*> \verbatim +*> WORK is COMPLEX array, dimension (MAX(1,LWORK)) +*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. +*> \endverbatim +*> +*> \param[in] LWORK +*> \verbatim +*> LWORK is INTEGER +*> The dimension of the array WORK. +*. LWORK >= N+NRHS-1 +*> For optimal performance LWORK >= NB*( N+NRHS+1 ), +*> where NB is the optimal block size for CGEQP3RK returned +*> by ILAENV. Minimal block size MINNB=2. +*> +*> NOTE: The decision, whether to use unblocked BLAS 2 +*> or blocked BLAS 3 code is based not only on the dimension +*> LWORK of the availbale workspace WORK, but also also on the +*> matrix A dimension N via crossover point NX returned +*> by ILAENV. (For N less than NX, unblocked code should be +*> used.) +*> +*> If LWORK = -1, then a workspace query is assumed; +*> the routine only calculates the optimal size of the WORK +*> array, returns this value as the first entry of the WORK +*> array, and no error message related to LWORK is issued +*> by XERBLA. +*> \endverbatim +*> +*> \param[out] RWORK +*> \verbatim +*> RWORK is REAL array, dimension (2*N) +*> \endverbatim +*> +*> \param[out] IWORK +*> \verbatim +*> IWORK is INTEGER array, dimension (N-1). +*> Is a work array. ( IWORK is used to store indices +*> of "bad" columns for norm downdating in the residual +*> matrix in the blocked step auxiliary subroutine CLAQP3RK ). +*> \endverbatim +*> +*> \param[out] INFO +*> \verbatim +*> INFO is INTEGER +*> 1) INFO = 0: successful exit. +*> 2) INFO < 0: if INFO = -i, the i-th argument had an +*> illegal value. +*> 3) If INFO = j_1, where 1 <= j_1 <= N, then NaN was +*> detected and the routine stops the computation. +*> The j_1-th column of the matrix A or the j_1-th +*> element of array TAU contains the first occurrence +*> of NaN in the factorization step K+1 ( when K columns +*> have been factorized ). +*> +*> On exit: +*> K is set to the number of +*> factorized columns without +*> exception. +*> MAXC2NRMK is set to NaN. +*> RELMAXC2NRMK is set to NaN. +*> TAU(K+1:min(M,N)) is not set and contains undefined +*> elements. If j_1=K+1, TAU(K+1) +*> may contain NaN. +*> 4) If INFO = j_2, where N+1 <= j_2 <= 2*N, then no NaN +*> was detected, but +Inf (or -Inf) was detected and +*> the routine continues the computation until completion. +*> The (j_2-N)-th column of the matrix A contains the first +*> occurrence of +Inf (or -Inf) in the factorization +*> step K+1 ( when K columns have been factorized ). +*> \endverbatim +* +* Authors: +* ======== +* +*> \author Univ. of Tennessee +*> \author Univ. of California Berkeley +*> \author Univ. of Colorado Denver +*> \author NAG Ltd. +* +*> \ingroup geqp3rk +* +*> \par Further Details: +* ===================== +* +*> \verbatim +*> CGEQP3RK is based on the same BLAS3 Householder QR factorization +*> algorithm with column pivoting as in CGEQP3 routine which uses +*> CLARFG routine to generate Householder reflectors +*> for QR factorization. +*> +*> We can also write: +*> +*> A = A_approx(K) + A_residual(K) +*> +*> The low rank approximation matrix A(K)_approx from +*> the truncated QR factorization of rank K of the matrix A is: +*> +*> A(K)_approx = Q(K) * ( R(K)_approx ) * P(K)**T +*> ( 0 0 ) +*> +*> = Q(K) * ( R11(K) R12(K) ) * P(K)**T +*> ( 0 0 ) +*> +*> The residual A_residual(K) of the matrix A is: +*> +*> A_residual(K) = Q(K) * ( 0 0 ) * P(K)**T = +*> ( 0 R(K)_residual ) +*> +*> = Q(K) * ( 0 0 ) * P(K)**T +*> ( 0 R22(K) ) +*> +*> The truncated (rank K) factorization guarantees that +*> the maximum column 2-norm of A_residual(K) is less than +*> or equal to MAXC2NRMK up to roundoff error. +*> +*> NOTE: An approximation of the null vectors +*> of A can be easily computed from R11(K) +*> and R12(K): +*> +*> Null( A(K) )_approx = P * ( inv(R11(K)) * R12(K) ) +*> ( -I ) +*> +*> \endverbatim +* +*> \par References: +* ================ +*> [1] A Level 3 BLAS QR factorization algorithm with column pivoting developed in 1996. +*> G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain. +*> X. Sun, Computer Science Dept., Duke University, USA. +*> C. H. Bischof, Math. and Comp. Sci. Div., Argonne National Lab, USA. +*> A BLAS-3 version of the QR factorization with column pivoting. +*> LAPACK Working Note 114 +*> \htmlonly +*> https://www.netlib.org/lapack/lawnspdf/lawn114.pdf +*> \endhtmlonly +*> and in +*> SIAM J. Sci. Comput., 19(5):1486-1494, Sept. 1998. +*> \htmlonly +*> https://doi.org/10.1137/S1064827595296732 +*> \endhtmlonly +*> +*> [2] A partial column norm updating strategy developed in 2006. +*> Z. Drmac and Z. Bujanovic, Dept. of Math., University of Zagreb, Croatia. +*> On the failure of rank revealing QR factorization software – a case study. +*> LAPACK Working Note 176. +*> \htmlonly +*> http://www.netlib.org/lapack/lawnspdf/lawn176.pdf +*> \endhtmlonly +*> and in +*> ACM Trans. Math. Softw. 35, 2, Article 12 (July 2008), 28 pages. +*> \htmlonly +*> https://doi.org/10.1145/1377612.1377616 +*> \endhtmlonly +* +*> \par Contributors: +* ================== +*> +*> \verbatim +*> +*> November 2023, Igor Kozachenko, James Demmel, +*> Computer Science Division, +*> University of California, Berkeley +*> +*> \endverbatim +* +* ===================================================================== + SUBROUTINE CGEQP3RK( M, N, NRHS, KMAX, ABSTOL, RELTOL, A, LDA, + $ K, MAXC2NRMK, RELMAXC2NRMK, JPIV, TAU, + $ WORK, LWORK, RWORK, IWORK, INFO ) + IMPLICIT NONE +* +* -- LAPACK computational routine -- +* -- LAPACK is a software package provided by Univ. of Tennessee, -- +* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- +* +* .. Scalar Arguments .. + INTEGER INFO, K, KF, KMAX, LDA, LWORK, M, N, NRHS + REAL ABSTOL, MAXC2NRMK, RELMAXC2NRMK, RELTOL +* .. +* .. Array Arguments .. + INTEGER IWORK( * ), JPIV( * ) + REAL RWORK( * ) + COMPLEX A( LDA, * ), TAU( * ), WORK( * ) +* .. +* +* ===================================================================== +* +* .. Parameters .. + INTEGER INB, INBMIN, IXOVER + PARAMETER ( INB = 1, INBMIN = 2, IXOVER = 3 ) + REAL ZERO, ONE, TWO + PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0, TWO = 2.0E+0 ) + COMPLEX CZERO + PARAMETER ( CZERO = ( 0.0E+0, 0.0E+0 ) ) +* .. +* .. Local Scalars .. + LOGICAL LQUERY, DONE + INTEGER IINFO, IOFFSET, IWS, J, JB, JBF, JMAXB, JMAX, + $ JMAXC2NRM, KP1, LWKOPT, MINMN, N_SUB, NB, + $ NBMIN, NX + REAL EPS, HUGEVAL, MAXC2NRM, SAFMIN +* .. +* .. External Subroutines .. + EXTERNAL CLAQP2RK, CLAQP3RK, XERBLA +* .. +* .. External Functions .. + LOGICAL SISNAN + INTEGER ISAMAX, ILAENV + REAL SLAMCH, SCNRM2 + EXTERNAL SISNAN, SLAMCH, SCNRM2, ISAMAX, ILAENV +* .. +* .. Intrinsic Functions .. + INTRINSIC CMPLX, MAX, MIN +* .. +* .. Executable Statements .. +* +* Test input arguments +* ==================== +* + INFO = 0 + LQUERY = ( LWORK.EQ.-1 ) + IF( M.LT.0 ) THEN + INFO = -1 + ELSE IF( N.LT.0 ) THEN + INFO = -2 + ELSE IF( NRHS.LT.0 ) THEN + INFO = -3 + ELSE IF( KMAX.LT.0 ) THEN + INFO = -4 + ELSE IF( SISNAN( ABSTOL ) ) THEN + INFO = -5 + ELSE IF( SISNAN( RELTOL ) ) THEN + INFO = -6 + ELSE IF( LDA.LT.MAX( 1, M ) ) THEN + INFO = -8 + END IF +* +* If the input parameters M, N, NRHS, KMAX, LDA are valid: +* a) Test the input workspace size LWORK for the minimum +* size requirement IWS. +* b) Determine the optimal block size NB and optimal +* workspace size LWKOPT to be returned in WORK(1) +* in case of (1) LWORK < IWS, (2) LQUERY = .TRUE., +* (3) when routine exits. +* Here, IWS is the miminum workspace required for unblocked +* code. +* + IF( INFO.EQ.0 ) THEN + MINMN = MIN( M, N ) + IF( MINMN.EQ.0 ) THEN + IWS = 1 + LWKOPT = 1 + ELSE +* +* Minimal workspace size in case of using only unblocked +* BLAS 2 code in CLAQP2RK. +* 1) CLAQP2RK: N+NRHS-1 to use in WORK array that is used +* in CLARF subroutine inside CLAQP2RK to apply an +* elementary reflector from the left. +* TOTAL_WORK_SIZE = 3*N + NRHS - 1 +* + IWS = N + NRHS - 1 +* +* Assign to NB optimal block size. +* + NB = ILAENV( INB, 'CGEQP3RK', ' ', M, N, -1, -1 ) +* +* A formula for the optimal workspace size in case of using +* both unblocked BLAS 2 in CLAQP2RK and blocked BLAS 3 code +* in CLAQP3RK. +* 1) CGEQP3RK, CLAQP2RK, CLAQP3RK: 2*N to store full and +* partial column 2-norms. +* 2) CLAQP2RK: N+NRHS-1 to use in WORK array that is used +* in CLARF subroutine to apply an elementary reflector +* from the left. +* 3) CLAQP3RK: NB*(N+NRHS) to use in the work array F that +* is used to apply a block reflector from +* the left. +* 4) CLAQP3RK: NB to use in the auxilixary array AUX. +* Sizes (2) and ((3) + (4)) should intersect, therefore +* TOTAL_WORK_SIZE = 2*N + NB*( N+NRHS+1 ), given NBMIN=2. +* + LWKOPT = 2*N + NB*( N+NRHS+1 ) + END IF + WORK( 1 ) = CMPLX( LWKOPT ) +* + IF( ( LWORK.LT.IWS ) .AND. .NOT.LQUERY ) THEN + INFO = -15 + END IF + END IF +* +* NOTE: The optimal workspace size is returned in WORK(1), if +* the input parameters M, N, NRHS, KMAX, LDA are valid. +* + IF( INFO.NE.0 ) THEN + CALL XERBLA( 'CGEQP3RK', -INFO ) + RETURN + ELSE IF( LQUERY ) THEN + RETURN + END IF +* +* Quick return if possible for M=0 or N=0. +* + IF( MINMN.EQ.0 ) THEN + K = 0 + MAXC2NRMK = ZERO + RELMAXC2NRMK = ZERO + WORK( 1 ) = CMPLX( LWKOPT ) + RETURN + END IF +* +* ================================================================== +* +* Initialize column pivot array JPIV. +* + DO J = 1, N + JPIV( J ) = J + END DO +* +* ================================================================== +* +* Initialize storage for partial and exact column 2-norms. +* a) The elements WORK(1:N) are used to store partial column +* 2-norms of the matrix A, and may decrease in each computation +* step; initialize to the values of complete columns 2-norms. +* b) The elements WORK(N+1:2*N) are used to store complete column +* 2-norms of the matrix A, they are not changed during the +* computation; initialize the values of complete columns 2-norms. +* + DO J = 1, N + RWORK( J ) = SCNRM2( M, A( 1, J ), 1 ) + RWORK( N+J ) = RWORK( J ) + END DO +* +* ================================================================== +* +* Compute the pivot column index and the maximum column 2-norm +* for the whole original matrix stored in A(1:M,1:N). +* + KP1 = ISAMAX( N, RWORK( 1 ), 1 ) +* +* ==================================================================. +* + IF( SISNAN( MAXC2NRM ) ) THEN +* +* Check if the matrix A contains NaN, set INFO parameter +* to the column number where the first NaN is found and return +* from the routine. +* + K = 0 + INFO = KP1 +* +* Set MAXC2NRMK and RELMAXC2NRMK to NaN. +* + MAXC2NRMK = MAXC2NRM + RELMAXC2NRMK = MAXC2NRM +* +* Array TAU is not set and contains undefined elements. +* + WORK( 1 ) = CMPLX( LWKOPT ) + RETURN + END IF +* +* =================================================================== +* + IF( MAXC2NRM.EQ.ZERO ) THEN +* +* Check is the matrix A is a zero matrix, set array TAU and +* return from the routine. +* + K = 0 + MAXC2NRMK = ZERO + RELMAXC2NRMK = ZERO +* + DO J = 1, MINMN + TAU( J ) = CZERO + END DO +* + WORK( 1 ) = CMPLX( LWKOPT ) + RETURN +* + END IF +* +* =================================================================== +* + HUGEVAL = SLAMCH( 'Overflow' ) +* + IF( MAXC2NRM.GT.HUGEVAL ) THEN +* +* Check if the matrix A contains +Inf or -Inf, set INFO parameter +* to the column number, where the first +/-Inf is found plus N, +* and continue the computation. +* + INFO = N + KP1 +* + END IF +* +* ================================================================== +* +* Quick return if possible for the case when the first +* stopping criterion is satisfied, i.e. KMAX = 0. +* + IF( KMAX.EQ.0 ) THEN + K = 0 + MAXC2NRMK = MAXC2NRM + RELMAXC2NRMK = ONE + DO J = 1, MINMN + TAU( J ) = CZERO + END DO + WORK( 1 ) = CMPLX( LWKOPT ) + RETURN + END IF +* +* ================================================================== +* + EPS = SLAMCH('Epsilon') +* +* Adjust ABSTOL +* + IF( ABSTOL.GE.ZERO ) THEN + SAFMIN = SLAMCH('Safe minimum') + ABSTOL = MAX( ABSTOL, TWO*SAFMIN ) + END IF +* +* Adjust RELTOL +* + IF( RELTOL.GE.ZERO ) THEN + RELTOL = MAX( RELTOL, EPS ) + END IF +* +* =================================================================== +* +* JMAX is the maximum index of the column to be factorized, +* which is also limited by the first stopping criterion KMAX. +* + JMAX = MIN( KMAX, MINMN ) +* +* =================================================================== +* +* Quick return if possible for the case when the second or third +* stopping criterion for the whole original matrix is satified, +* i.e. MAXC2NRM <= ABSTOL or RELMAXC2NRM <= RELTOL +* (which is ONE <= RELTOL). +* + IF( MAXC2NRM.LE.ABSTOL .OR. ONE.LE.RELTOL ) THEN +* + K = 0 + MAXC2NRMK = MAXC2NRM + RELMAXC2NRMK = ONE +* + DO J = 1, MINMN + TAU( J ) = CZERO + END DO +* + WORK( 1 ) = CMPLX( LWKOPT ) + RETURN + END IF +* +* ================================================================== +* Factorize columns +* ================================================================== +* +* Determine the block size. +* + NBMIN = 2 + NX = 0 +* + IF( ( NB.GT.1 ) .AND. ( NB.LT.MINMN ) ) THEN +* +* Determine when to cross over from blocked to unblocked code. +* (for N less than NX, unblocked code should be used). +* + NX = MAX( 0, ILAENV( IXOVER, 'CGEQP3RK', ' ', M, N, -1, -1 ) ) +* + IF( NX.LT.MINMN ) THEN +* +* Determine if workspace is large enough for blocked code. +* + IF( LWORK.LT.LWKOPT ) THEN +* +* Not enough workspace to use optimal block size that +* is currently stored in NB. +* Reduce NB and determine the minimum value of NB. +* + NB = ( LWORK-2*N ) / ( N+1 ) + NBMIN = MAX( 2, ILAENV( INBMIN, 'CGEQP3RK', ' ', M, N, + $ -1, -1 ) ) +* + END IF + END IF + END IF +* +* ================================================================== +* +* DONE is the boolean flag to rerpresent the case when the +* factorization completed in the block factorization routine, +* before the end of the block. +* + DONE = .FALSE. +* +* J is the column index. +* + J = 1 +* +* (1) Use blocked code initially. +* +* JMAXB is the maximum column index of the block, when the +* blocked code is used, is also limited by the first stopping +* criterion KMAX. +* + JMAXB = MIN( KMAX, MINMN - NX ) +* + IF( NB.GE.NBMIN .AND. NB.LT.JMAX .AND. JMAXB.GT.0 ) THEN +* +* Loop over the column blocks of the matrix A(1:M,1:JMAXB). Here: +* J is the column index of a column block; +* JB is the column block size to pass to block factorization +* routine in a loop step; +* JBF is the number of columns that were actually factorized +* that was returned by the block factorization routine +* in a loop step, JBF <= JB; +* N_SUB is the number of columns in the submatrix; +* IOFFSET is the number of rows that should not be factorized. +* + DO WHILE( J.LE.JMAXB ) +* + JB = MIN( NB, JMAXB-J+1 ) + N_SUB = N-J+1 + IOFFSET = J-1 +* +* Factorize JB columns among the columns A(J:N). +* + CALL CLAQP3RK( M, N_SUB, NRHS, IOFFSET, JB, ABSTOL, + $ RELTOL, KP1, MAXC2NRM, A( 1, J ), LDA, + $ DONE, JBF, MAXC2NRMK, RELMAXC2NRMK, + $ JPIV( J ), TAU( J ), + $ RWORK( J ), RWORK( N+J ), + $ WORK( 1 ), WORK( JB+1 ), + $ N+NRHS-J+1, IWORK, IINFO ) +* +* Set INFO on the first occurence of Inf. +* + IF( IINFO.GT.N_SUB .AND. INFO.EQ.0 ) THEN + INFO = 2*IOFFSET + IINFO + END IF +* + IF( DONE ) THEN +* +* Either the submatrix is zero before the end of the +* column block, or ABSTOL or RELTOL criterion is +* satisfied before the end of the column block, we can +* return from the routine. Perform the following before +* returning: +* a) Set the number of factorized columns K, +* K = IOFFSET + JBF from the last call of blocked +* routine. +* NOTE: 1) MAXC2NRMK and RELMAXC2NRMK are returned +* by the block factorization routine; +* 2) The remaining TAUs are set to ZERO by the +* block factorization routine. +* + K = IOFFSET + JBF +* +* Set INFO on the first occurrence of NaN, NaN takes +* prcedence over Inf. +* + IF( IINFO.LE.N_SUB .AND. IINFO.GT.0 ) THEN + INFO = IOFFSET + IINFO + END IF +* +* Return from the routine. +* + WORK( 1 ) = CMPLX( LWKOPT ) +* + RETURN +* + END IF +* + J = J + JBF +* + END DO +* + END IF +* +* Use unblocked code to factor the last or only block. +* J = JMAX+1 means we factorized the maximum possible number of +* columns, that is in ELSE clause we need to compute +* the MAXC2NORM and RELMAXC2NORM to return after we processed +* the blocks. +* + IF( J.LE.JMAX ) THEN +* +* N_SUB is the number of columns in the submatrix; +* IOFFSET is the number of rows that should not be factorized. +* + N_SUB = N-J+1 + IOFFSET = J-1 +* + CALL CLAQP2RK( M, N_SUB, NRHS, IOFFSET, JMAX-J+1, + $ ABSTOL, RELTOL, KP1, MAXC2NRM, A( 1, J ), LDA, + $ KF, MAXC2NRMK, RELMAXC2NRMK, JPIV( J ), + $ TAU( J ), RWORK( J ), RWORK( N+J ), + $ WORK( 1 ), IINFO ) +* +* ABSTOL or RELTOL criterion is satisfied when the number of +* the factorized columns KF is smaller then the number +* of columns JMAX-J+1 supplied to be factorized by the +* unblocked routine, we can return from +* the routine. Perform the following before returning: +* a) Set the number of factorized columns K, +* b) MAXC2NRMK and RELMAXC2NRMK are returned by the +* unblocked factorization routine above. +* + K = J - 1 + KF +* +* Set INFO on the first exception occurence. +* +* Set INFO on the first exception occurence of Inf or NaN, +* (NaN takes precedence over Inf). +* + IF( IINFO.GT.N_SUB .AND. INFO.EQ.0 ) THEN + INFO = 2*IOFFSET + IINFO + ELSE IF( IINFO.LE.N_SUB .AND. IINFO.GT.0 ) THEN + INFO = IOFFSET + IINFO + END IF +* + ELSE +* +* Compute the return values for blocked code. +* +* Set the number of factorized columns if the unblocked routine +* was not called. +* + K = JMAX +* +* If there exits a residual matrix after the blocked code: +* 1) compute the values of MAXC2NRMK, RELMAXC2NRMK of the +* residual matrix, otherwise set them to ZERO; +* 2) Set TAU(K+1:MINMN) to ZERO. +* + IF( K.LT.MINMN ) THEN + JMAXC2NRM = K + ISAMAX( N-K, RWORK( K+1 ), 1 ) + MAXC2NRMK = RWORK( JMAXC2NRM ) + IF( K.EQ.0 ) THEN + RELMAXC2NRMK = ONE + ELSE + RELMAXC2NRMK = MAXC2NRMK / MAXC2NRM + END IF +* + DO J = K + 1, MINMN + TAU( J ) = CZERO + END DO +* + ELSE + MAXC2NRMK = ZERO + RELMAXC2NRMK = ZERO +* + END IF +* +* END IF( J.LE.JMAX ) THEN +* + END IF +* + WORK( 1 ) = CMPLX( LWKOPT ) +* + RETURN +* +* End of CGEQP3RK +* + END diff --git a/SRC/claqp2rk.f b/SRC/claqp2rk.f new file mode 100755 index 0000000000..073ad0f88d --- /dev/null +++ b/SRC/claqp2rk.f @@ -0,0 +1,726 @@ +*> \brief \b CLAQP2RK computes truncated QR factorization with column pivoting of a complex matrix block using Level 2 BLAS and overwrites a complex m-by-nrhs matrix B with Q**H * B. +* +* =========== DOCUMENTATION =========== +* +* Online html documentation available at +* http://www.netlib.org/lapack/explore-html/ +* +*> \htmlonly +*> Download CLAQP2RK + dependencies +*> +*> [TGZ] +*> +*> [ZIP] +*> +*> [TXT] +*> \endhtmlonly +* +* Definition: +* =========== +* +* SUBROUTINE CLAQP2RK( M, N, NRHS, IOFFSET, KMAX, ABSTOL, RELTOL, +* $ KP1, MAXC2NRM, A, LDA, K, MAXC2NRMK, +* $ RELMAXC2NRMK, JPIV, TAU, VN1, VN2, WORK, +* $ INFO ) +* IMPLICIT NONE +* +* .. Scalar Arguments .. +* INTEGER INFO, IOFFSET, KP1, K, KMAX, LDA, M, N, NRHS +* REAL ABSTOL, MAXC2NRM, MAXC2NRMK, RELMAXC2NRMK, +* $ RELTOL +* .. +* .. Array Arguments .. +* INTEGER JPIV( * ) +* REAL VN1( * ), VN2( * ) +* COMPLEX A( LDA, * ), TAU( * ), WORK( * ) +* $ +* .. +* +* +*> \par Purpose: +* ============= +*> +*> \verbatim +*> +*> CLAQP2RK computes a truncated (rank K) or full rank Householder QR +*> factorization with column pivoting of the complex matrix +*> block A(IOFFSET+1:M,1:N) as +*> +*> A * P(K) = Q(K) * R(K). +*> +*> The routine uses Level 2 BLAS. The block A(1:IOFFSET,1:N) +*> is accordingly pivoted, but not factorized. +*> +*> The routine also overwrites the right-hand-sides matrix block B +*> stored in A(IOFFSET+1:M,N+1:N+NRHS) with Q(K)**H * B. +*> \endverbatim +* +* Arguments: +* ========== +* +*> \param[in] M +*> \verbatim +*> M is INTEGER +*> The number of rows of the matrix A. M >= 0. +*> \endverbatim +*> +*> \param[in] N +*> \verbatim +*> N is INTEGER +*> The number of columns of the matrix A. N >= 0. +*> \endverbatim +*> +*> \param[in] NRHS +*> \verbatim +*> NRHS is INTEGER +*> The number of right hand sides, i.e., the number of +*> columns of the matrix B. NRHS >= 0. +*> \endverbatim +*> +*> \param[in] IOFFSET +*> \verbatim +*> IOFFSET is INTEGER +*> The number of rows of the matrix A that must be pivoted +*> but not factorized. IOFFSET >= 0. +*> +*> IOFFSET also represents the number of columns of the whole +*> original matrix A_orig that have been factorized +*> in the previous steps. +*> \endverbatim +*> +*> \param[in] KMAX +*> \verbatim +*> KMAX is INTEGER +*> +*> The first factorization stopping criterion. KMAX >= 0. +*> +*> The maximum number of columns of the matrix A to factorize, +*> i.e. the maximum factorization rank. +*> +*> a) If KMAX >= min(M-IOFFSET,N), then this stopping +*> criterion is not used, factorize columns +*> depending on ABSTOL and RELTOL. +*> +*> b) If KMAX = 0, then this stopping criterion is +*> satisfied on input and the routine exits immediately. +*> This means that the factorization is not performed, +*> the matrices A and B and the arrays TAU, IPIV +*> are not modified. +*> \endverbatim +*> +*> \param[in] ABSTOL +*> \verbatim +*> ABSTOL is REAL, cannot be NaN. +*> +*> The second factorization stopping criterion. +*> +*> The absolute tolerance (stopping threshold) for +*> maximum column 2-norm of the residual matrix. +*> The algorithm converges (stops the factorization) when +*> the maximum column 2-norm of the residual matrix +*> is less than or equal to ABSTOL. +*> +*> a) If ABSTOL < 0.0, then this stopping criterion is not +*> used, the routine factorizes columns depending +*> on KMAX and RELTOL. +*> This includes the case ABSTOL = -Inf. +*> +*> b) If 0.0 <= ABSTOL then the input value +*> of ABSTOL is used. +*> \endverbatim +*> +*> \param[in] RELTOL +*> \verbatim +*> RELTOL is REAL, cannot be NaN. +*> +*> The third factorization stopping criterion. +*> +*> The tolerance (stopping threshold) for the ratio of the +*> maximum column 2-norm of the residual matrix to the maximum +*> column 2-norm of the original matrix A_orig. The algorithm +*> converges (stops the factorization), when this ratio is +*> less than or equal to RELTOL. +*> +*> a) If RELTOL < 0.0, then this stopping criterion is not +*> used, the routine factorizes columns depending +*> on KMAX and ABSTOL. +*> This includes the case RELTOL = -Inf. +*> +*> d) If 0.0 <= RELTOL then the input value of RELTOL +*> is used. +*> \endverbatim +*> +*> \param[in] KP1 +*> \verbatim +*> KP1 is INTEGER +*> The index of the column with the maximum 2-norm in +*> the whole original matrix A_orig determined in the +*> main routine CGEQP3RK. 1 <= KP1 <= N_orig_mat. +*> \endverbatim +*> +*> \param[in] MAXC2NRM +*> \verbatim +*> MAXC2NRM is REAL +*> The maximum column 2-norm of the whole original +*> matrix A_orig computed in the main routine CGEQP3RK. +*> MAXC2NRM >= 0. +*> \endverbatim +*> +*> \param[in,out] A +*> \verbatim +*> A is COMPLEX array, dimension (LDA,N+NRHS) +*> On entry: +*> the M-by-N matrix A and M-by-NRHS matrix B, as in +*> +*> N NRHS +*> array_A = M [ mat_A, mat_B ] +*> +*> On exit: +*> 1. The elements in block A(IOFFSET+1:M,1:K) below +*> the diagonal together with the array TAU represent +*> the orthogonal matrix Q(K) as a product of elementary +*> reflectors. +*> 2. The upper triangular block of the matrix A stored +*> in A(IOFFSET+1:M,1:K) is the triangular factor obtained. +*> 3. The block of the matrix A stored in A(1:IOFFSET,1:N) +*> has been accordingly pivoted, but not factorized. +*> 4. The rest of the array A, block A(IOFFSET+1:M,K+1:N+NRHS). +*> The left part A(IOFFSET+1:M,K+1:N) of this block +*> contains the residual of the matrix A, and, +*> if NRHS > 0, the right part of the block +*> A(IOFFSET+1:M,N+1:N+NRHS) contains the block of +*> the right-hand-side matrix B. Both these blocks have been +*> updated by multiplication from the left by Q(K)**H. +*> \endverbatim +*> +*> \param[in] LDA +*> \verbatim +*> LDA is INTEGER +*> The leading dimension of the array A. LDA >= max(1,M). +*> \endverbatim +*> +*> \param[out] K +*> \verbatim +*> K is INTEGER +*> Factorization rank of the matrix A, i.e. the rank of +*> the factor R, which is the same as the number of non-zero +*> rows of the factor R. 0 <= K <= min(M-IOFFSET,KMAX,N). +*> +*> K also represents the number of non-zero Householder +*> vectors. +*> \endverbatim +*> +*> \param[out] MAXC2NRMK +*> \verbatim +*> MAXC2NRMK is REAL +*> The maximum column 2-norm of the residual matrix, +*> when the factorization stopped at rank K. MAXC2NRMK >= 0. +*> \endverbatim +*> +*> \param[out] RELMAXC2NRMK +*> \verbatim +*> RELMAXC2NRMK is REAL +*> The ratio MAXC2NRMK / MAXC2NRM of the maximum column +*> 2-norm of the residual matrix (when the factorization +*> stopped at rank K) to the maximum column 2-norm of the +*> whole original matrix A. RELMAXC2NRMK >= 0. +*> \endverbatim +*> +*> \param[out] JPIV +*> \verbatim +*> JPIV is INTEGER array, dimension (N) +*> Column pivot indices, for 1 <= j <= N, column j +*> of the matrix A was interchanged with column JPIV(j). +*> \endverbatim +*> +*> \param[out] TAU +*> \verbatim +*> TAU is COMPLEX array, dimension (min(M-IOFFSET,N)) +*> The scalar factors of the elementary reflectors. +*> \endverbatim +*> +*> \param[in,out] VN1 +*> \verbatim +*> VN1 is REAL array, dimension (N) +*> The vector with the partial column norms. +*> \endverbatim +*> +*> \param[in,out] VN2 +*> \verbatim +*> VN2 is REAL array, dimension (N) +*> The vector with the exact column norms. +*> \endverbatim +*> +*> \param[out] WORK +*> \verbatim +*> WORK is COMPLEX array, dimension (N-1) +*> Used in CLARF subroutine to apply an elementary +*> reflector from the left. +*> \endverbatim +*> +*> \param[out] INFO +*> \verbatim +*> INFO is INTEGER +*> 1) INFO = 0: successful exit. +*> 2) If INFO = j_1, where 1 <= j_1 <= N, then NaN was +*> detected and the routine stops the computation. +*> The j_1-th column of the matrix A or the j_1-th +*> element of array TAU contains the first occurrence +*> of NaN in the factorization step K+1 ( when K columns +*> have been factorized ). +*> +*> On exit: +*> K is set to the number of +*> factorized columns without +*> exception. +*> MAXC2NRMK is set to NaN. +*> RELMAXC2NRMK is set to NaN. +*> TAU(K+1:min(M,N)) is not set and contains undefined +*> elements. If j_1=K+1, TAU(K+1) +*> may contain NaN. +*> 3) If INFO = j_2, where N+1 <= j_2 <= 2*N, then no NaN +*> was detected, but +Inf (or -Inf) was detected and +*> the routine continues the computation until completion. +*> The (j_2-N)-th column of the matrix A contains the first +*> occurrence of +Inf (or -Inf) in the factorization +*> step K+1 ( when K columns have been factorized ). +*> \endverbatim +* +* Authors: +* ======== +* +*> \author Univ. of Tennessee +*> \author Univ. of California Berkeley +*> \author Univ. of Colorado Denver +*> \author NAG Ltd. +* +*> \ingroup laqp2rk +* +*> \par References: +* ================ +*> [1] A Level 3 BLAS QR factorization algorithm with column pivoting developed in 1996. +*> G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain. +*> X. Sun, Computer Science Dept., Duke University, USA. +*> C. H. Bischof, Math. and Comp. Sci. Div., Argonne National Lab, USA. +*> A BLAS-3 version of the QR factorization with column pivoting. +*> LAPACK Working Note 114 +*> \htmlonly +*> https://www.netlib.org/lapack/lawnspdf/lawn114.pdf +*> \endhtmlonly +*> and in +*> SIAM J. Sci. Comput., 19(5):1486-1494, Sept. 1998. +*> \htmlonly +*> https://doi.org/10.1137/S1064827595296732 +*> \endhtmlonly +*> +*> [2] A partial column norm updating strategy developed in 2006. +*> Z. Drmac and Z. Bujanovic, Dept. of Math., University of Zagreb, Croatia. +*> On the failure of rank revealing QR factorization software – a case study. +*> LAPACK Working Note 176. +*> \htmlonly +*> http://www.netlib.org/lapack/lawnspdf/lawn176.pdf +*> \endhtmlonly +*> and in +*> ACM Trans. Math. Softw. 35, 2, Article 12 (July 2008), 28 pages. +*> \htmlonly +*> https://doi.org/10.1145/1377612.1377616 +*> \endhtmlonly +* +*> \par Contributors: +* ================== +*> +*> \verbatim +*> +*> November 2023, Igor Kozachenko, James Demmel, +*> Computer Science Division, +*> University of California, Berkeley +*> +*> \endverbatim +* +* ===================================================================== + SUBROUTINE CLAQP2RK( M, N, NRHS, IOFFSET, KMAX, ABSTOL, RELTOL, + $ KP1, MAXC2NRM, A, LDA, K, MAXC2NRMK, + $ RELMAXC2NRMK, JPIV, TAU, VN1, VN2, WORK, + $ INFO ) + IMPLICIT NONE +* +* -- LAPACK auxiliary routine -- +* -- LAPACK is a software package provided by Univ. of Tennessee, -- +* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- +* +* .. Scalar Arguments .. + INTEGER INFO, IOFFSET, KP1, K, KMAX, LDA, M, N, NRHS + REAL ABSTOL, MAXC2NRM, MAXC2NRMK, RELMAXC2NRMK, + $ RELTOL +* .. +* .. Array Arguments .. + INTEGER JPIV( * ) + REAL VN1( * ), VN2( * ) + COMPLEX A( LDA, * ), TAU( * ), WORK( * ) +* .. +* +* ===================================================================== +* +* .. Parameters .. + REAL ZERO, ONE + PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) + COMPLEX CZERO, CONE + PARAMETER ( CZERO = ( 0.0E+0, 0.0E+0 ), + $ CONE = ( 1.0E+0, 0.0E+0 ) ) +* .. +* .. Local Scalars .. + INTEGER I, ITEMP, J, JMAXC2NRM, KK, KP, MINMNFACT, + $ MINMNUPDT + REAL HUGEVAL, TAUNAN, TEMP, TEMP2, TOL3Z + COMPLEX AIKK +* .. +* .. External Subroutines .. + EXTERNAL CLARF, CLARFG, CSWAP +* .. +* .. Intrinsic Functions .. + INTRINSIC ABS, REAL, CONJG, IMAG, MAX, MIN, SQRT +* .. +* .. External Functions .. + LOGICAL SISNAN + INTEGER ISAMAX + REAL SLAMCH, SCNRM2 + EXTERNAL SISNAN, SLAMCH, ISAMAX, SCNRM2 +* .. +* .. Executable Statements .. +* +* Initialize INFO +* + INFO = 0 +* +* MINMNFACT in the smallest dimension of the submatrix +* A(IOFFSET+1:M,1:N) to be factorized. +* +* MINMNUPDT is the smallest dimension +* of the subarray A(IOFFSET+1:M,1:N+NRHS) to be udated, which +* contains the submatrices A(IOFFSET+1:M,1:N) and +* B(IOFFSET+1:M,1:NRHS) as column blocks. +* + MINMNFACT = MIN( M-IOFFSET, N ) + MINMNUPDT = MIN( M-IOFFSET, N+NRHS ) + KMAX = MIN( KMAX, MINMNFACT ) + TOL3Z = SQRT( SLAMCH( 'Epsilon' ) ) + HUGEVAL = SLAMCH( 'Overflow' ) +* +* Compute the factorization, KK is the lomn loop index. +* + DO KK = 1, KMAX +* + I = IOFFSET + KK +* + IF( I.EQ.1 ) THEN +* +* ============================================================ +* +* We are at the first column of the original whole matrix A, +* therefore we use the computed KP1 and MAXC2NRM from the +* main routine. +* + KP = KP1 +* +* ============================================================ +* + ELSE +* +* ============================================================ +* +* Determine the pivot column in KK-th step, i.e. the index +* of the column with the maximum 2-norm in the +* submatrix A(I:M,K:N). +* + KP = ( KK-1 ) + ISAMAX( N-KK+1, VN1( KK ), 1 ) +* +* Determine the maximum column 2-norm and the relative maximum +* column 2-norm of the submatrix A(I:M,KK:N) in step KK. +* RELMAXC2NRMK will be computed later, after somecondition +* checks on MAXC2NRMK. +* + MAXC2NRMK = VN1( KP ) +* +* ============================================================ +* +* Check if the submatrix A(I:M,KK:N) contains NaN, and set +* INFO parameter to the column number, where the first NaN +* is found and return from the routine. +* We need to check the condition only if the +* column index (same as row index) of the original whole +* matrix is larger than 1, since the condition for whole +* original matrix is checked in the main routine. +* + IF( SISNAN( MAXC2NRMK ) ) THEN +* +* Set K, the number of factorized columns. +* that are not zero. +* + K = KK - 1 + INFO = K + KP +* +* Set RELMAXC2NRMK to NaN. +* + RELMAXC2NRMK = MAXC2NRMK +* +* Array TAU(K+1:MINMNFACT) is not set and contains +* undefined elements. +* + RETURN + END IF +* +* ============================================================ +* +* Quick return, if the submatrix A(I:M,KK:N) is +* a zero matrix. +* We need to check the condition only if the +* column index (same as row index) of the original whole +* matrix is larger than 1, since the condition for whole +* original matrix is checked in the main routine. +* + IF( MAXC2NRMK.EQ.ZERO ) THEN +* +* Set K, the number of factorized columns. +* that are not zero. +* + K = KK - 1 + RELMAXC2NRMK = ZERO +* +* Set TAUs corresponding to the columns that were not +* factorized to ZERO, i.e. set TAU(KK:MINMNFACT) to CZERO. +* + DO J = KK, MINMNFACT + TAU( J ) = CZERO + END DO +* +* Return from the routine. +* + RETURN +* + END IF +* +* ============================================================ +* +* Check if the submatrix A(I:M,KK:N) contains Inf, +* set INFO parameter to the column number, where +* the first Inf is found plus N, and continue +* the computation. +* We need to check the condition only if the +* column index (same as row index) of the original whole +* matrix is larger than 1, since the condition for whole +* original matrix is checked in the main routine. +* + IF( INFO.EQ.0 .AND. MAXC2NRMK.GT.HUGEVAL ) THEN + INFO = N + KK - 1 + KP + END IF +* +* ============================================================ +* +* Test for the second and third stopping criteria. +* NOTE: There is no need to test for ABSTOL >= ZERO, since +* MAXC2NRMK is non-negative. Similarly, there is no need +* to test for RELTOL >= ZERO, since RELMAXC2NRMK is +* non-negative. +* We need to check the condition only if the +* column index (same as row index) of the original whole +* matrix is larger than 1, since the condition for whole +* original matrix is checked in the main routine. + + RELMAXC2NRMK = MAXC2NRMK / MAXC2NRM +* + IF( MAXC2NRMK.LE.ABSTOL .OR. RELMAXC2NRMK.LE.RELTOL ) THEN +* +* Set K, the number of factorized columns. +* + K = KK - 1 +* +* Set TAUs corresponding to the columns that were not +* factorized to ZERO, i.e. set TAU(KK:MINMNFACT) to CZERO. +* + DO J = KK, MINMNFACT + TAU( J ) = CZERO + END DO +* +* Return from the routine. +* + RETURN +* + END IF +* +* ============================================================ +* +* End ELSE of IF(I.EQ.1) +* + END IF +* +* =============================================================== +* +* If the pivot column is not the first column of the +* subblock A(1:M,KK:N): +* 1) swap the KK-th column and the KP-th pivot column +* in A(1:M,1:N); +* 2) copy the KK-th element into the KP-th element of the partial +* and exact 2-norm vectors VN1 and VN2. ( Swap is not needed +* for VN1 and VN2 since we use the element with the index +* larger than KK in the next loop step.) +* 3) Save the pivot interchange with the indices relative to the +* the original matrix A, not the block A(1:M,1:N). +* + IF( KP.NE.KK ) THEN + CALL CSWAP( M, A( 1, KP ), 1, A( 1, KK ), 1 ) + VN1( KP ) = VN1( KK ) + VN2( KP ) = VN2( KK ) + ITEMP = JPIV( KP ) + JPIV( KP ) = JPIV( KK ) + JPIV( KK ) = ITEMP + END IF +* +* Generate elementary reflector H(KK) using the column A(I:M,KK), +* if the column has more than one element, otherwise +* the elementary reflector would be an identity matrix, +* and TAU(KK) = CZERO. +* + IF( I.LT.M ) THEN + CALL CLARFG( M-I+1, A( I, KK ), A( I+1, KK ), 1, + $ TAU( KK ) ) + ELSE + TAU( KK ) = CZERO + END IF +* +* Check if TAU(KK) contains NaN, set INFO parameter +* to the column number where NaN is found and return from +* the routine. +* NOTE: There is no need to check TAU(KK) for Inf, +* since CLARFG cannot produce TAU(KK) or Householder vector +* below the diagonal containing Inf. Only BETA on the diagonal, +* returned by CLARFG can contain Inf, which requires +* TAU(KK) to contain NaN. Therefore, this case of generating Inf +* by CLARFG is covered by checking TAU(KK) for NaN. +* + IF( SISNAN( REAL( TAU(KK) ) ) ) THEN + TAUNAN = REAL( TAU(KK) ) + ELSE IF( SISNAN( IMAG( TAU(KK) ) ) ) THEN + TAUNAN = IMAG( TAU(KK) ) + ELSE + TAUNAN = ZERO + END IF +* + IF( SISNAN( TAUNAN ) ) THEN + K = KK - 1 + INFO = KK +* +* Set MAXC2NRMK and RELMAXC2NRMK to NaN. +* + MAXC2NRMK = TAUNAN + RELMAXC2NRMK = TAUNAN +* +* Array TAU(KK:MINMNFACT) is not set and contains +* undefined elements, except the first element TAU(KK) = NaN. +* + RETURN + END IF +* +* Apply H(KK)**H to A(I:M,KK+1:N+NRHS) from the left. +* ( If M >= N, then at KK = N there is no residual matrix, +* i.e. no columns of A to update, only columns of B. +* If M < N, then at KK = M-IOFFSET, I = M and we have a +* one-row residual matrix in A and the elementary +* reflector is a unit matrix, TAU(KK) = CZERO, i.e. no update +* is needed for the residual matrix in A and the +* right-hand-side-matrix in B. +* Therefore, we update only if +* KK < MINMNUPDT = min(M-IOFFSET, N+NRHS) +* condition is satisfied, not only KK < N+NRHS ) +* + IF( KK.LT.MINMNUPDT ) THEN + AIKK = A( I, KK ) + A( I, KK ) = CONE + CALL CLARF( 'Left', M-I+1, N+NRHS-KK, A( I, KK ), 1, + $ CONJG( TAU( KK ) ), A( I, KK+1 ), LDA, + $ WORK( 1 ) ) + A( I, KK ) = AIKK + END IF +* + IF( KK.LT.MINMNFACT ) THEN +* +* Update the partial column 2-norms for the residual matrix, +* only if the residual matrix A(I+1:M,KK+1:N) exists, i.e. +* when KK < min(M-IOFFSET, N). +* + DO J = KK + 1, N + IF( VN1( J ).NE.ZERO ) THEN +* +* NOTE: The following lines follow from the analysis in +* Lapack Working Note 176. +* + TEMP = ONE - ( ABS( A( I, J ) ) / VN1( J ) )**2 + TEMP = MAX( TEMP, ZERO ) + TEMP2 = TEMP*( VN1( J ) / VN2( J ) )**2 + IF( TEMP2 .LE. TOL3Z ) THEN +* +* Compute the column 2-norm for the partial +* column A(I+1:M,J) by explicitly computing it, +* and store it in both partial 2-norm vector VN1 +* and exact column 2-norm vector VN2. +* + VN1( J ) = SCNRM2( M-I, A( I+1, J ), 1 ) + VN2( J ) = VN1( J ) +* + ELSE +* +* Update the column 2-norm for the partial +* column A(I+1:M,J) by removing one +* element A(I,J) and store it in partial +* 2-norm vector VN1. +* + VN1( J ) = VN1( J )*SQRT( TEMP ) +* + END IF + END IF + END DO +* + END IF +* +* End factorization loop +* + END DO +* +* If we reached this point, all colunms have been factorized, +* i.e. no condition was triggered to exit the routine. +* Set the number of factorized columns. +* + K = KMAX +* +* We reached the end of the loop, i.e. all KMAX columns were +* factorized, we need to set MAXC2NRMK and RELMAXC2NRMK before +* we return. +* + IF( K.LT.MINMNFACT ) THEN +* + JMAXC2NRM = K + ISAMAX( N-K, VN1( K+1 ), 1 ) + MAXC2NRMK = VN1( JMAXC2NRM ) +* + IF( K.EQ.0 ) THEN + RELMAXC2NRMK = ONE + ELSE + RELMAXC2NRMK = MAXC2NRMK / MAXC2NRM + END IF +* + ELSE + MAXC2NRMK = ZERO + RELMAXC2NRMK = ZERO + END IF +* +* We reached the end of the loop, i.e. all KMAX columns were +* factorized, set TAUs corresponding to the columns that were +* not factorized to ZERO, i.e. TAU(K+1:MINMNFACT) set to CZERO. +* + DO J = K + 1, MINMNFACT + TAU( J ) = CZERO + END DO +* + RETURN +* +* End of CLAQP2RK +* + END diff --git a/SRC/claqp3rk.f b/SRC/claqp3rk.f new file mode 100755 index 0000000000..af5e856457 --- /dev/null +++ b/SRC/claqp3rk.f @@ -0,0 +1,947 @@ +*> \brief \b CLAQP3RK computes a step of truncated QR factorization with column pivoting of a complex m-by-n matrix A using Level 3 BLAS and overwrites a complex m-by-nrhs matrix B with Q**H * B. +* +* =========== DOCUMENTATION =========== +* +* Online html documentation available at +* http://www.netlib.org/lapack/explore-html/ +* +*> \htmlonly +*> Download CLAQP3RK + dependencies +*> +*> [TGZ] +*> +*> [ZIP] +*> +*> [TXT] +*> \endhtmlonly +* +* Definition: +* =========== +* +* SUBROUTINE CLAQP3RK( M, N, NRHS, IOFFSET, NB, ABSTOL, +* $ RELTOL, KP1, MAXC2NRM, A, LDA, DONE, KB, +* $ MAXC2NRMK, RELMAXC2NRMK, JPIV, TAU, +* $ VN1, VN2, AUXV, F, LDF, IWORK, INFO ) +* IMPLICIT NONE +* LOGICAL DONE +* INTEGER INFO, IOFFSET, KB, KP1, LDA, LDF, M, N, +* $ NB, NRHS +* REAL ABSTOL, MAXC2NRM, MAXC2NRMK, RELMAXC2NRMK, +* $ RELTOL +* .. +* .. Array Arguments .. +* INTEGER IWORK( * ), JPIV( * ) +* REAL VN1( * ), VN2( * ) +* COMPLEX*16 A( LDA, * ), AUXV( * ), F( LDF, * ), TAU( * ) +* .. +* +* +*> \par Purpose: +* ============= +*> +*> \verbatim +*> +*> CLAQP3RK computes a step of truncated QR factorization with column +*> pivoting of a complex M-by-N matrix A block A(IOFFSET+1:M,1:N) +*> by using Level 3 BLAS as +*> +*> A * P(KB) = Q(KB) * R(KB). +*> +*> The routine tries to factorize NB columns from A starting from +*> the row IOFFSET+1 and updates the residual matrix with BLAS 3 +*> xGEMM. The number of actually factorized columns is returned +*> is smaller than NB. +*> +*> Block A(1:IOFFSET,1:N) is accordingly pivoted, but not factorized. +*> +*> The routine also overwrites the right-hand-sides B matrix stored +*> in A(IOFFSET+1:M,1:N+1:N+NRHS) with Q(KB)**H * B. +*> +*> Cases when the number of factorized columns KB < NB: +*> +*> (1) In some cases, due to catastrophic cancellations, it cannot +*> factorize all NB columns and need to update the residual matrix. +*> Hence, the actual number of factorized columns in the block returned +*> in KB is smaller than NB. The logical DONE is returned as FALSE. +*> The factorization of the whole original matrix A_orig must proceed +*> with the next block. +*> +*> (2) Whenever the stopping criterion ABSTOL or RELTOL is satisfied, +*> the factorization of the whole original matrix A_orig is stopped, +*> the logical DONE is returned as TRUE. The number of factorized +*> columns which is smaller than NB is returned in KB. +*> +*> (3) In case both stopping criteria ABSTOL or RELTOL are not used, +*> and when the residual matrix is a zero matrix in some factorization +*> step KB, the factorization of the whole original matrix A_orig is +*> stopped, the logical DONE is returned as TRUE. The number of +*> factorized columns which is smaller than NB is returned in KB. +*> +*> (4) Whenever NaN is detected in the matrix A or in the array TAU, +*> the factorization of the whole original matrix A_orig is stopped, +*> the logical DONE is returned as TRUE. The number of factorized +*> columns which is smaller than NB is returned in KB. The INFO +*> parameter is set to the column index of the first NaN occurrence. +*> +*> \endverbatim +* +* Arguments: +* ========== +* +*> \param[in] M +*> \verbatim +*> M is INTEGER +*> The number of rows of the matrix A. M >= 0. +*> \endverbatim +*> +*> \param[in] N +*> \verbatim +*> N is INTEGER +*> The number of columns of the matrix A. N >= 0 +*> \endverbatim +*> +*> \param[in] NRHS +*> \verbatim +*> NRHS is INTEGER +*> The number of right hand sides, i.e., the number of +*> columns of the matrix B. NRHS >= 0. +*> \endverbatim +*> +*> \param[in] IOFFSET +*> \verbatim +*> IOFFSET is INTEGER +*> The number of rows of the matrix A that must be pivoted +*> but not factorized. IOFFSET >= 0. +*> +*> IOFFSET also represents the number of columns of the whole +*> original matrix A_orig that have been factorized +*> in the previous steps. +*> \endverbatim +*> +*> \param[in] NB +*> \verbatim +*> NB is INTEGER +*> Factorization block size, i.e the number of columns +*> to factorize in the matrix A. 0 <= NB +*> +*> If NB = 0, then the routine exits immediately. +*> This means that the factorization is not performed, +*> the matrices A and B and the arrays TAU, IPIV +*> are not modified. +*> \endverbatim +*> +*> \param[in] ABSTOL +*> \verbatim +*> ABSTOL is REAL, cannot be NaN. +*> +*> The absolute tolerance (stopping threshold) for +*> maximum column 2-norm of the residual matrix. +*> The algorithm converges (stops the factorization) when +*> the maximum column 2-norm of the residual matrix +*> is less than or equal to ABSTOL. +*> +*> a) If ABSTOL < 0.0, then this stopping criterion is not +*> used, the routine factorizes columns depending +*> on NB and RELTOL. +*> This includes the case ABSTOL = -Inf. +*> +*> b) If 0.0 <= ABSTOL then the input value +*> of ABSTOL is used. +*> \endverbatim +*> +*> \param[in] RELTOL +*> \verbatim +*> RELTOL is REAL, cannot be NaN. +*> +*> The tolerance (stopping threshold) for the ratio of the +*> maximum column 2-norm of the residual matrix to the maximum +*> column 2-norm of the original matrix A_orig. The algorithm +*> converges (stops the factorization), when this ratio is +*> less than or equal to RELTOL. +*> +*> a) If RELTOL < 0.0, then this stopping criterion is not +*> used, the routine factorizes columns depending +*> on NB and ABSTOL. +*> This includes the case RELTOL = -Inf. +*> +*> d) If 0.0 <= RELTOL then the input value of RELTOL +*> is used. +*> \endverbatim +*> +*> \param[in] KP1 +*> \verbatim +*> KP1 is INTEGER +*> The index of the column with the maximum 2-norm in +*> the whole original matrix A_orig determined in the +*> main routine CGEQP3RK. 1 <= KP1 <= N_orig. +*> \endverbatim +*> +*> \param[in] MAXC2NRM +*> \verbatim +*> MAXC2NRM is REAL +*> The maximum column 2-norm of the whole original +*> matrix A_orig computed in the main routine CGEQP3RK. +*> MAXC2NRM >= 0. +*> \endverbatim +*> +*> \param[in,out] A +*> \verbatim +*> A is COMPLEX array, dimension (LDA,N+NRHS) +*> On entry: +*> the M-by-N matrix A and M-by-NRHS matrix B, as in +*> +*> N NRHS +*> array_A = M [ mat_A, mat_B ] +*> +*> On exit: +*> 1. The elements in block A(IOFFSET+1:M,1:KB) below +*> the diagonal together with the array TAU represent +*> the orthogonal matrix Q(KB) as a product of elementary +*> reflectors. +*> 2. The upper triangular block of the matrix A stored +*> in A(IOFFSET+1:M,1:KB) is the triangular factor obtained. +*> 3. The block of the matrix A stored in A(1:IOFFSET,1:N) +*> has been accordingly pivoted, but not factorized. +*> 4. The rest of the array A, block A(IOFFSET+1:M,KB+1:N+NRHS). +*> The left part A(IOFFSET+1:M,KB+1:N) of this block +*> contains the residual of the matrix A, and, +*> if NRHS > 0, the right part of the block +*> A(IOFFSET+1:M,N+1:N+NRHS) contains the block of +*> the right-hand-side matrix B. Both these blocks have been +*> updated by multiplication from the left by Q(KB)**H. +*> \endverbatim +*> +*> \param[in] LDA +*> \verbatim +*> LDA is INTEGER +*> The leading dimension of the array A. LDA >= max(1,M). +*> \endverbatim +*> +*> \param[out] +*> \verbatim +*> DONE is LOGICAL +*> TRUE: a) if the factorization completed before processing +*> all min(M-IOFFSET,NB,N) columns due to ABSTOL +*> or RELTOL criterion, +*> b) if the factorization completed before processing +*> all min(M-IOFFSET,NB,N) columns due to the +*> residual matrix being a ZERO matrix. +*> c) when NaN was detected in the matrix A +*> or in the array TAU. +*> FALSE: otherwise. +*> \endverbatim +*> +*> \param[out] KB +*> \verbatim +*> KB is INTEGER +*> Factorization rank of the matrix A, i.e. the rank of +*> the factor R, which is the same as the number of non-zero +*> rows of the factor R. 0 <= KB <= min(M-IOFFSET,NB,N). +*> +*> KB also represents the number of non-zero Householder +*> vectors. +*> \endverbatim +*> +*> \param[out] MAXC2NRMK +*> \verbatim +*> MAXC2NRMK is REAL +*> The maximum column 2-norm of the residual matrix, +*> when the factorization stopped at rank KB. MAXC2NRMK >= 0. +*> \endverbatim +*> +*> \param[out] RELMAXC2NRMK +*> \verbatim +*> RELMAXC2NRMK is REAL +*> The ratio MAXC2NRMK / MAXC2NRM of the maximum column +*> 2-norm of the residual matrix (when the factorization +*> stopped at rank KB) to the maximum column 2-norm of the +*> original matrix A_orig. RELMAXC2NRMK >= 0. +*> \endverbatim +*> +*> \param[out] JPIV +*> \verbatim +*> JPIV is INTEGER array, dimension (N) +*> Column pivot indices, for 1 <= j <= N, column j +*> of the matrix A was interchanged with column JPIV(j). +*> \endverbatim +*> +*> \param[out] TAU +*> \verbatim +*> TAU is COMPLEX array, dimension (min(M-IOFFSET,N)) +*> The scalar factors of the elementary reflectors. +*> \endverbatim +*> +*> \param[in,out] VN1 +*> \verbatim +*> VN1 is REAL array, dimension (N) +*> The vector with the partial column norms. +*> \endverbatim +*> +*> \param[in,out] VN2 +*> \verbatim +*> VN2 is REAL array, dimension (N) +*> The vector with the exact column norms. +*> \endverbatim +*> +*> \param[out] AUXV +*> \verbatim +*> AUXV is COMPLEX array, dimension (NB) +*> Auxiliary vector. +*> \endverbatim +*> +*> \param[out] F +*> \verbatim +*> F is COMPLEX array, dimension (LDF,NB) +*> Matrix F**H = L*(Y**H)*A. +*> \endverbatim +*> +*> \param[in] LDF +*> \verbatim +*> LDF is INTEGER +*> The leading dimension of the array F. LDF >= max(1,N+NRHS). +*> \endverbatim +*> +*> \param[out] IWORK +*> \verbatim +*> IWORK is INTEGER array, dimension (N-1). +*> Is a work array. ( IWORK is used to store indices +*> of "bad" columns for norm downdating in the residual +*> matrix ). +*> \endverbatim +*> +*> \param[out] INFO +*> \verbatim +*> INFO is INTEGER +*> 1) INFO = 0: successful exit. +*> 2) If INFO = j_1, where 1 <= j_1 <= N, then NaN was +*> detected and the routine stops the computation. +*> The j_1-th column of the matrix A or the j_1-th +*> element of array TAU contains the first occurrence +*> of NaN in the factorization step KB+1 ( when KB columns +*> have been factorized ). +*> +*> On exit: +*> KB is set to the number of +*> factorized columns without +*> exception. +*> MAXC2NRMK is set to NaN. +*> RELMAXC2NRMK is set to NaN. +*> TAU(KB+1:min(M,N)) is not set and contains undefined +*> elements. If j_1=KB+1, TAU(KB+1) +*> may contain NaN. +*> 3) If INFO = j_2, where N+1 <= j_2 <= 2*N, then no NaN +*> was detected, but +Inf (or -Inf) was detected and +*> the routine continues the computation until completion. +*> The (j_2-N)-th column of the matrix A contains the first +*> occurrence of +Inf (or -Inf) in the actorization +*> step KB+1 ( when KB columns have been factorized ). +*> \endverbatim +* +* Authors: +* ======== +* +*> \author Univ. of Tennessee +*> \author Univ. of California Berkeley +*> \author Univ. of Colorado Denver +*> \author NAG Ltd. +* +*> \ingroup laqp3rk +* +*> \par References: +* ================ +*> [1] A Level 3 BLAS QR factorization algorithm with column pivoting developed in 1996. +*> G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain. +*> X. Sun, Computer Science Dept., Duke University, USA. +*> C. H. Bischof, Math. and Comp. Sci. Div., Argonne National Lab, USA. +*> A BLAS-3 version of the QR factorization with column pivoting. +*> LAPACK Working Note 114 +*> \htmlonly +*> https://www.netlib.org/lapack/lawnspdf/lawn114.pdf +*> \endhtmlonly +*> and in +*> SIAM J. Sci. Comput., 19(5):1486-1494, Sept. 1998. +*> \htmlonly +*> https://doi.org/10.1137/S1064827595296732 +*> \endhtmlonly +*> +*> [2] A partial column norm updating strategy developed in 2006. +*> Z. Drmac and Z. Bujanovic, Dept. of Math., University of Zagreb, Croatia. +*> On the failure of rank revealing QR factorization software – a case study. +*> LAPACK Working Note 176. +*> \htmlonly +*> http://www.netlib.org/lapack/lawnspdf/lawn176.pdf +*> \endhtmlonly +*> and in +*> ACM Trans. Math. Softw. 35, 2, Article 12 (July 2008), 28 pages. +*> \htmlonly +*> https://doi.org/10.1145/1377612.1377616 +*> \endhtmlonly +* +*> \par Contributors: +* ================== +*> +*> \verbatim +*> +*> November 2023, Igor Kozachenko, James Demmel, +*> Computer Science Division, +*> University of California, Berkeley +*> +*> \endverbatim +* +* ===================================================================== + SUBROUTINE CLAQP3RK( M, N, NRHS, IOFFSET, NB, ABSTOL, + $ RELTOL, KP1, MAXC2NRM, A, LDA, DONE, KB, + $ MAXC2NRMK, RELMAXC2NRMK, JPIV, TAU, + $ VN1, VN2, AUXV, F, LDF, IWORK, INFO ) + IMPLICIT NONE +* +* -- LAPACK auxiliary routine -- +* -- LAPACK is a software package provided by Univ. of Tennessee, -- +* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- +* +* .. Scalar Arguments .. + LOGICAL DONE + INTEGER INFO, IOFFSET, KB, KP1, LDA, LDF, M, N, + $ NB, NRHS + REAL ABSTOL, MAXC2NRM, MAXC2NRMK, RELMAXC2NRMK, + $ RELTOL +* .. +* .. Array Arguments .. + INTEGER IWORK( * ), JPIV( * ) + REAL VN1( * ), VN2( * ) + COMPLEX A( LDA, * ), AUXV( * ), F( LDF, * ), TAU( * ) +* .. +* +* ===================================================================== +* +* .. Parameters .. + REAL ZERO, ONE + PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) + COMPLEX CZERO, CONE + PARAMETER ( CZERO = ( 0.0E+0, 0.0E+0 ), + $ CONE = ( 1.0E+0, 0.0E+0 ) ) +* .. +* .. Local Scalars .. + INTEGER ITEMP, J, K, MINMNFACT, MINMNUPDT, + $ LSTICC, KP, I, IF + REAL HUGEVAL, TAUNAN, TEMP, TEMP2, TOL3Z + COMPLEX AIK +* .. +* .. External Subroutines .. + EXTERNAL CGEMM, CGEMV, CLARFG, CSWAP +* .. +* .. Intrinsic Functions .. + INTRINSIC ABS, REAL, CONJG, IMAG, MAX, MIN, SQRT +* .. +* .. External Functions .. + LOGICAL SISNAN + INTEGER ISAMAX + REAL SLAMCH, SCNRM2 + EXTERNAL SISNAN, SLAMCH, ISAMAX, SCNRM2 +* .. +* .. Executable Statements .. +* +* Initialize INFO +* + INFO = 0 +* +* MINMNFACT in the smallest dimension of the submatrix +* A(IOFFSET+1:M,1:N) to be factorized. +* + MINMNFACT = MIN( M-IOFFSET, N ) + MINMNUPDT = MIN( M-IOFFSET, N+NRHS ) + NB = MIN( NB, MINMNFACT ) + TOL3Z = SQRT( SLAMCH( 'Epsilon' ) ) + HUGEVAL = SLAMCH( 'Overflow' ) +* +* Compute factorization in a while loop over NB columns, +* K is the column index in the block A(1:M,1:N). +* + K = 0 + LSTICC = 0 + DONE = .FALSE. +* + DO WHILE ( K.LT.NB .AND. LSTICC.EQ.0 ) + K = K + 1 + I = IOFFSET + K +* + IF( I.EQ.1 ) THEN +* +* We are at the first column of the original whole matrix A_orig, +* therefore we use the computed KP1 and MAXC2NRM from the +* main routine. +* + KP = KP1 +* + ELSE +* +* Determine the pivot column in K-th step, i.e. the index +* of the column with the maximum 2-norm in the +* submatrix A(I:M,K:N). +* + KP = ( K-1 ) + ISAMAX( N-K+1, VN1( K ), 1 ) +* +* Determine the maximum column 2-norm and the relative maximum +* column 2-norm of the submatrix A(I:M,K:N) in step K. +* + MAXC2NRMK = VN1( KP ) +* +* ============================================================ +* +* Check if the submatrix A(I:M,K:N) contains NaN, set +* INFO parameter to the column number, where the first NaN +* is found and return from the routine. +* We need to check the condition only if the +* column index (same as row index) of the original whole +* matrix is larger than 1, since the condition for whole +* original matrix is checked in the main routine. +* + IF( SISNAN( MAXC2NRMK ) ) THEN +* + DONE = .TRUE. +* +* Set KB, the number of factorized partial columns +* that are non-zero in each step in the block, +* i.e. the rank of the factor R. +* Set IF, the number of processed rows in the block, which +* is the same as the number of processed rows in +* the original whole matrix A_orig. +* + KB = K - 1 + IF = I - 1 + INFO = KB + KP +* +* Set RELMAXC2NRMK to NaN. +* + RELMAXC2NRMK = MAXC2NRMK +* +* There is no need to apply the block reflector to the +* residual of the matrix A stored in A(KB+1:M,KB+1:N), +* since the submatrix contains NaN and we stop +* the computation. +* But, we need to apply the block reflector to the residual +* right hand sides stored in A(KB+1:M,N+1:N+NRHS), if the +* residual right hand sides exist. This occurs +* when ( NRHS != 0 AND KB <= (M-IOFFSET) ): +* +* A(I+1:M,N+1:N+NRHS) := A(I+1:M,N+1:N+NRHS) - +* A(I+1:M,1:KB) * F(N+1:N+NRHS,1:KB)**H. + + IF( NRHS.GT.0 .AND. KB.LT.(M-IOFFSET) ) THEN + CALL CGEMM( 'No transpose', 'Conjugate transpose', + $ M-IF, NRHS, KB, -CONE, A( IF+1, 1 ), LDA, + $ F( N+1, 1 ), LDF, CONE, A( IF+1, N+1 ), LDA ) + END IF +* +* There is no need to recompute the 2-norm of the +* difficult columns, since we stop the factorization. +* +* Array TAU(KF+1:MINMNFACT) is not set and contains +* undefined elements. +* +* Return from the routine. +* + RETURN + END IF +* +* Quick return, if the submatrix A(I:M,K:N) is +* a zero matrix. We need to check it only if the column index +* (same as row index) is larger than 1, since the condition +* for the whole original matrix A_orig is checked in the main +* routine. +* + IF( MAXC2NRMK.EQ.ZERO ) THEN +* + DONE = .TRUE. +* +* Set KB, the number of factorized partial columns +* that are non-zero in each step in the block, +* i.e. the rank of the factor R. +* Set IF, the number of processed rows in the block, which +* is the same as the number of processed rows in +* the original whole matrix A_orig. +* + KB = K - 1 + IF = I - 1 + RELMAXC2NRMK = ZERO +* +* There is no need to apply the block reflector to the +* residual of the matrix A stored in A(KB+1:M,KB+1:N), +* since the submatrix is zero and we stop the computation. +* But, we need to apply the block reflector to the residual +* right hand sides stored in A(KB+1:M,N+1:N+NRHS), if the +* residual right hand sides exist. This occurs +* when ( NRHS != 0 AND KB <= (M-IOFFSET) ): +* +* A(I+1:M,N+1:N+NRHS) := A(I+1:M,N+1:N+NRHS) - +* A(I+1:M,1:KB) * F(N+1:N+NRHS,1:KB)**H. +* + IF( NRHS.GT.0 .AND. KB.LT.(M-IOFFSET) ) THEN + CALL CGEMM( 'No transpose', 'Conjugate transpose', + $ M-IF, NRHS, KB, -CONE, A( IF+1, 1 ), LDA, + $ F( N+1, 1 ), LDF, CONE, A( IF+1, N+1 ), LDA ) + END IF +* +* There is no need to recompute the 2-norm of the +* difficult columns, since we stop the factorization. +* +* Set TAUs corresponding to the columns that were not +* factorized to ZERO, i.e. set TAU(KB+1:MINMNFACT) = CZERO, +* which is equivalent to seting TAU(K:MINMNFACT) = CZERO. +* + DO J = K, MINMNFACT + TAU( J ) = CZERO + END DO +* +* Return from the routine. +* + RETURN +* + END IF +* +* ============================================================ +* +* Check if the submatrix A(I:M,K:N) contains Inf, +* set INFO parameter to the column number, where +* the first Inf is found plus N, and continue +* the computation. +* We need to check the condition only if the +* column index (same as row index) of the original whole +* matrix is larger than 1, since the condition for whole +* original matrix is checked in the main routine. +* + IF( INFO.EQ.0 .AND. MAXC2NRMK.GT.HUGEVAL ) THEN + INFO = N + K - 1 + KP + END IF +* +* ============================================================ +* +* Test for the second and third tolerance stopping criteria. +* NOTE: There is no need to test for ABSTOL.GE.ZERO, since +* MAXC2NRMK is non-negative. Similarly, there is no need +* to test for RELTOL.GE.ZERO, since RELMAXC2NRMK is +* non-negative. +* We need to check the condition only if the +* column index (same as row index) of the original whole +* matrix is larger than 1, since the condition for whole +* original matrix is checked in the main routine. +* + RELMAXC2NRMK = MAXC2NRMK / MAXC2NRM +* + IF( MAXC2NRMK.LE.ABSTOL .OR. RELMAXC2NRMK.LE.RELTOL ) THEN +* + DONE = .TRUE. +* +* Set KB, the number of factorized partial columns +* that are non-zero in each step in the block, +* i.e. the rank of the factor R. +* Set IF, the number of processed rows in the block, which +* is the same as the number of processed rows in +* the original whole matrix A_orig; +* + KB = K - 1 + IF = I - 1 +* +* Apply the block reflector to the residual of the +* matrix A and the residual of the right hand sides B, if +* the residual matrix and and/or the residual of the right +* hand sides exist, i.e. if the submatrix +* A(I+1:M,KB+1:N+NRHS) exists. This occurs when +* KB < MINMNUPDT = min( M-IOFFSET, N+NRHS ): +* +* A(IF+1:M,K+1:N+NRHS) := A(IF+1:M,KB+1:N+NRHS) - +* A(IF+1:M,1:KB) * F(KB+1:N+NRHS,1:KB)**H. +* + IF( KB.LT.MINMNUPDT ) THEN + CALL CGEMM( 'No transpose', 'Conjugate transpose', + $ M-IF, N+NRHS-KB, KB,-CONE, A( IF+1, 1 ), LDA, + $ F( KB+1, 1 ), LDF, CONE, A( IF+1, KB+1 ), LDA ) + END IF +* +* There is no need to recompute the 2-norm of the +* difficult columns, since we stop the factorization. +* +* Set TAUs corresponding to the columns that were not +* factorized to ZERO, i.e. set TAU(KB+1:MINMNFACT) = CZERO, +* which is equivalent to seting TAU(K:MINMNFACT) = CZERO. +* + DO J = K, MINMNFACT + TAU( J ) = CZERO + END DO +* +* Return from the routine. +* + RETURN +* + END IF +* +* ============================================================ +* +* End ELSE of IF(I.EQ.1) +* + END IF +* +* =============================================================== +* +* If the pivot column is not the first column of the +* subblock A(1:M,K:N): +* 1) swap the K-th column and the KP-th pivot column +* in A(1:M,1:N); +* 2) swap the K-th row and the KP-th row in F(1:N,1:K-1) +* 3) copy the K-th element into the KP-th element of the partial +* and exact 2-norm vectors VN1 and VN2. (Swap is not needed +* for VN1 and VN2 since we use the element with the index +* larger than K in the next loop step.) +* 4) Save the pivot interchange with the indices relative to the +* the original matrix A_orig, not the block A(1:M,1:N). +* + IF( KP.NE.K ) THEN + CALL CSWAP( M, A( 1, KP ), 1, A( 1, K ), 1 ) + CALL CSWAP( K-1, F( KP, 1 ), LDF, F( K, 1 ), LDF ) + VN1( KP ) = VN1( K ) + VN2( KP ) = VN2( K ) + ITEMP = JPIV( KP ) + JPIV( KP ) = JPIV( K ) + JPIV( K ) = ITEMP + END IF +* +* Apply previous Householder reflectors to column K: +* A(I:M,K) := A(I:M,K) - A(I:M,1:K-1)*F(K,1:K-1)**H. +* + IF( K.GT.1 ) THEN + DO J = 1, K - 1 + F( K, J ) = CONJG( F( K, J ) ) + END DO + CALL CGEMV( 'No transpose', M-I+1, K-1, -CONE, A( I, 1 ), + $ LDA, F( K, 1 ), LDF, CONE, A( I, K ), 1 ) + DO J = 1, K - 1 + F( K, J ) = CONJG( F( K, J ) ) + END DO + END IF +* +* Generate elementary reflector H(k) using the column A(I:M,K). +* + IF( I.LT.M ) THEN + CALL CLARFG( M-I+1, A( I, K ), A( I+1, K ), 1, TAU( K ) ) + ELSE + TAU( K ) = CZERO + END IF +* +* Check if TAU(K) contains NaN, set INFO parameter +* to the column number where NaN is found and return from +* the routine. +* NOTE: There is no need to check TAU(K) for Inf, +* since CLARFG cannot produce TAU(KK) or Householder vector +* below the diagonal containing Inf. Only BETA on the diagonal, +* returned by CLARFG can contain Inf, which requires +* TAU(K) to contain NaN. Therefore, this case of generating Inf +* by CLARFG is covered by checking TAU(K) for NaN. +* + IF( SISNAN( REAL( TAU(K) ) ) ) THEN + TAUNAN = REAL( TAU(K) ) + ELSE IF( SISNAN( IMAG( TAU(K) ) ) ) THEN + TAUNAN = IMAG( TAU(K) ) + ELSE + TAUNAN = ZERO + END IF +* + IF( SISNAN( TAUNAN ) ) THEN +* + DONE = .TRUE. +* +* Set KB, the number of factorized partial columns +* that are non-zero in each step in the block, +* i.e. the rank of the factor R. +* Set IF, the number of processed rows in the block, which +* is the same as the number of processed rows in +* the original whole matrix A_orig. +* + KB = K - 1 + IF = I - 1 + INFO = K +* +* Set MAXC2NRMK and RELMAXC2NRMK to NaN. +* + MAXC2NRMK = TAUNAN + RELMAXC2NRMK = TAUNAN +* +* There is no need to apply the block reflector to the +* residual of the matrix A stored in A(KB+1:M,KB+1:N), +* since the submatrix contains NaN and we stop +* the computation. +* But, we need to apply the block reflector to the residual +* right hand sides stored in A(KB+1:M,N+1:N+NRHS), if the +* residual right hand sides exist. This occurs +* when ( NRHS != 0 AND KB <= (M-IOFFSET) ): +* +* A(I+1:M,N+1:N+NRHS) := A(I+1:M,N+1:N+NRHS) - +* A(I+1:M,1:KB) * F(N+1:N+NRHS,1:KB)**H. +* + IF( NRHS.GT.0 .AND. KB.LT.(M-IOFFSET) ) THEN + CALL CGEMM( 'No transpose', 'Conjugate transpose', + $ M-IF, NRHS, KB, -CONE, A( IF+1, 1 ), LDA, + $ F( N+1, 1 ), LDF, CONE, A( IF+1, N+1 ), LDA ) + END IF +* +* There is no need to recompute the 2-norm of the +* difficult columns, since we stop the factorization. +* +* Array TAU(KF+1:MINMNFACT) is not set and contains +* undefined elements. +* +* Return from the routine. +* + RETURN + END IF +* +* =============================================================== +* + AIK = A( I, K ) + A( I, K ) = CONE +* +* =============================================================== +* +* Compute the current K-th column of F: +* 1) F(K+1:N,K) := tau(K) * A(I:M,K+1:N)**H * A(I:M,K). +* + IF( K.LT.N+NRHS ) THEN + CALL CGEMV( 'Conjugate transpose', M-I+1, N+NRHS-K, + $ TAU( K ), A( I, K+1 ), LDA, A( I, K ), 1, + $ CZERO, F( K+1, K ), 1 ) + END IF +* +* 2) Zero out elements above and on the diagonal of the +* column K in matrix F, i.e elements F(1:K,K). +* + DO J = 1, K + F( J, K ) = CZERO + END DO +* +* 3) Incremental updating of the K-th column of F: +* F(1:N,K) := F(1:N,K) - tau(K) * F(1:N,1:K-1) * A(I:M,1:K-1)**H +* * A(I:M,K). +* + IF( K.GT.1 ) THEN + CALL CGEMV( 'Conjugate Transpose', M-I+1, K-1, -TAU( K ), + $ A( I, 1 ), LDA, A( I, K ), 1, CZERO, + $ AUXV( 1 ), 1 ) +* + CALL CGEMV( 'No transpose', N+NRHS, K-1, CONE, + $ F( 1, 1 ), LDF, AUXV( 1 ), 1, CONE, + $ F( 1, K ), 1 ) + END IF +* +* =============================================================== +* +* Update the current I-th row of A: +* A(I,K+1:N+NRHS) := A(I,K+1:N+NRHS) +* - A(I,1:K)*F(K+1:N+NRHS,1:K)**H. +* + IF( K.LT.N+NRHS ) THEN + CALL CGEMM( 'No transpose', 'Conjugate transpose', + $ 1, N+NRHS-K, K, -CONE, A( I, 1 ), LDA, + $ F( K+1, 1 ), LDF, CONE, A( I, K+1 ), LDA ) + END IF +* + A( I, K ) = AIK +* +* Update the partial column 2-norms for the residual matrix, +* only if the residual matrix A(I+1:M,K+1:N) exists, i.e. +* when K < MINMNFACT = min( M-IOFFSET, N ). +* + IF( K.LT.MINMNFACT ) THEN +* + DO J = K + 1, N + IF( VN1( J ).NE.ZERO ) THEN +* +* NOTE: The following lines follow from the analysis in +* Lapack Working Note 176. +* + TEMP = ABS( A( I, J ) ) / VN1( J ) + TEMP = MAX( ZERO, ( ONE+TEMP )*( ONE-TEMP ) ) + TEMP2 = TEMP*( VN1( J ) / VN2( J ) )**2 + IF( TEMP2.LE.TOL3Z ) THEN +* +* At J-index, we have a difficult column for the +* update of the 2-norm. Save the index of the previous +* difficult column in IWORK(J-1). +* NOTE: ILSTCC > 1, threfore we can use IWORK only +* with N-1 elements, where the elements are +* shifted by 1 to the left. +* + IWORK( J-1 ) = LSTICC +* +* Set the index of the last difficult column LSTICC. +* + LSTICC = J +* + ELSE + VN1( J ) = VN1( J )*SQRT( TEMP ) + END IF + END IF + END DO +* + END IF +* +* End of while loop. +* + END DO +* +* Now, afler the loop: +* Set KB, the number of factorized columns in the block; +* Set IF, the number of processed rows in the block, which +* is the same as the number of processed rows in +* the original whole matrix A_orig, IF = IOFFSET + KB. +* + KB = K + IF = I +* +* Apply the block reflector to the residual of the matrix A +* and the residual of the right hand sides B, if the residual +* matrix and and/or the residual of the right hand sides +* exist, i.e. if the submatrix A(I+1:M,KB+1:N+NRHS) exists. +* This occurs when KB < MINMNUPDT = min( M-IOFFSET, N+NRHS ): +* +* A(IF+1:M,K+1:N+NRHS) := A(IF+1:M,KB+1:N+NRHS) - +* A(IF+1:M,1:KB) * F(KB+1:N+NRHS,1:KB)**H. +* + IF( KB.LT.MINMNUPDT ) THEN + CALL CGEMM( 'No transpose', 'Conjugate transpose', + $ M-IF, N+NRHS-KB, KB, -CONE, A( IF+1, 1 ), LDA, + $ F( KB+1, 1 ), LDF, CONE, A( IF+1, KB+1 ), LDA ) + END IF +* +* Recompute the 2-norm of the difficult columns. +* Loop over the index of the difficult columns from the largest +* to the smallest index. +* + DO WHILE( LSTICC.GT.0 ) +* +* LSTICC is the index of the last difficult column is greater +* than 1. +* ITEMP is the index of the previous difficult column. +* + ITEMP = IWORK( LSTICC-1 ) +* +* Compute the 2-norm explicilty for the last difficult column and +* save it in the partial and exact 2-norm vectors VN1 and VN2. +* +* NOTE: The computation of VN1( LSTICC ) relies on the fact that +* SCNRM2 does not fail on vectors with norm below the value of +* SQRT(SLAMCH('S')) +* + VN1( LSTICC ) = SCNRM2( M-IF, A( IF+1, LSTICC ), 1 ) + VN2( LSTICC ) = VN1( LSTICC ) +* +* Downdate the index of the last difficult column to +* the index of the previous difficult column. +* + LSTICC = ITEMP +* + END DO +* + RETURN +* +* End of CLAQP3RK +* + END diff --git a/SRC/dgeqp3rk.f b/SRC/dgeqp3rk.f new file mode 100755 index 0000000000..ace97b712b --- /dev/null +++ b/SRC/dgeqp3rk.f @@ -0,0 +1,1081 @@ +*> \brief \b DGEQP3RK computes a truncated Householder QR factorization with column pivoting of a real m-by-n matrix A by using Level 3 BLAS and overwrites a real m-by-nrhs matrix B with Q**T * B. +* +* =========== DOCUMENTATION =========== +* +* Online html documentation available at +* http://www.netlib.org/lapack/explore-html/ +* +*> \htmlonly +*> Download DGEQP3RK + dependencies +*> +*> [TGZ] +*> +*> [ZIP] +*> +*> [TXT] +*> \endhtmlonly +* +* Definition: +* =========== +* +* SUBROUTINE DGEQP3RK( M, N, NRHS, KMAX, ABSTOL, RELTOL, A, LDA, +* $ K, MAXC2NRMK, RELMAXC2NRMK, JPIV, TAU, +* $ WORK, LWORK, IWORK, INFO ) +* IMPLICIT NONE +* +* .. Scalar Arguments .. +* INTEGER INFO, K, KMAX, LDA, LWORK, M, N, NRHS +* DOUBLE PRECISION ABSTOL, MAXC2NRMK, RELMAXC2NRMK, RELTOL +* .. +* .. Array Arguments .. +* INTEGER IWORK( * ), JPIV( * ) +* DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * ) +* .. +* +* +*> \par Purpose: +* ============= +*> +*> \verbatim +*> +*> DGEQP3RK performs two tasks simultaneously: +*> +*> Task 1: The routine computes a truncated (rank K) or full rank +*> Householder QR factorization with column pivoting of a real +*> M-by-N matrix A using Level 3 BLAS. K is the number of columns +*> that were factorized, i.e. factorization rank of the +*> factor R, K <= min(M,N). +*> +*> A * P(K) = Q(K) * R(K) = +*> +*> = Q(K) * ( R11(K) R12(K) ) = Q(K) * ( R(K)_approx ) +*> ( 0 R22(K) ) ( 0 R(K)_residual ), +*> +*> where: +*> +*> P(K) is an N-by-N permutation matrix; +*> Q(K) is an M-by-M orthogonal matrix; +*> R(K)_approx = ( R11(K), R12(K) ) is a rank K approximation of the +*> full rank factor R with K-by-K upper-triangular +*> R11(K) and K-by-N rectangular R12(K). The diagonal +*> entries of R11(K) appear in non-increasing order +*> of absolute value, and absolute values of all of +*> them exceed the maximum column 2-norm of R22(K) +*> up to roundoff error. +*> R(K)_residual = R22(K) is the residual of a rank K approximation +*> of the full rank factor R. It is a +*> an (M-K)-by-(N-K) rectangular matrix; +*> 0 is a an (M-K)-by-K zero matrix. +*> +*> Task 2: At the same time, the routine overwrites a real M-by-NRHS +*> matrix B with Q(K)**T * B using Level 3 BLAS. +*> +*> ===================================================================== +*> +*> The matrices A and B are stored on input in the array A as +*> the left and right blocks A(1:M,1:N) and A(1:M, N+1:N+NRHS) +*> respectively. +*> +*> N NRHS +*> array_A = M [ mat_A, mat_B ] +*> +*> The truncation criteria (i.e. when to stop the factorization) +*> can be any of the following: +*> +*> 1) The input parameter KMAX, the maximum number of columns +*> KMAX to factorize, i.e. the factorization rank is limited +*> to KMAX. If KMAX >= min(M,N), the criterion is not used. +*> +*> 2) The input parameter ABSTOL, the absolute tolerance for +*> the maximum column 2-norm of the residual matrix R22(K). This +*> means that the factorization stops if this norm is less or +*> equal to ABSTOL. If ABSTOL < 0.0, the criterion is not used. +*> +*> 3) The input parameter RELTOL, the tolerance for the maximum +*> column 2-norm matrix of the residual matrix R22(K) divided +*> by the maximum column 2-norm of the original matrix A, which +*> is equal to abs(R(1,1)). This means that the factorization stops +*> when the ratio of the maximum column 2-norm of R22(K) to +*> the maximum column 2-norm of A is less than or equal to RELTOL. +*> If RELTOL < 0.0, the criterion is not used. +*> +*> 4) In case both stopping criteria ABSTOL or RELTOL are not used, +*> and when the residual matrix R22(K) is a zero matrix in some +*> factorization step K. ( This stopping criterion is implicit. ) +*> +*> The algorithm stops when any of these conditions is first +*> satisfied, otherwise the whole matrix A is factorized. +*> +*> To factorize the whole matrix A, use the values +*> KMAX >= min(M,N), ABSTOL < 0.0 and RELTOL < 0.0. +*> +*> The routine returns: +*> a) Q(K), R(K)_approx = ( R11(K), R12(K) ), +*> R(K)_residual = R22(K), P(K), i.e. the resulting matrices +*> of the factorization; P(K) is represented by JPIV, +*> ( if K = min(M,N), R(K)_approx is the full factor R, +*> and there is no residual matrix R(K)_residual); +*> b) K, the number of columns that were factorized, +*> i.e. factorization rank; +*> c) MAXC2NRMK, the maximum column 2-norm of the residual +*> matrix R(K)_residual = R22(K), +*> ( if K = min(M,N), MAXC2NRMK = 0.0 ); +*> d) RELMAXC2NRMK equals MAXC2NRMK divided by MAXC2NRM, the maximum +*> column 2-norm of the original matrix A, which is equal +*> to abs(R(1,1)), ( if K = min(M,N), RELMAXC2NRMK = 0.0 ); +*> e) Q(K)**T * B, the matrix B with the orthogonal +*> transformation Q(K)**T applied on the left. +*> +*> The N-by-N permutation matrix P(K) is stored in a compact form in +*> the integer array JPIV. For 1 <= j <= N, column j +*> of the matrix A was interchanged with column JPIV(j). +*> +*> The M-by-M orthogonal matrix Q is represented as a product +*> of elementary Householder reflectors +*> +*> Q(K) = H(1) * H(2) * . . . * H(K), +*> +*> where K is the number of columns that were factorized. +*> +*> Each H(j) has the form +*> +*> H(j) = I - tau * v * v**T, +*> +*> where 1 <= j <= K and +*> I is an M-by-M identity matrix, +*> tau is a real scalar, +*> v is a real vector with v(1:j-1) = 0 and v(j) = 1. +*> +*> v(j+1:M) is stored on exit in A(j+1:M,j) and tau in TAU(j). +*> +*> See the Further Details section for more information. +*> \endverbatim +* +* Arguments: +* ========== +* +*> \param[in] M +*> \verbatim +*> M is INTEGER +*> The number of rows of the matrix A. M >= 0. +*> \endverbatim +*> +*> \param[in] N +*> \verbatim +*> N is INTEGER +*> The number of columns of the matrix A. N >= 0. +*> \endverbatim +*> +*> \param[in] NRHS +*> \verbatim +*> NRHS is INTEGER +*> The number of right hand sides, i.e. the number of +*> columns of the matrix B. NRHS >= 0. +*> \endverbatim +*> +*> \param[in] KMAX +*> \verbatim +*> KMAX is INTEGER +*> +*> The first factorization stopping criterion. KMAX >= 0. +*> +*> The maximum number of columns of the matrix A to factorize, +*> i.e. the maximum factorization rank. +*> +*> a) If KMAX >= min(M,N), then this stopping criterion +*> is not used, the routine factorizes columns +*> depending on ABSTOL and RELTOL. +*> +*> b) If KMAX = 0, then this stopping criterion is +*> satisfied on input and the routine exits immediately. +*> This means that the factorization is not performed, +*> the matrices A and B are not modified, and +*> the matrix A is itself the residual. +*> \endverbatim +*> +*> \param[in] ABSTOL +*> \verbatim +*> ABSTOL is DOUBLE PRECISION +*> +*> The second factorization stopping criterion, cannot be NaN. +*> +*> The absolute tolerance (stopping threshold) for +*> maximum column 2-norm of the residual matrix R22(K). +*> The algorithm converges (stops the factorization) when +*> the maximum column 2-norm of the residual matrix R22(K) +*> is less than or equal to ABSTOL. Let SAFMIN = DLAMCH('S'). +*> +*> a) If ABSTOL is NaN, then no computation is performed +*> and an error message ( INFO = -5 ) is issued +*> by XERBLA. +*> +*> b) If ABSTOL < 0.0, then this stopping criterion is not +*> used, the routine factorizes columns depending +*> on KMAX and RELTOL. +*> This includes the case ABSTOL = -Inf. +*> +*> c) If 0.0 <= ABSTOL < 2*SAFMIN, then ABSTOL = 2*SAFMIN +*> is used. This includes the case ABSTOL = -0.0. +*> +*> d) If 2*SAFMIN <= ABSTOL then the input value +*> of ABSTOL is used. +*> +*> Let MAXC2NRM be the maximum column 2-norm of the +*> whole original matrix A. +*> If ABSTOL chosen above is >= MAXC2NRM, then this +*> stopping criterion is satisfied on input and routine exits +*> immediately after MAXC2NRM is computed. The routine +*> returns MAXC2NRM in MAXC2NORMK, +*> and 1.0 in RELMAXC2NORMK. +*> This includes the case ABSTOL = +Inf. This means that the +*> factorization is not performed, the matrices A and B are not +*> modified, and the matrix A is itself the residual. +*> \endverbatim +*> +*> \param[in] RELTOL +*> \verbatim +*> RELTOL is DOUBLE PRECISION +*> +*> The third factorization stopping criterion, cannot be NaN. +*> +*> The tolerance (stopping threshold) for the ratio +*> abs(R(K+1,K+1))/abs(R(1,1)) of the maximum column 2-norm of +*> the residual matrix R22(K) to the maximum column 2-norm of +*> the original matrix A. The algorithm converges (stops the +*> factorization), when abs(R(K+1,K+1))/abs(R(1,1)) A is less +*> than or equal to RELTOL. Let EPS = DLAMCH('E'). +*> +*> a) If RELTOL is NaN, then no computation is performed +*> and an error message ( INFO = -6 ) is issued +*> by XERBLA. +*> +*> b) If RELTOL < 0.0, then this stopping criterion is not +*> used, the routine factorizes columns depending +*> on KMAX and ABSTOL. +*> This includes the case RELTOL = -Inf. +*> +*> c) If 0.0 <= RELTOL < EPS, then RELTOL = EPS is used. +*> This includes the case RELTOL = -0.0. +*> +*> d) If EPS <= RELTOL then the input value of RELTOL +*> is used. +*> +*> Let MAXC2NRM be the maximum column 2-norm of the +*> whole original matrix A. +*> If RELTOL chosen above is >= 1.0, then this stopping +*> criterion is satisfied on input and routine exits +*> immediately after MAXC2NRM is computed. +*> The routine returns MAXC2NRM in MAXC2NORMK, +*> and 1.0 in RELMAXC2NORMK. +*> This includes the case RELTOL = +Inf. This means that the +*> factorization is not performed, the matrices A and B are not +*> modified, and the matrix A is itself the residual. +*> +*> NOTE: We recommend that RELTOL satisfy +*> min( max(M,N)*EPS, sqrt(EPS) ) <= RELTOL +*> \endverbatim +*> +*> \param[in,out] A +*> \verbatim +*> A is DOUBLE PRECISION array, dimension (LDA,N+NRHS) +*> +*> On entry: +*> +*> a) The subarray A(1:M,1:N) contains the M-by-N matrix A. +*> b) The subarray A(1:M,N+1:N+NRHS) contains the M-by-NRHS +*> matrix B. +*> +*> N NRHS +*> array_A = M [ mat_A, mat_B ] +*> +*> On exit: +*> +*> a) The subarray A(1:M,1:N) contains parts of the factors +*> of the matrix A: +*> +*> 1) If K = 0, A(1:M,1:N) contains the original matrix A. +*> 2) If K > 0, A(1:M,1:N) contains parts of the +*> factors: +*> +*> 1. The elements below the diagonal of the subarray +*> A(1:M,1:K) together with TAU(1:K) represent the +*> orthogonal matrix Q(K) as a product of K Householder +*> elementary reflectors. +*> +*> 2. The elements on and above the diagonal of +*> the subarray A(1:K,1:N) contain K-by-N +*> upper-trapezoidal matrix +*> R(K)_approx = ( R11(K), R12(K) ). +*> NOTE: If K=min(M,N), i.e. full rank factorization, +*> then R_approx(K) is the full factor R which +*> is upper-trapezoidal. If, in addition, M>=N, +*> then R is upper-triangular. +*> +*> 3. The subarray A(K+1:M,K+1:N) contains (M-K)-by-(N-K) +*> rectangular matrix R(K)_residual = R22(K). +*> +*> b) If NRHS > 0, the subarray A(1:M,N+1:N+NRHS) contains +*> the M-by-NRHS product Q(K)**T * B. +*> \endverbatim +*> +*> \param[in] LDA +*> \verbatim +*> LDA is INTEGER +*> The leading dimension of the array A. LDA >= max(1,M). +*> This is the leading dimension for both matrices, A and B. +*> \endverbatim +*> +*> \param[out] K +*> \verbatim +*> K is INTEGER +*> Factorization rank of the matrix A, i.e. the rank of +*> the factor R, which is the same as the number of non-zero +*> rows of the factor R. 0 <= K <= min(M,KMAX,N). +*> +*> K also represents the number of non-zero Householder +*> vectors. +*> +*> NOTE: If K = 0, a) the arrays A and B are not modified; +*> b) the array TAU(1:min(M,N)) is set to ZERO, +*> if the matrix A does not contain NaN, +*> otherwise the elements TAU(1:min(M,N)) +*> are undefined; +*> c) the elements of the array JPIV are set +*> as follows: for j = 1:N, JPIV(j) = j. +*> \endverbatim +*> +*> \param[out] MAXC2NRMK +*> \verbatim +*> MAXC2NRMK is DOUBLE PRECISION +*> The maximum column 2-norm of the residual matrix R22(K), +*> when the factorization stopped at rank K. MAXC2NRMK >= 0. +*> +*> a) If K = 0, i.e. the factorization was not performed, +*> the matrix A was not modified and is itself a residual +*> matrix, then MAXC2NRMK equals the maximum column 2-norm +*> of the original matrix A. +*> +*> b) If 0 < K < min(M,N), then MAXC2NRMK is returned. +*> +*> c) If K = min(M,N), i.e. the whole matrix A was +*> factorized and there is no residual matrix, +*> then MAXC2NRMK = 0.0. +*> +*> NOTE: MAXC2NRMK in the factorization step K would equal +*> R(K+1,K+1) in the next factorization step K+1. +*> \endverbatim +*> +*> \param[out] RELMAXC2NRMK +*> \verbatim +*> RELMAXC2NRMK is DOUBLE PRECISION +*> The ratio MAXC2NRMK / MAXC2NRM of the maximum column +*> 2-norm of the residual matrix R22(K) (when the factorization +*> stopped at rank K) to the maximum column 2-norm of the +*> whole original matrix A. RELMAXC2NRMK >= 0. +*> +*> a) If K = 0, i.e. the factorization was not performed, +*> the matrix A was not modified and is itself a residual +*> matrix, then RELMAXC2NRMK = 1.0. +*> +*> b) If 0 < K < min(M,N), then +*> RELMAXC2NRMK = MAXC2NRMK / MAXC2NRM is returned. +*> +*> c) If K = min(M,N), i.e. the whole matrix A was +*> factorized and there is no residual matrix, +*> then RELMAXC2NRMK = 0.0. +*> +*> NOTE: RELMAXC2NRMK in the factorization step K would equal +*> abs(R(K+1,K+1))/abs(R(1,1)) in the next factorization +*> step K+1. +*> \endverbatim +*> +*> \param[out] JPIV +*> \verbatim +*> JPIV is INTEGER array, dimension (N) +*> Column pivot indices. For 1 <= j <= N, column j +*> of the matrix A was interchanged with column JPIV(j). +*> +*> The elements of the array JPIV(1:N) are always set +*> by the routine, for example, even when no columns +*> were factorized, i.e. when K = 0, the elements are +*> set as JPIV(j) = j for j = 1:N. +*> \endverbatim +*> +*> \param[out] TAU +*> \verbatim +*> TAU is DOUBLE PRECISION array, dimension (min(M,N)) +*> The scalar factors of the elementary reflectors. +*> +*> If 0 < K <= min(M,N), only the elements TAU(1:K) of +*> the array TAU are modified by the factorization. +*> After the factorization computed, if no NaN was found +*> during the factorization, the remaining elements +*> TAU(K+1:min(M,N)) are set to zero, otherwise the +*> elements TAU(K+1:min(M,N)) are not set and therefore +*> undefined. +*> ( If K = 0, all elements of TAU are set to zero, if +*> the matrix A does not contain NaN. ) +*> \endverbatim +*> +*> \param[out] WORK +*> \verbatim +*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)) +*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. +*> \endverbatim +*> +*> \param[in] LWORK +*> \verbatim +*> LWORK is INTEGER +*> The dimension of the array WORK. +*. LWORK >= (3*N + NRHS - 1) +*> For optimal performance LWORK >= (2*N + NB*( N+NRHS+1 )), +*> where NB is the optimal block size for DGEQP3RK returned +*> by ILAENV. Minimal block size MINNB=2. +*> +*> NOTE: The decision, whether to use unblocked BLAS 2 +*> or blocked BLAS 3 code is based not only on the dimension +*> LWORK of the availbale workspace WORK, but also also on the +*> matrix A dimension N via crossover point NX returned +*> by ILAENV. (For N less than NX, unblocked code should be +*> used.) +*> +*> If LWORK = -1, then a workspace query is assumed; +*> the routine only calculates the optimal size of the WORK +*> array, returns this value as the first entry of the WORK +*> array, and no error message related to LWORK is issued +*> by XERBLA. +*> \endverbatim +*> +*> \param[out] IWORK +*> \verbatim +*> IWORK is INTEGER array, dimension (N-1). +*> Is a work array. ( IWORK is used to store indices +*> of "bad" columns for norm downdating in the residual +*> matrix in the blocked step auxiliary subroutine DLAQP3RK ). +*> \endverbatim +*> +*> \param[out] INFO +*> \verbatim +*> INFO is INTEGER +*> 1) INFO = 0: successful exit. +*> 2) INFO < 0: if INFO = -i, the i-th argument had an +*> illegal value. +*> 3) If INFO = j_1, where 1 <= j_1 <= N, then NaN was +*> detected and the routine stops the computation. +*> The j_1-th column of the matrix A or the j_1-th +*> element of array TAU contains the first occurrence +*> of NaN in the factorization step K+1 ( when K columns +*> have been factorized ). +*> +*> On exit: +*> K is set to the number of +*> factorized columns without +*> exception. +*> MAXC2NRMK is set to NaN. +*> RELMAXC2NRMK is set to NaN. +*> TAU(K+1:min(M,N)) is not set and contains undefined +*> elements. If j_1=K+1, TAU(K+1) +*> may contain NaN. +*> 4) If INFO = j_2, where N+1 <= j_2 <= 2*N, then no NaN +*> was detected, but +Inf (or -Inf) was detected and +*> the routine continues the computation until completion. +*> The (j_2-N)-th column of the matrix A contains the first +*> occurrence of +Inf (or -Inf) in the factorization +*> step K+1 ( when K columns have been factorized ). +*> \endverbatim +* +* Authors: +* ======== +* +*> \author Univ. of Tennessee +*> \author Univ. of California Berkeley +*> \author Univ. of Colorado Denver +*> \author NAG Ltd. +* +*> \ingroup geqp3rk +* +*> \par Further Details: +* ===================== +* +*> \verbatim +*> DGEQP3RK is based on the same BLAS3 Householder QR factorization +*> algorithm with column pivoting as in DGEQP3 routine which uses +*> DLARFG routine to generate Householder reflectors +*> for QR factorization. +*> +*> We can also write: +*> +*> A = A_approx(K) + A_residual(K) +*> +*> The low rank approximation matrix A(K)_approx from +*> the truncated QR factorization of rank K of the matrix A is: +*> +*> A(K)_approx = Q(K) * ( R(K)_approx ) * P(K)**T +*> ( 0 0 ) +*> +*> = Q(K) * ( R11(K) R12(K) ) * P(K)**T +*> ( 0 0 ) +*> +*> The residual A_residual(K) of the matrix A is: +*> +*> A_residual(K) = Q(K) * ( 0 0 ) * P(K)**T = +*> ( 0 R(K)_residual ) +*> +*> = Q(K) * ( 0 0 ) * P(K)**T +*> ( 0 R22(K) ) +*> +*> The truncated (rank K) factorization guarantees that +*> the maximum column 2-norm of A_residual(K) is less than +*> or equal to MAXC2NRMK up to roundoff error. +*> +*> NOTE: An approximation of the null vectors +*> of A can be easily computed from R11(K) +*> and R12(K): +*> +*> Null( A(K) )_approx = P * ( inv(R11(K)) * R12(K) ) +*> ( -I ) +*> +*> \endverbatim +* +*> \par References: +* ================ +*> [1] A Level 3 BLAS QR factorization algorithm with column pivoting developed in 1996. +*> G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain. +*> X. Sun, Computer Science Dept., Duke University, USA. +*> C. H. Bischof, Math. and Comp. Sci. Div., Argonne National Lab, USA. +*> A BLAS-3 version of the QR factorization with column pivoting. +*> LAPACK Working Note 114 +*> \htmlonly +*> https://www.netlib.org/lapack/lawnspdf/lawn114.pdf +*> \endhtmlonly +*> and in +*> SIAM J. Sci. Comput., 19(5):1486-1494, Sept. 1998. +*> \htmlonly +*> https://doi.org/10.1137/S1064827595296732 +*> \endhtmlonly +*> +*> [2] A partial column norm updating strategy developed in 2006. +*> Z. Drmac and Z. Bujanovic, Dept. of Math., University of Zagreb, Croatia. +*> On the failure of rank revealing QR factorization software – a case study. +*> LAPACK Working Note 176. +*> \htmlonly +*> http://www.netlib.org/lapack/lawnspdf/lawn176.pdf +*> \endhtmlonly +*> and in +*> ACM Trans. Math. Softw. 35, 2, Article 12 (July 2008), 28 pages. +*> \htmlonly +*> https://doi.org/10.1145/1377612.1377616 +*> \endhtmlonly +* +*> \par Contributors: +* ================== +*> +*> \verbatim +*> +*> November 2023, Igor Kozachenko, James Demmel, +*> Computer Science Division, +*> University of California, Berkeley +*> +*> \endverbatim +* +* ===================================================================== + SUBROUTINE DGEQP3RK( M, N, NRHS, KMAX, ABSTOL, RELTOL, A, LDA, + $ K, MAXC2NRMK, RELMAXC2NRMK, JPIV, TAU, + $ WORK, LWORK, IWORK, INFO ) + IMPLICIT NONE +* +* -- LAPACK computational routine -- +* -- LAPACK is a software package provided by Univ. of Tennessee, -- +* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- +* +* .. Scalar Arguments .. + INTEGER INFO, K, KF, KMAX, LDA, LWORK, M, N, NRHS + DOUBLE PRECISION ABSTOL, MAXC2NRMK, RELMAXC2NRMK, RELTOL +* .. +* .. Array Arguments .. + INTEGER IWORK( * ), JPIV( * ) + DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * ) +* .. +* +* ===================================================================== +* +* .. Parameters .. + INTEGER INB, INBMIN, IXOVER + PARAMETER ( INB = 1, INBMIN = 2, IXOVER = 3 ) + DOUBLE PRECISION ZERO, ONE, TWO + PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 ) +* .. +* .. Local Scalars .. + LOGICAL LQUERY, DONE + INTEGER IINFO, IOFFSET, IWS, J, JB, JBF, JMAXB, JMAX, + $ JMAXC2NRM, KP1, LWKOPT, MINMN, N_SUB, NB, + $ NBMIN, NX + DOUBLE PRECISION EPS, HUGEVAL, MAXC2NRM, SAFMIN +* .. +* .. External Subroutines .. + EXTERNAL DLAQP2RK, DLAQP3RK, XERBLA +* .. +* .. External Functions .. + LOGICAL DISNAN + INTEGER IDAMAX, ILAENV + DOUBLE PRECISION DLAMCH, DNRM2 + EXTERNAL DISNAN, DLAMCH, DNRM2, IDAMAX, ILAENV +* .. +* .. Intrinsic Functions .. + INTRINSIC DBLE, MAX, MIN +* .. +* .. Executable Statements .. +* +* Test input arguments +* ==================== +* + INFO = 0 + LQUERY = ( LWORK.EQ.-1 ) + IF( M.LT.0 ) THEN + INFO = -1 + ELSE IF( N.LT.0 ) THEN + INFO = -2 + ELSE IF( NRHS.LT.0 ) THEN + INFO = -3 + ELSE IF( KMAX.LT.0 ) THEN + INFO = -4 + ELSE IF( DISNAN( ABSTOL ) ) THEN + INFO = -5 + ELSE IF( DISNAN( RELTOL ) ) THEN + INFO = -6 + ELSE IF( LDA.LT.MAX( 1, M ) ) THEN + INFO = -8 + END IF +* +* If the input parameters M, N, NRHS, KMAX, LDA are valid: +* a) Test the input workspace size LWORK for the minimum +* size requirement IWS. +* b) Determine the optimal block size NB and optimal +* workspace size LWKOPT to be returned in WORK(1) +* in case of (1) LWORK < IWS, (2) LQUERY = .TRUE., +* (3) when routine exits. +* Here, IWS is the miminum workspace required for unblocked +* code. +* + IF( INFO.EQ.0 ) THEN + MINMN = MIN( M, N ) + IF( MINMN.EQ.0 ) THEN + IWS = 1 + LWKOPT = 1 + ELSE +* +* Minimal workspace size in case of using only unblocked +* BLAS 2 code in DLAQP2RK. +* 1) DGEQP3RK and DLAQP2RK: 2*N to store full and partial +* column 2-norms. +* 2) DLAQP2RK: N+NRHS-1 to use in WORK array that is used +* in DLARF subroutine inside DLAQP2RK to apply an +* elementary reflector from the left. +* TOTAL_WORK_SIZE = 3*N + NRHS - 1 +* + IWS = 3*N + NRHS - 1 +* +* Assign to NB optimal block size. +* + NB = ILAENV( INB, 'DGEQP3RK', ' ', M, N, -1, -1 ) +* +* A formula for the optimal workspace size in case of using +* both unblocked BLAS 2 in DLAQP2RK and blocked BLAS 3 code +* in DLAQP3RK. +* 1) DGEQP3RK, DLAQP2RK, DLAQP3RK: 2*N to store full and +* partial column 2-norms. +* 2) DLAQP2RK: N+NRHS-1 to use in WORK array that is used +* in DLARF subroutine to apply an elementary reflector +* from the left. +* 3) DLAQP3RK: NB*(N+NRHS) to use in the work array F that +* is used to apply a block reflector from +* the left. +* 4) DLAQP3RK: NB to use in the auxilixary array AUX. +* Sizes (2) and ((3) + (4)) should intersect, therefore +* TOTAL_WORK_SIZE = 2*N + NB*( N+NRHS+1 ), given NBMIN=2. +* + LWKOPT = 2*N + NB*( N+NRHS+1 ) + END IF + WORK( 1 ) = DBLE( LWKOPT ) +* + IF( ( LWORK.LT.IWS ) .AND. .NOT.LQUERY ) THEN + INFO = -15 + END IF + END IF +* +* NOTE: The optimal workspace size is returned in WORK(1), if +* the input parameters M, N, NRHS, KMAX, LDA are valid. +* + IF( INFO.NE.0 ) THEN + CALL XERBLA( 'DGEQP3RK', -INFO ) + RETURN + ELSE IF( LQUERY ) THEN + RETURN + END IF +* +* Quick return if possible for M=0 or N=0. +* + IF( MINMN.EQ.0 ) THEN + K = 0 + MAXC2NRMK = ZERO + RELMAXC2NRMK = ZERO + WORK( 1 ) = DBLE( LWKOPT ) + RETURN + END IF +* +* ================================================================== +* +* Initialize column pivot array JPIV. +* + DO J = 1, N + JPIV( J ) = J + END DO +* +* ================================================================== +* +* Initialize storage for partial and exact column 2-norms. +* a) The elements WORK(1:N) are used to store partial column +* 2-norms of the matrix A, and may decrease in each computation +* step; initialize to the values of complete columns 2-norms. +* b) The elements WORK(N+1:2*N) are used to store complete column +* 2-norms of the matrix A, they are not changed during the +* computation; initialize the values of complete columns 2-norms. +* + DO J = 1, N + WORK( J ) = DNRM2( M, A( 1, J ), 1 ) + WORK( N+J ) = WORK( J ) + END DO +* +* ================================================================== +* +* Compute the pivot column index and the maximum column 2-norm +* for the whole original matrix stored in A(1:M,1:N). +* + KP1 = IDAMAX( N, WORK( 1 ), 1 ) + MAXC2NRM = WORK( KP1 ) +* +* ==================================================================. +* + IF( DISNAN( MAXC2NRM ) ) THEN +* +* Check if the matrix A contains NaN, set INFO parameter +* to the column number where the first NaN is found and return +* from the routine. +* + K = 0 + INFO = KP1 +* +* Set MAXC2NRMK and RELMAXC2NRMK to NaN. +* + MAXC2NRMK = MAXC2NRM + RELMAXC2NRMK = MAXC2NRM +* +* Array TAU is not set and contains undefined elements. +* + WORK( 1 ) = DBLE( LWKOPT ) + RETURN + END IF +* +* =================================================================== +* + IF( MAXC2NRM.EQ.ZERO ) THEN +* +* Check is the matrix A is a zero matrix, set array TAU and +* return from the routine. +* + K = 0 + MAXC2NRMK = ZERO + RELMAXC2NRMK = ZERO +* + DO J = 1, MINMN + TAU( J ) = ZERO + END DO +* + WORK( 1 ) = DBLE( LWKOPT ) + RETURN +* + END IF +* +* =================================================================== +* + HUGEVAL = DLAMCH( 'Overflow' ) +* + IF( MAXC2NRM.GT.HUGEVAL ) THEN +* +* Check if the matrix A contains +Inf or -Inf, set INFO parameter +* to the column number, where the first +/-Inf is found plus N, +* and continue the computation. +* + INFO = N + KP1 +* + END IF +* +* ================================================================== +* +* Quick return if possible for the case when the first +* stopping criterion is satisfied, i.e. KMAX = 0. +* + IF( KMAX.EQ.0 ) THEN + K = 0 + MAXC2NRMK = MAXC2NRM + RELMAXC2NRMK = ONE + DO J = 1, MINMN + TAU( J ) = ZERO + END DO + WORK( 1 ) = DBLE( LWKOPT ) + RETURN + END IF +* +* ================================================================== +* + EPS = DLAMCH('Epsilon') +* +* Adjust ABSTOL +* + IF( ABSTOL.GE.ZERO ) THEN + SAFMIN = DLAMCH('Safe minimum') + ABSTOL = MAX( ABSTOL, TWO*SAFMIN ) + END IF +* +* Adjust RELTOL +* + IF( RELTOL.GE.ZERO ) THEN + RELTOL = MAX( RELTOL, EPS ) + END IF +* +* =================================================================== +* +* JMAX is the maximum index of the column to be factorized, +* which is also limited by the first stopping criterion KMAX. +* + JMAX = MIN( KMAX, MINMN ) +* +* =================================================================== +* +* Quick return if possible for the case when the second or third +* stopping criterion for the whole original matrix is satified, +* i.e. MAXC2NRM <= ABSTOL or RELMAXC2NRM <= RELTOL +* (which is ONE <= RELTOL). +* + IF( MAXC2NRM.LE.ABSTOL .OR. ONE.LE.RELTOL ) THEN +* + K = 0 + MAXC2NRMK = MAXC2NRM + RELMAXC2NRMK = ONE +* + DO J = 1, MINMN + TAU( J ) = ZERO + END DO +* + WORK( 1 ) = DBLE( LWKOPT ) + RETURN + END IF +* +* ================================================================== +* Factorize columns +* ================================================================== +* +* Determine the block size. +* + NBMIN = 2 + NX = 0 +* + IF( ( NB.GT.1 ) .AND. ( NB.LT.MINMN ) ) THEN +* +* Determine when to cross over from blocked to unblocked code. +* (for N less than NX, unblocked code should be used). +* + NX = MAX( 0, ILAENV( IXOVER, 'DGEQP3RK', ' ', M, N, -1, -1 )) +* + IF( NX.LT.MINMN ) THEN +* +* Determine if workspace is large enough for blocked code. +* + IF( LWORK.LT.LWKOPT ) THEN +* +* Not enough workspace to use optimal block size that +* is currently stored in NB. +* Reduce NB and determine the minimum value of NB. +* + NB = ( LWORK-2*N ) / ( N+1 ) + NBMIN = MAX( 2, ILAENV( INBMIN, 'DGEQP3RK', ' ', M, N, + $ -1, -1 ) ) +* + END IF + END IF + END IF +* +* ================================================================== +* +* DONE is the boolean flag to rerpresent the case when the +* factorization completed in the block factorization routine, +* before the end of the block. +* + DONE = .FALSE. +* +* J is the column index. +* + J = 1 +* +* (1) Use blocked code initially. +* +* JMAXB is the maximum column index of the block, when the +* blocked code is used, is also limited by the first stopping +* criterion KMAX. +* + JMAXB = MIN( KMAX, MINMN - NX ) +* + IF( NB.GE.NBMIN .AND. NB.LT.JMAX .AND. JMAXB.GT.0 ) THEN +* +* Loop over the column blocks of the matrix A(1:M,1:JMAXB). Here: +* J is the column index of a column block; +* JB is the column block size to pass to block factorization +* routine in a loop step; +* JBF is the number of columns that were actually factorized +* that was returned by the block factorization routine +* in a loop step, JBF <= JB; +* N_SUB is the number of columns in the submatrix; +* IOFFSET is the number of rows that should not be factorized. +* + DO WHILE( J.LE.JMAXB ) +* + JB = MIN( NB, JMAXB-J+1 ) + N_SUB = N-J+1 + IOFFSET = J-1 +* +* Factorize JB columns among the columns A(J:N). +* + CALL DLAQP3RK( M, N_SUB, NRHS, IOFFSET, JB, ABSTOL, + $ RELTOL, KP1, MAXC2NRM, A( 1, J ), LDA, + $ DONE, JBF, MAXC2NRMK, RELMAXC2NRMK, + $ JPIV( J ), TAU( J ), + $ WORK( J ), WORK( N+J ), + $ WORK( 2*N+1 ), WORK( 2*N+JB+1 ), + $ N+NRHS-J+1, IWORK, IINFO ) +* +* Set INFO on the first occurence of Inf. +* + IF( IINFO.GT.N_SUB .AND. INFO.EQ.0 ) THEN + INFO = 2*IOFFSET + IINFO + END IF +* + IF( DONE ) THEN +* +* Either the submatrix is zero before the end of the +* column block, or ABSTOL or RELTOL criterion is +* satisfied before the end of the column block, we can +* return from the routine. Perform the following before +* returning: +* a) Set the number of factorized columns K, +* K = IOFFSET + JBF from the last call of blocked +* routine. +* NOTE: 1) MAXC2NRMK and RELMAXC2NRMK are returned +* by the block factorization routine; +* 2) The remaining TAUs are set to ZERO by the +* block factorization routine. +* + K = IOFFSET + JBF +* +* Set INFO on the first occurrence of NaN, NaN takes +* prcedence over Inf. +* + IF( IINFO.LE.N_SUB .AND. IINFO.GT.0 ) THEN + INFO = IOFFSET + IINFO + END IF +* +* Return from the routine. +* + WORK( 1 ) = DBLE( LWKOPT ) +* + RETURN +* + END IF +* + J = J + JBF +* + END DO +* + END IF +* +* Use unblocked code to factor the last or only block. +* J = JMAX+1 means we factorized the maximum possible number of +* columns, that is in ELSE clause we need to compute +* the MAXC2NORM and RELMAXC2NORM to return after we processed +* the blocks. +* + IF( J.LE.JMAX ) THEN +* +* N_SUB is the number of columns in the submatrix; +* IOFFSET is the number of rows that should not be factorized. +* + N_SUB = N-J+1 + IOFFSET = J-1 +* + CALL DLAQP2RK( M, N_SUB, NRHS, IOFFSET, JMAX-J+1, + $ ABSTOL, RELTOL, KP1, MAXC2NRM, A( 1, J ), LDA, + $ KF, MAXC2NRMK, RELMAXC2NRMK, JPIV( J ), + $ TAU( J ), WORK( J ), WORK( N+J ), + $ WORK( 2*N+1 ), IINFO ) +* +* ABSTOL or RELTOL criterion is satisfied when the number of +* the factorized columns KF is smaller then the number +* of columns JMAX-J+1 supplied to be factorized by the +* unblocked routine, we can return from +* the routine. Perform the following before returning: +* a) Set the number of factorized columns K, +* b) MAXC2NRMK and RELMAXC2NRMK are returned by the +* unblocked factorization routine above. +* + K = J - 1 + KF +* +* Set INFO on the first exception occurence. +* +* Set INFO on the first exception occurence of Inf or NaN, +* (NaN takes precedence over Inf). +* + IF( IINFO.GT.N_SUB .AND. INFO.EQ.0 ) THEN + INFO = 2*IOFFSET + IINFO + ELSE IF( IINFO.LE.N_SUB .AND. IINFO.GT.0 ) THEN + INFO = IOFFSET + IINFO + END IF +* + ELSE +* +* Compute the return values for blocked code. +* +* Set the number of factorized columns if the unblocked routine +* was not called. +* + K = JMAX +* +* If there exits a residual matrix after the blocked code: +* 1) compute the values of MAXC2NRMK, RELMAXC2NRMK of the +* residual matrix, otherwise set them to ZERO; +* 2) Set TAU(K+1:MINMN) to ZERO. +* + IF( K.LT.MINMN ) THEN + JMAXC2NRM = K + IDAMAX( N-K, WORK( K+1 ), 1 ) + MAXC2NRMK = WORK( JMAXC2NRM ) + IF( K.EQ.0 ) THEN + RELMAXC2NRMK = ONE + ELSE + RELMAXC2NRMK = MAXC2NRMK / MAXC2NRM + END IF +* + DO J = K + 1, MINMN + TAU( J ) = ZERO + END DO +* + END IF +* +* END IF( J.LE.JMAX ) THEN +* + END IF +* + WORK( 1 ) = DBLE( LWKOPT ) +* + RETURN +* +* End of DGEQP3RK +* + END diff --git a/SRC/dlaqp2rk.f b/SRC/dlaqp2rk.f new file mode 100755 index 0000000000..b5a84d0de1 --- /dev/null +++ b/SRC/dlaqp2rk.f @@ -0,0 +1,713 @@ +*> \brief \b DLAQP2RK computes truncated QR factorization with column pivoting of a real matrix block using Level 2 BLAS and overwrites a real m-by-nrhs matrix B with Q**T * B. +* +* =========== DOCUMENTATION =========== +* +* Online html documentation available at +* http://www.netlib.org/lapack/explore-html/ +* +*> \htmlonly +*> Download DLAQP2RK + dependencies +*> +*> [TGZ] +*> +*> [ZIP] +*> +*> [TXT] +*> \endhtmlonly +* +* Definition: +* =========== +* +* SUBROUTINE DLAQP2RK( M, N, NRHS, IOFFSET, KMAX, ABSTOL, RELTOL, +* $ KP1, MAXC2NRM, A, LDA, K, MAXC2NRMK, +* $ RELMAXC2NRMK, JPIV, TAU, VN1, VN2, WORK, +* $ INFO ) +* IMPLICIT NONE +* +* .. Scalar Arguments .. +* INTEGER INFO, IOFFSET, KP1, K, KMAX, LDA, M, N, NRHS +* DOUBLE PRECISION ABSTOL, MAXC2NRM, MAXC2NRMK, RELMAXC2NRMK, +* $ RELTOL +* .. +* .. Array Arguments .. +* INTEGER JPIV( * ) +* DOUBLE PRECISION A( LDA, * ), TAU( * ), VN1( * ), VN2( * ), +* $ WORK( * ) +* .. +* +* +*> \par Purpose: +* ============= +*> +*> \verbatim +*> +*> DLAQP2RK computes a truncated (rank K) or full rank Householder QR +*> factorization with column pivoting of a real matrix +*> block A(IOFFSET+1:M,1:N) as +*> +*> A * P(K) = Q(K) * R(K). +*> +*> The routine uses Level 2 BLAS. The block A(1:IOFFSET,1:N) +*> is accordingly pivoted, but not factorized. +*> +*> The routine also overwrites the right-hand-sides matrix block B +*> stored in A(IOFFSET+1:M,N+1:N+NRHS) with Q(K)**T * B. +*> \endverbatim +* +* Arguments: +* ========== +* +*> \param[in] M +*> \verbatim +*> M is INTEGER +*> The number of rows of the matrix A. M >= 0. +*> \endverbatim +*> +*> \param[in] N +*> \verbatim +*> N is INTEGER +*> The number of columns of the matrix A. N >= 0. +*> \endverbatim +*> +*> \param[in] NRHS +*> \verbatim +*> NRHS is INTEGER +*> The number of right hand sides, i.e., the number of +*> columns of the matrix B. NRHS >= 0. +*> \endverbatim +*> +*> \param[in] IOFFSET +*> \verbatim +*> IOFFSET is INTEGER +*> The number of rows of the matrix A that must be pivoted +*> but not factorized. IOFFSET >= 0. +*> +*> IOFFSET also represents the number of columns of the whole +*> original matrix A_orig that have been factorized +*> in the previous steps. +*> \endverbatim +*> +*> \param[in] KMAX +*> \verbatim +*> KMAX is INTEGER +*> +*> The first factorization stopping criterion. KMAX >= 0. +*> +*> The maximum number of columns of the matrix A to factorize, +*> i.e. the maximum factorization rank. +*> +*> a) If KMAX >= min(M-IOFFSET,N), then this stopping +*> criterion is not used, factorize columns +*> depending on ABSTOL and RELTOL. +*> +*> b) If KMAX = 0, then this stopping criterion is +*> satisfied on input and the routine exits immediately. +*> This means that the factorization is not performed, +*> the matrices A and B and the arrays TAU, IPIV +*> are not modified. +*> \endverbatim +*> +*> \param[in] ABSTOL +*> \verbatim +*> ABSTOL is DOUBLE PRECISION, cannot be NaN. +*> +*> The second factorization stopping criterion. +*> +*> The absolute tolerance (stopping threshold) for +*> maximum column 2-norm of the residual matrix. +*> The algorithm converges (stops the factorization) when +*> the maximum column 2-norm of the residual matrix +*> is less than or equal to ABSTOL. +*> +*> a) If ABSTOL < 0.0, then this stopping criterion is not +*> used, the routine factorizes columns depending +*> on KMAX and RELTOL. +*> This includes the case ABSTOL = -Inf. +*> +*> b) If 0.0 <= ABSTOL then the input value +*> of ABSTOL is used. +*> \endverbatim +*> +*> \param[in] RELTOL +*> \verbatim +*> RELTOL is DOUBLE PRECISION, cannot be NaN. +*> +*> The third factorization stopping criterion. +*> +*> The tolerance (stopping threshold) for the ratio of the +*> maximum column 2-norm of the residual matrix to the maximum +*> column 2-norm of the original matrix A_orig. The algorithm +*> converges (stops the factorization), when this ratio is +*> less than or equal to RELTOL. +*> +*> a) If RELTOL < 0.0, then this stopping criterion is not +*> used, the routine factorizes columns depending +*> on KMAX and ABSTOL. +*> This includes the case RELTOL = -Inf. +*> +*> d) If 0.0 <= RELTOL then the input value of RELTOL +*> is used. +*> \endverbatim +*> +*> \param[in] KP1 +*> \verbatim +*> KP1 is INTEGER +*> The index of the column with the maximum 2-norm in +*> the whole original matrix A_orig determined in the +*> main routine DGEQP3RK. 1 <= KP1 <= N_orig_mat. +*> \endverbatim +*> +*> \param[in] MAXC2NRM +*> \verbatim +*> MAXC2NRM is DOUBLE PRECISION +*> The maximum column 2-norm of the whole original +*> matrix A_orig computed in the main routine DGEQP3RK. +*> MAXC2NRM >= 0. +*> \endverbatim +*> +*> \param[in,out] A +*> \verbatim +*> A is DOUBLE PRECISION array, dimension (LDA,N+NRHS) +*> On entry: +*> the M-by-N matrix A and M-by-NRHS matrix B, as in +*> +*> N NRHS +*> array_A = M [ mat_A, mat_B ] +*> +*> On exit: +*> 1. The elements in block A(IOFFSET+1:M,1:K) below +*> the diagonal together with the array TAU represent +*> the orthogonal matrix Q(K) as a product of elementary +*> reflectors. +*> 2. The upper triangular block of the matrix A stored +*> in A(IOFFSET+1:M,1:K) is the triangular factor obtained. +*> 3. The block of the matrix A stored in A(1:IOFFSET,1:N) +*> has been accordingly pivoted, but not factorized. +*> 4. The rest of the array A, block A(IOFFSET+1:M,K+1:N+NRHS). +*> The left part A(IOFFSET+1:M,K+1:N) of this block +*> contains the residual of the matrix A, and, +*> if NRHS > 0, the right part of the block +*> A(IOFFSET+1:M,N+1:N+NRHS) contains the block of +*> the right-hand-side matrix B. Both these blocks have been +*> updated by multiplication from the left by Q(K)**T. +*> \endverbatim +*> +*> \param[in] LDA +*> \verbatim +*> LDA is INTEGER +*> The leading dimension of the array A. LDA >= max(1,M). +*> \endverbatim +*> +*> \param[out] K +*> \verbatim +*> K is INTEGER +*> Factorization rank of the matrix A, i.e. the rank of +*> the factor R, which is the same as the number of non-zero +*> rows of the factor R. 0 <= K <= min(M-IOFFSET,KMAX,N). +*> +*> K also represents the number of non-zero Householder +*> vectors. +*> \endverbatim +*> +*> \param[out] MAXC2NRMK +*> \verbatim +*> MAXC2NRMK is DOUBLE PRECISION +*> The maximum column 2-norm of the residual matrix, +*> when the factorization stopped at rank K. MAXC2NRMK >= 0. +*> \endverbatim +*> +*> \param[out] RELMAXC2NRMK +*> \verbatim +*> RELMAXC2NRMK is DOUBLE PRECISION +*> The ratio MAXC2NRMK / MAXC2NRM of the maximum column +*> 2-norm of the residual matrix (when the factorization +*> stopped at rank K) to the maximum column 2-norm of the +*> whole original matrix A. RELMAXC2NRMK >= 0. +*> \endverbatim +*> +*> \param[out] JPIV +*> \verbatim +*> JPIV is INTEGER array, dimension (N) +*> Column pivot indices, for 1 <= j <= N, column j +*> of the matrix A was interchanged with column JPIV(j). +*> \endverbatim +*> +*> \param[out] TAU +*> \verbatim +*> TAU is DOUBLE PRECISION array, dimension (min(M-IOFFSET,N)) +*> The scalar factors of the elementary reflectors. +*> \endverbatim +*> +*> \param[in,out] VN1 +*> \verbatim +*> VN1 is DOUBLE PRECISION array, dimension (N) +*> The vector with the partial column norms. +*> \endverbatim +*> +*> \param[in,out] VN2 +*> \verbatim +*> VN2 is DOUBLE PRECISION array, dimension (N) +*> The vector with the exact column norms. +*> \endverbatim +*> +*> \param[out] WORK +*> \verbatim +*> WORK is DOUBLE PRECISION array, dimension (N-1) +*> Used in DLARF subroutine to apply an elementary +*> reflector from the left. +*> \endverbatim +*> +*> \param[out] INFO +*> \verbatim +*> INFO is INTEGER +*> 1) INFO = 0: successful exit. +*> 2) If INFO = j_1, where 1 <= j_1 <= N, then NaN was +*> detected and the routine stops the computation. +*> The j_1-th column of the matrix A or the j_1-th +*> element of array TAU contains the first occurrence +*> of NaN in the factorization step K+1 ( when K columns +*> have been factorized ). +*> +*> On exit: +*> K is set to the number of +*> factorized columns without +*> exception. +*> MAXC2NRMK is set to NaN. +*> RELMAXC2NRMK is set to NaN. +*> TAU(K+1:min(M,N)) is not set and contains undefined +*> elements. If j_1=K+1, TAU(K+1) +*> may contain NaN. +*> 3) If INFO = j_2, where N+1 <= j_2 <= 2*N, then no NaN +*> was detected, but +Inf (or -Inf) was detected and +*> the routine continues the computation until completion. +*> The (j_2-N)-th column of the matrix A contains the first +*> occurrence of +Inf (or -Inf) in the factorization +*> step K+1 ( when K columns have been factorized ). +*> \endverbatim +* +* Authors: +* ======== +* +*> \author Univ. of Tennessee +*> \author Univ. of California Berkeley +*> \author Univ. of Colorado Denver +*> \author NAG Ltd. +* +*> \ingroup laqp2rk +* +*> \par References: +* ================ +*> [1] A Level 3 BLAS QR factorization algorithm with column pivoting developed in 1996. +*> G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain. +*> X. Sun, Computer Science Dept., Duke University, USA. +*> C. H. Bischof, Math. and Comp. Sci. Div., Argonne National Lab, USA. +*> A BLAS-3 version of the QR factorization with column pivoting. +*> LAPACK Working Note 114 +*> \htmlonly +*> https://www.netlib.org/lapack/lawnspdf/lawn114.pdf +*> \endhtmlonly +*> and in +*> SIAM J. Sci. Comput., 19(5):1486-1494, Sept. 1998. +*> \htmlonly +*> https://doi.org/10.1137/S1064827595296732 +*> \endhtmlonly +*> +*> [2] A partial column norm updating strategy developed in 2006. +*> Z. Drmac and Z. Bujanovic, Dept. of Math., University of Zagreb, Croatia. +*> On the failure of rank revealing QR factorization software – a case study. +*> LAPACK Working Note 176. +*> \htmlonly +*> http://www.netlib.org/lapack/lawnspdf/lawn176.pdf +*> \endhtmlonly +*> and in +*> ACM Trans. Math. Softw. 35, 2, Article 12 (July 2008), 28 pages. +*> \htmlonly +*> https://doi.org/10.1145/1377612.1377616 +*> \endhtmlonly +* +*> \par Contributors: +* ================== +*> +*> \verbatim +*> +*> November 2023, Igor Kozachenko, James Demmel, +*> Computer Science Division, +*> University of California, Berkeley +*> +*> \endverbatim +* +* ===================================================================== + SUBROUTINE DLAQP2RK( M, N, NRHS, IOFFSET, KMAX, ABSTOL, RELTOL, + $ KP1, MAXC2NRM, A, LDA, K, MAXC2NRMK, + $ RELMAXC2NRMK, JPIV, TAU, VN1, VN2, WORK, + $ INFO ) + IMPLICIT NONE +* +* -- LAPACK auxiliary routine -- +* -- LAPACK is a software package provided by Univ. of Tennessee, -- +* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- +* +* .. Scalar Arguments .. + INTEGER INFO, IOFFSET, KP1, K, KMAX, LDA, M, N, NRHS + DOUBLE PRECISION ABSTOL, MAXC2NRM, MAXC2NRMK, RELMAXC2NRMK, + $ RELTOL +* .. +* .. Array Arguments .. + INTEGER JPIV( * ) + DOUBLE PRECISION A( LDA, * ), TAU( * ), VN1( * ), VN2( * ), + $ WORK( * ) +* .. +* +* ===================================================================== +* +* .. Parameters .. + DOUBLE PRECISION ZERO, ONE + PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) +* .. +* .. Local Scalars .. + INTEGER I, ITEMP, J, JMAXC2NRM, KK, KP, MINMNFACT, + $ MINMNUPDT + DOUBLE PRECISION AIKK, HUGEVAL, TEMP, TEMP2, TOL3Z +* .. +* .. External Subroutines .. + EXTERNAL DLARF, DLARFG, DSWAP +* .. +* .. Intrinsic Functions .. + INTRINSIC ABS, MAX, MIN, SQRT +* .. +* .. External Functions .. + LOGICAL DISNAN + INTEGER IDAMAX + DOUBLE PRECISION DLAMCH, DNRM2 + EXTERNAL DISNAN, DLAMCH, IDAMAX, DNRM2 +* .. +* .. Executable Statements .. +* +* Initialize INFO +* + INFO = 0 +* +* MINMNFACT in the smallest dimension of the submatrix +* A(IOFFSET+1:M,1:N) to be factorized. +* +* MINMNUPDT is the smallest dimension +* of the subarray A(IOFFSET+1:M,1:N+NRHS) to be udated, which +* contains the submatrices A(IOFFSET+1:M,1:N) and +* B(IOFFSET+1:M,1:NRHS) as column blocks. +* + MINMNFACT = MIN( M-IOFFSET, N ) + MINMNUPDT = MIN( M-IOFFSET, N+NRHS ) + KMAX = MIN( KMAX, MINMNFACT ) + TOL3Z = SQRT( DLAMCH( 'Epsilon' ) ) + HUGEVAL = DLAMCH( 'Overflow' ) +* +* Compute the factorization, KK is the lomn loop index. +* + DO KK = 1, KMAX +* + I = IOFFSET + KK +* + IF( I.EQ.1 ) THEN +* +* ============================================================ +* +* We are at the first column of the original whole matrix A, +* therefore we use the computed KP1 and MAXC2NRM from the +* main routine. +* + + KP = KP1 +* +* ============================================================ +* + ELSE +* +* ============================================================ +* +* Determine the pivot column in KK-th step, i.e. the index +* of the column with the maximum 2-norm in the +* submatrix A(I:M,K:N). +* + KP = ( KK-1 ) + IDAMAX( N-KK+1, VN1( KK ), 1 ) +* +* Determine the maximum column 2-norm and the relative maximum +* column 2-norm of the submatrix A(I:M,KK:N) in step KK. +* RELMAXC2NRMK will be computed later, after somecondition +* checks on MAXC2NRMK. +* + MAXC2NRMK = VN1( KP ) +* +* ============================================================ +* +* Check if the submatrix A(I:M,KK:N) contains NaN, and set +* INFO parameter to the column number, where the first NaN +* is found and return from the routine. +* We need to check the condition only if the +* column index (same as row index) of the original whole +* matrix is larger than 1, since the condition for whole +* original matrix is checked in the main routine. +* + IF( DISNAN( MAXC2NRMK ) ) THEN +* +* Set K, the number of factorized columns. +* that are not zero. +* + K = KK - 1 + INFO = K + KP +* +* Set RELMAXC2NRMK to NaN. +* + RELMAXC2NRMK = MAXC2NRMK +* +* Array TAU(K+1:MINMNFACT) is not set and contains +* undefined elements. +* + RETURN + END IF +* +* ============================================================ +* +* Quick return, if the submatrix A(I:M,KK:N) is +* a zero matrix. +* We need to check the condition only if the +* column index (same as row index) of the original whole +* matrix is larger than 1, since the condition for whole +* original matrix is checked in the main routine. +* + IF( MAXC2NRMK.EQ.ZERO ) THEN +* +* Set K, the number of factorized columns. +* that are not zero. +* + K = KK - 1 + RELMAXC2NRMK = ZERO +* +* Set TAUs corresponding to the columns that were not +* factorized to ZERO, i.e. set TAU(KK:MINMNFACT) to ZERO. +* + DO J = KK, MINMNFACT + TAU( J ) = ZERO + END DO +* +* Return from the routine. +* + RETURN +* + END IF +* +* ============================================================ +* +* Check if the submatrix A(I:M,KK:N) contains Inf, +* set INFO parameter to the column number, where +* the first Inf is found plus N, and continue +* the computation. +* We need to check the condition only if the +* column index (same as row index) of the original whole +* matrix is larger than 1, since the condition for whole +* original matrix is checked in the main routine. +* + IF( INFO.EQ.0 .AND. MAXC2NRMK.GT.HUGEVAL ) THEN + INFO = N + KK - 1 + KP + END IF +* +* ============================================================ +* +* Test for the second and third stopping criteria. +* NOTE: There is no need to test for ABSTOL >= ZERO, since +* MAXC2NRMK is non-negative. Similarly, there is no need +* to test for RELTOL >= ZERO, since RELMAXC2NRMK is +* non-negative. +* We need to check the condition only if the +* column index (same as row index) of the original whole +* matrix is larger than 1, since the condition for whole +* original matrix is checked in the main routine. + + RELMAXC2NRMK = MAXC2NRMK / MAXC2NRM +* + IF( MAXC2NRMK.LE.ABSTOL .OR. RELMAXC2NRMK.LE.RELTOL ) THEN +* +* Set K, the number of factorized columns. +* + K = KK - 1 +* +* Set TAUs corresponding to the columns that were not +* factorized to ZERO, i.e. set TAU(KK:MINMNFACT) to ZERO. +* + DO J = KK, MINMNFACT + TAU( J ) = ZERO + END DO +* +* Return from the routine. +* + RETURN +* + END IF +* +* ============================================================ +* +* End ELSE of IF(I.EQ.1) +* + END IF +* +* =============================================================== +* +* If the pivot column is not the first column of the +* subblock A(1:M,KK:N): +* 1) swap the KK-th column and the KP-th pivot column +* in A(1:M,1:N); +* 2) copy the KK-th element into the KP-th element of the partial +* and exact 2-norm vectors VN1 and VN2. ( Swap is not needed +* for VN1 and VN2 since we use the element with the index +* larger than KK in the next loop step.) +* 3) Save the pivot interchange with the indices relative to the +* the original matrix A, not the block A(1:M,1:N). +* + IF( KP.NE.KK ) THEN + CALL DSWAP( M, A( 1, KP ), 1, A( 1, KK ), 1 ) + VN1( KP ) = VN1( KK ) + VN2( KP ) = VN2( KK ) + ITEMP = JPIV( KP ) + JPIV( KP ) = JPIV( KK ) + JPIV( KK ) = ITEMP + END IF +* +* Generate elementary reflector H(KK) using the column A(I:M,KK), +* if the column has more than one element, otherwise +* the elementary reflector would be an identity matrix, +* and TAU(KK) = ZERO. +* + IF( I.LT.M ) THEN + CALL DLARFG( M-I+1, A( I, KK ), A( I+1, KK ), 1, + $ TAU( KK ) ) + ELSE + TAU( KK ) = ZERO + END IF +* +* Check if TAU(KK) contains NaN, set INFO parameter +* to the column number where NaN is found and return from +* the routine. +* NOTE: There is no need to check TAU(KK) for Inf, +* since DLARFG cannot produce TAU(KK) or Householder vector +* below the diagonal containing Inf. Only BETA on the diagonal, +* returned by DLARFG can contain Inf, which requires +* TAU(KK) to contain NaN. Therefore, this case of generating Inf +* by DLARFG is covered by checking TAU(KK) for NaN. +* + IF( DISNAN( TAU(KK) ) ) THEN + K = KK - 1 + INFO = KK +* +* Set MAXC2NRMK and RELMAXC2NRMK to NaN. +* + MAXC2NRMK = TAU( KK ) + RELMAXC2NRMK = TAU( KK ) +* +* Array TAU(KK:MINMNFACT) is not set and contains +* undefined elements, except the first element TAU(KK) = NaN. +* + RETURN + END IF +* +* Apply H(KK)**T to A(I:M,KK+1:N+NRHS) from the left. +* ( If M >= N, then at KK = N there is no residual matrix, +* i.e. no columns of A to update, only columns of B. +* If M < N, then at KK = M-IOFFSET, I = M and we have a +* one-row residual matrix in A and the elementary +* reflector is a unit matrix, TAU(KK) = ZERO, i.e. no update +* is needed for the residual matrix in A and the +* right-hand-side-matrix in B. +* Therefore, we update only if +* KK < MINMNUPDT = min(M-IOFFSET, N+NRHS) +* condition is satisfied, not only KK < N+NRHS ) +* + IF( KK.LT.MINMNUPDT ) THEN + AIKK = A( I, KK ) + A( I, KK ) = ONE + CALL DLARF( 'Left', M-I+1, N+NRHS-KK, A( I, KK ), 1, + $ TAU( KK ), A( I, KK+1 ), LDA, WORK( 1 ) ) + A( I, KK ) = AIKK + END IF +* + IF( KK.LT.MINMNFACT ) THEN +* +* Update the partial column 2-norms for the residual matrix, +* only if the residual matrix A(I+1:M,KK+1:N) exists, i.e. +* when KK < min(M-IOFFSET, N). +* + DO J = KK + 1, N + IF( VN1( J ).NE.ZERO ) THEN +* +* NOTE: The following lines follow from the analysis in +* Lapack Working Note 176. +* + TEMP = ONE - ( ABS( A( I, J ) ) / VN1( J ) )**2 + TEMP = MAX( TEMP, ZERO ) + TEMP2 = TEMP*( VN1( J ) / VN2( J ) )**2 + IF( TEMP2 .LE. TOL3Z ) THEN +* +* Compute the column 2-norm for the partial +* column A(I+1:M,J) by explicitly computing it, +* and store it in both partial 2-norm vector VN1 +* and exact column 2-norm vector VN2. +* + VN1( J ) = DNRM2( M-I, A( I+1, J ), 1 ) + VN2( J ) = VN1( J ) +* + ELSE +* +* Update the column 2-norm for the partial +* column A(I+1:M,J) by removing one +* element A(I,J) and store it in partial +* 2-norm vector VN1. +* + VN1( J ) = VN1( J )*SQRT( TEMP ) +* + END IF + END IF + END DO +* + END IF +* +* End factorization loop +* + END DO +* +* If we reached this point, all colunms have been factorized, +* i.e. no condition was triggered to exit the routine. +* Set the number of factorized columns. +* + K = KMAX +* +* We reached the end of the loop, i.e. all KMAX columns were +* factorized, we need to set MAXC2NRMK and RELMAXC2NRMK before +* we return. +* + IF( K.LT.MINMNFACT ) THEN +* + JMAXC2NRM = K + IDAMAX( N-K, VN1( K+1 ), 1 ) + MAXC2NRMK = VN1( JMAXC2NRM ) +* + IF( K.EQ.0 ) THEN + RELMAXC2NRMK = ONE + ELSE + RELMAXC2NRMK = MAXC2NRMK / MAXC2NRM + END IF +* + ELSE + MAXC2NRMK = ZERO + RELMAXC2NRMK = ZERO + END IF +* +* We reached the end of the loop, i.e. all KMAX columns were +* factorized, set TAUs corresponding to the columns that were +* not factorized to ZERO, i.e. TAU(K+1:MINMNFACT) set to ZERO. +* + DO J = K + 1, MINMNFACT + TAU( J ) = ZERO + END DO +* + RETURN +* +* End of DLAQP2RK +* + END diff --git a/SRC/dlaqp3rk.f b/SRC/dlaqp3rk.f new file mode 100755 index 0000000000..39e617d0e1 --- /dev/null +++ b/SRC/dlaqp3rk.f @@ -0,0 +1,935 @@ +*> \brief \b DLAQP3RK computes a step of truncated QR factorization with column pivoting of a real m-by-n matrix A using Level 3 BLAS and overwrites a real m-by-nrhs matrix B with Q**T * B. +* +* =========== DOCUMENTATION =========== +* +* Online html documentation available at +* http://www.netlib.org/lapack/explore-html/ +* +*> \htmlonly +*> Download DLAQP3RK + dependencies +*> +*> [TGZ] +*> +*> [ZIP] +*> +*> [TXT] +*> \endhtmlonly +* +* Definition: +* =========== +* +* SUBROUTINE DLAQP3RK( M, N, NRHS, IOFFSET, NB, ABSTOL, +* $ RELTOL, KP1, MAXC2NRM, A, LDA, DONE, KB, +* $ MAXC2NRMK, RELMAXC2NRMK, JPIV, TAU, +* $ VN1, VN2, AUXV, F, LDF, IWORK, INFO ) +* IMPLICIT NONE +* LOGICAL DONE +* INTEGER INFO, IOFFSET, KB, KP1, LDA, LDF, M, N, +* $ NB, NRHS +* DOUBLE PRECISION ABSTOL, MAXC2NRM, MAXC2NRMK, RELMAXC2NRMK, +* $ RELTOL +* +* .. Scalar Arguments .. +* LOGICAL DONE +* INTEGER KB, LDA, LDF, M, N, NB, NRHS, IOFFSET +* DOUBLE PRECISION ABSTOL, MAXC2NRM, MAXC2NRMK, RELMAXC2NRMK, +* $ RELTOL +* .. +* .. Array Arguments .. +* INTEGER IWORK( * ), JPIV( * ) +* DOUBLE PRECISION A( LDA, * ), AUXV( * ), F( LDF, * ), TAU( * ), +* $ VN1( * ), VN2( * ) +* .. +* +* +*> \par Purpose: +* ============= +*> +*> \verbatim +*> +*> DLAQP3RK computes a step of truncated QR factorization with column +*> pivoting of a real M-by-N matrix A block A(IOFFSET+1:M,1:N) +*> by using Level 3 BLAS as +*> +*> A * P(KB) = Q(KB) * R(KB). +*> +*> The routine tries to factorize NB columns from A starting from +*> the row IOFFSET+1 and updates the residual matrix with BLAS 3 +*> xGEMM. The number of actually factorized columns is returned +*> is smaller than NB. +*> +*> Block A(1:IOFFSET,1:N) is accordingly pivoted, but not factorized. +*> +*> The routine also overwrites the right-hand-sides B matrix stored +*> in A(IOFFSET+1:M,1:N+1:N+NRHS) with Q(KB)**T * B. +*> +*> Cases when the number of factorized columns KB < NB: +*> +*> (1) In some cases, due to catastrophic cancellations, it cannot +*> factorize all NB columns and need to update the residual matrix. +*> Hence, the actual number of factorized columns in the block returned +*> in KB is smaller than NB. The logical DONE is returned as FALSE. +*> The factorization of the whole original matrix A_orig must proceed +*> with the next block. +*> +*> (2) Whenever the stopping criterion ABSTOL or RELTOL is satisfied, +*> the factorization of the whole original matrix A_orig is stopped, +*> the logical DONE is returned as TRUE. The number of factorized +*> columns which is smaller than NB is returned in KB. +*> +*> (3) In case both stopping criteria ABSTOL or RELTOL are not used, +*> and when the residual matrix is a zero matrix in some factorization +*> step KB, the factorization of the whole original matrix A_orig is +*> stopped, the logical DONE is returned as TRUE. The number of +*> factorized columns which is smaller than NB is returned in KB. +*> +*> (4) Whenever NaN is detected in the matrix A or in the array TAU, +*> the factorization of the whole original matrix A_orig is stopped, +*> the logical DONE is returned as TRUE. The number of factorized +*> columns which is smaller than NB is returned in KB. The INFO +*> parameter is set to the column index of the first NaN occurrence. +*> +*> \endverbatim +* +* Arguments: +* ========== +* +*> \param[in] M +*> \verbatim +*> M is INTEGER +*> The number of rows of the matrix A. M >= 0. +*> \endverbatim +*> +*> \param[in] N +*> \verbatim +*> N is INTEGER +*> The number of columns of the matrix A. N >= 0 +*> \endverbatim +*> +*> \param[in] NRHS +*> \verbatim +*> NRHS is INTEGER +*> The number of right hand sides, i.e., the number of +*> columns of the matrix B. NRHS >= 0. +*> \endverbatim +*> +*> \param[in] IOFFSET +*> \verbatim +*> IOFFSET is INTEGER +*> The number of rows of the matrix A that must be pivoted +*> but not factorized. IOFFSET >= 0. +*> +*> IOFFSET also represents the number of columns of the whole +*> original matrix A_orig that have been factorized +*> in the previous steps. +*> \endverbatim +*> +*> \param[in] NB +*> \verbatim +*> NB is INTEGER +*> Factorization block size, i.e the number of columns +*> to factorize in the matrix A. 0 <= NB +*> +*> If NB = 0, then the routine exits immediately. +*> This means that the factorization is not performed, +*> the matrices A and B and the arrays TAU, IPIV +*> are not modified. +*> \endverbatim +*> +*> \param[in] ABSTOL +*> \verbatim +*> ABSTOL is DOUBLE PRECISION, cannot be NaN. +*> +*> The absolute tolerance (stopping threshold) for +*> maximum column 2-norm of the residual matrix. +*> The algorithm converges (stops the factorization) when +*> the maximum column 2-norm of the residual matrix +*> is less than or equal to ABSTOL. +*> +*> a) If ABSTOL < 0.0, then this stopping criterion is not +*> used, the routine factorizes columns depending +*> on NB and RELTOL. +*> This includes the case ABSTOL = -Inf. +*> +*> b) If 0.0 <= ABSTOL then the input value +*> of ABSTOL is used. +*> \endverbatim +*> +*> \param[in] RELTOL +*> \verbatim +*> RELTOL is DOUBLE PRECISION, cannot be NaN. +*> +*> The tolerance (stopping threshold) for the ratio of the +*> maximum column 2-norm of the residual matrix to the maximum +*> column 2-norm of the original matrix A_orig. The algorithm +*> converges (stops the factorization), when this ratio is +*> less than or equal to RELTOL. +*> +*> a) If RELTOL < 0.0, then this stopping criterion is not +*> used, the routine factorizes columns depending +*> on NB and ABSTOL. +*> This includes the case RELTOL = -Inf. +*> +*> d) If 0.0 <= RELTOL then the input value of RELTOL +*> is used. +*> \endverbatim +*> +*> \param[in] KP1 +*> \verbatim +*> KP1 is INTEGER +*> The index of the column with the maximum 2-norm in +*> the whole original matrix A_orig determined in the +*> main routine DGEQP3RK. 1 <= KP1 <= N_orig. +*> \endverbatim +*> +*> \param[in] MAXC2NRM +*> \verbatim +*> MAXC2NRM is DOUBLE PRECISION +*> The maximum column 2-norm of the whole original +*> matrix A_orig computed in the main routine DGEQP3RK. +*> MAXC2NRM >= 0. +*> \endverbatim +*> +*> \param[in,out] A +*> \verbatim +*> A is DOUBLE PRECISION array, dimension (LDA,N+NRHS) +*> On entry: +*> the M-by-N matrix A and M-by-NRHS matrix B, as in +*> +*> N NRHS +*> array_A = M [ mat_A, mat_B ] +*> +*> On exit: +*> 1. The elements in block A(IOFFSET+1:M,1:KB) below +*> the diagonal together with the array TAU represent +*> the orthogonal matrix Q(KB) as a product of elementary +*> reflectors. +*> 2. The upper triangular block of the matrix A stored +*> in A(IOFFSET+1:M,1:KB) is the triangular factor obtained. +*> 3. The block of the matrix A stored in A(1:IOFFSET,1:N) +*> has been accordingly pivoted, but not factorized. +*> 4. The rest of the array A, block A(IOFFSET+1:M,KB+1:N+NRHS). +*> The left part A(IOFFSET+1:M,KB+1:N) of this block +*> contains the residual of the matrix A, and, +*> if NRHS > 0, the right part of the block +*> A(IOFFSET+1:M,N+1:N+NRHS) contains the block of +*> the right-hand-side matrix B. Both these blocks have been +*> updated by multiplication from the left by Q(KB)**T. +*> \endverbatim +*> +*> \param[in] LDA +*> \verbatim +*> LDA is INTEGER +*> The leading dimension of the array A. LDA >= max(1,M). +*> \endverbatim +*> +*> \param[out] +*> \verbatim +*> DONE is LOGICAL +*> TRUE: a) if the factorization completed before processing +*> all min(M-IOFFSET,NB,N) columns due to ABSTOL +*> or RELTOL criterion, +*> b) if the factorization completed before processing +*> all min(M-IOFFSET,NB,N) columns due to the +*> residual matrix being a ZERO matrix. +*> c) when NaN was detected in the matrix A +*> or in the array TAU. +*> FALSE: otherwise. +*> \endverbatim +*> +*> \param[out] KB +*> \verbatim +*> KB is INTEGER +*> Factorization rank of the matrix A, i.e. the rank of +*> the factor R, which is the same as the number of non-zero +*> rows of the factor R. 0 <= KB <= min(M-IOFFSET,NB,N). +*> +*> KB also represents the number of non-zero Householder +*> vectors. +*> \endverbatim +*> +*> \param[out] MAXC2NRMK +*> \verbatim +*> MAXC2NRMK is DOUBLE PRECISION +*> The maximum column 2-norm of the residual matrix, +*> when the factorization stopped at rank KB. MAXC2NRMK >= 0. +*> \endverbatim +*> +*> \param[out] RELMAXC2NRMK +*> \verbatim +*> RELMAXC2NRMK is DOUBLE PRECISION +*> The ratio MAXC2NRMK / MAXC2NRM of the maximum column +*> 2-norm of the residual matrix (when the factorization +*> stopped at rank KB) to the maximum column 2-norm of the +*> original matrix A_orig. RELMAXC2NRMK >= 0. +*> \endverbatim +*> +*> \param[out] JPIV +*> \verbatim +*> JPIV is INTEGER array, dimension (N) +*> Column pivot indices, for 1 <= j <= N, column j +*> of the matrix A was interchanged with column JPIV(j). +*> \endverbatim +*> +*> \param[out] TAU +*> \verbatim +*> TAU is DOUBLE PRECISION array, dimension (min(M-IOFFSET,N)) +*> The scalar factors of the elementary reflectors. +*> \endverbatim +*> +*> \param[in,out] VN1 +*> \verbatim +*> VN1 is DOUBLE PRECISION array, dimension (N) +*> The vector with the partial column norms. +*> \endverbatim +*> +*> \param[in,out] VN2 +*> \verbatim +*> VN2 is DOUBLE PRECISION array, dimension (N) +*> The vector with the exact column norms. +*> \endverbatim +*> +*> \param[out] AUXV +*> \verbatim +*> AUXV is DOUBLE PRECISION array, dimension (NB) +*> Auxiliary vector. +*> \endverbatim +*> +*> \param[out] F +*> \verbatim +*> F is DOUBLE PRECISION array, dimension (LDF,NB) +*> Matrix F**T = L*(Y**T)*A. +*> \endverbatim +*> +*> \param[in] LDF +*> \verbatim +*> LDF is INTEGER +*> The leading dimension of the array F. LDF >= max(1,N+NRHS). +*> \endverbatim +*> +*> \param[out] IWORK +*> \verbatim +*> IWORK is INTEGER array, dimension (N-1). +*> Is a work array. ( IWORK is used to store indices +*> of "bad" columns for norm downdating in the residual +*> matrix ). +*> \endverbatim +*> +*> \param[out] INFO +*> \verbatim +*> INFO is INTEGER +*> 1) INFO = 0: successful exit. +*> 2) If INFO = j_1, where 1 <= j_1 <= N, then NaN was +*> detected and the routine stops the computation. +*> The j_1-th column of the matrix A or the j_1-th +*> element of array TAU contains the first occurrence +*> of NaN in the factorization step KB+1 ( when KB columns +*> have been factorized ). +*> +*> On exit: +*> KB is set to the number of +*> factorized columns without +*> exception. +*> MAXC2NRMK is set to NaN. +*> RELMAXC2NRMK is set to NaN. +*> TAU(KB+1:min(M,N)) is not set and contains undefined +*> elements. If j_1=KB+1, TAU(KB+1) +*> may contain NaN. +*> 3) If INFO = j_2, where N+1 <= j_2 <= 2*N, then no NaN +*> was detected, but +Inf (or -Inf) was detected and +*> the routine continues the computation until completion. +*> The (j_2-N)-th column of the matrix A contains the first +*> occurrence of +Inf (or -Inf) in the actorization +*> step KB+1 ( when KB columns have been factorized ). +*> \endverbatim +* +* Authors: +* ======== +* +*> \author Univ. of Tennessee +*> \author Univ. of California Berkeley +*> \author Univ. of Colorado Denver +*> \author NAG Ltd. +* +*> \ingroup laqp3rk +* +*> \par References: +* ================ +*> [1] A Level 3 BLAS QR factorization algorithm with column pivoting developed in 1996. +*> G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain. +*> X. Sun, Computer Science Dept., Duke University, USA. +*> C. H. Bischof, Math. and Comp. Sci. Div., Argonne National Lab, USA. +*> A BLAS-3 version of the QR factorization with column pivoting. +*> LAPACK Working Note 114 +*> \htmlonly +*> https://www.netlib.org/lapack/lawnspdf/lawn114.pdf +*> \endhtmlonly +*> and in +*> SIAM J. Sci. Comput., 19(5):1486-1494, Sept. 1998. +*> \htmlonly +*> https://doi.org/10.1137/S1064827595296732 +*> \endhtmlonly +*> +*> [2] A partial column norm updating strategy developed in 2006. +*> Z. Drmac and Z. Bujanovic, Dept. of Math., University of Zagreb, Croatia. +*> On the failure of rank revealing QR factorization software – a case study. +*> LAPACK Working Note 176. +*> \htmlonly +*> http://www.netlib.org/lapack/lawnspdf/lawn176.pdf +*> \endhtmlonly +*> and in +*> ACM Trans. Math. Softw. 35, 2, Article 12 (July 2008), 28 pages. +*> \htmlonly +*> https://doi.org/10.1145/1377612.1377616 +*> \endhtmlonly +* +*> \par Contributors: +* ================== +*> +*> \verbatim +*> +*> November 2023, Igor Kozachenko, James Demmel, +*> Computer Science Division, +*> University of California, Berkeley +*> +*> \endverbatim +* +* ===================================================================== + SUBROUTINE DLAQP3RK( M, N, NRHS, IOFFSET, NB, ABSTOL, + $ RELTOL, KP1, MAXC2NRM, A, LDA, DONE, KB, + $ MAXC2NRMK, RELMAXC2NRMK, JPIV, TAU, + $ VN1, VN2, AUXV, F, LDF, IWORK, INFO ) + IMPLICIT NONE +* +* -- LAPACK auxiliary routine -- +* -- LAPACK is a software package provided by Univ. of Tennessee, -- +* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- +* +* .. Scalar Arguments .. + LOGICAL DONE + INTEGER INFO, IOFFSET, KB, KP1, LDA, LDF, M, N, + $ NB, NRHS + DOUBLE PRECISION ABSTOL, MAXC2NRM, MAXC2NRMK, RELMAXC2NRMK, + $ RELTOL +* .. +* .. Array Arguments .. + INTEGER IWORK( * ), JPIV( * ) + DOUBLE PRECISION A( LDA, * ), AUXV( * ), F( LDF, * ), TAU( * ), + $ VN1( * ), VN2( * ) +* .. +* +* ===================================================================== +* +* .. Parameters .. + DOUBLE PRECISION ZERO, ONE + PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) +* .. +* .. Local Scalars .. + INTEGER ITEMP, J, K, MINMNFACT, MINMNUPDT, + $ LSTICC, KP, I, IF + DOUBLE PRECISION AIK, HUGEVAL, TEMP, TEMP2, TOL3Z +* .. +* .. External Subroutines .. + EXTERNAL DGEMM, DGEMV, DLARFG, DSWAP +* .. +* .. Intrinsic Functions .. + INTRINSIC ABS, MAX, MIN, SQRT +* .. +* .. External Functions .. + LOGICAL DISNAN + INTEGER IDAMAX + DOUBLE PRECISION DLAMCH, DNRM2 + EXTERNAL DISNAN, DLAMCH, IDAMAX, DNRM2 +* .. +* .. Executable Statements .. +* +* Initialize INFO +* + INFO = 0 +* +* MINMNFACT in the smallest dimension of the submatrix +* A(IOFFSET+1:M,1:N) to be factorized. +* + MINMNFACT = MIN( M-IOFFSET, N ) + MINMNUPDT = MIN( M-IOFFSET, N+NRHS ) + NB = MIN( NB, MINMNFACT ) + TOL3Z = SQRT( DLAMCH( 'Epsilon' ) ) + HUGEVAL = DLAMCH( 'Overflow' ) +* +* Compute factorization in a while loop over NB columns, +* K is the column index in the block A(1:M,1:N). +* + K = 0 + LSTICC = 0 + DONE = .FALSE. +* + DO WHILE ( K.LT.NB .AND. LSTICC.EQ.0 ) + K = K + 1 + I = IOFFSET + K +* + IF( I.EQ.1 ) THEN +* +* We are at the first column of the original whole matrix A_orig, +* therefore we use the computed KP1 and MAXC2NRM from the +* main routine. +* + KP = KP1 +* + ELSE +* +* Determine the pivot column in K-th step, i.e. the index +* of the column with the maximum 2-norm in the +* submatrix A(I:M,K:N). +* + KP = ( K-1 ) + IDAMAX( N-K+1, VN1( K ), 1 ) +* +* Determine the maximum column 2-norm and the relative maximum +* column 2-norm of the submatrix A(I:M,K:N) in step K. +* + MAXC2NRMK = VN1( KP ) +* +* ============================================================ +* +* Check if the submatrix A(I:M,K:N) contains NaN, set +* INFO parameter to the column number, where the first NaN +* is found and return from the routine. +* We need to check the condition only if the +* column index (same as row index) of the original whole +* matrix is larger than 1, since the condition for whole +* original matrix is checked in the main routine. +* + IF( DISNAN( MAXC2NRMK ) ) THEN +* + DONE = .TRUE. +* +* Set KB, the number of factorized partial columns +* that are non-zero in each step in the block, +* i.e. the rank of the factor R. +* Set IF, the number of processed rows in the block, which +* is the same as the number of processed rows in +* the original whole matrix A_orig. +* + KB = K - 1 + IF = I - 1 + INFO = KB + KP +* +* Set RELMAXC2NRMK to NaN. +* + RELMAXC2NRMK = MAXC2NRMK +* +* There is no need to apply the block reflector to the +* residual of the matrix A stored in A(KB+1:M,KB+1:N), +* since the submatrix contains NaN and we stop +* the computation. +* But, we need to apply the block reflector to the residual +* right hand sides stored in A(KB+1:M,N+1:N+NRHS), if the +* residual right hand sides exist. This occurs +* when ( NRHS != 0 AND KB <= (M-IOFFSET) ): +* +* A(I+1:M,N+1:N+NRHS) := A(I+1:M,N+1:N+NRHS) - +* A(I+1:M,1:KB) * F(N+1:N+NRHS,1:KB)**T. + + IF( NRHS.GT.0 .AND. KB.LT.(M-IOFFSET) ) THEN + CALL DGEMM( 'No transpose', 'Transpose', + $ M-IF, NRHS, KB, -ONE, A( IF+1, 1 ), LDA, + $ F( N+1, 1 ), LDF, ONE, A( IF+1, N+1 ), LDA ) + END IF +* +* There is no need to recompute the 2-norm of the +* difficult columns, since we stop the factorization. +* +* Array TAU(KF+1:MINMNFACT) is not set and contains +* undefined elements. +* +* Return from the routine. +* + RETURN + END IF +* +* Quick return, if the submatrix A(I:M,K:N) is +* a zero matrix. We need to check it only if the column index +* (same as row index) is larger than 1, since the condition +* for the whole original matrix A_orig is checked in the main +* routine. +* + IF( MAXC2NRMK.EQ.ZERO ) THEN +* + DONE = .TRUE. +* +* Set KB, the number of factorized partial columns +* that are non-zero in each step in the block, +* i.e. the rank of the factor R. +* Set IF, the number of processed rows in the block, which +* is the same as the number of processed rows in +* the original whole matrix A_orig. +* + KB = K - 1 + IF = I - 1 + RELMAXC2NRMK = ZERO +* +* There is no need to apply the block reflector to the +* residual of the matrix A stored in A(KB+1:M,KB+1:N), +* since the submatrix is zero and we stop the computation. +* But, we need to apply the block reflector to the residual +* right hand sides stored in A(KB+1:M,N+1:N+NRHS), if the +* residual right hand sides exist. This occurs +* when ( NRHS != 0 AND KB <= (M-IOFFSET) ): +* +* A(I+1:M,N+1:N+NRHS) := A(I+1:M,N+1:N+NRHS) - +* A(I+1:M,1:KB) * F(N+1:N+NRHS,1:KB)**T. +* + IF( NRHS.GT.0 .AND. KB.LT.(M-IOFFSET) ) THEN + CALL DGEMM( 'No transpose', 'Transpose', + $ M-IF, NRHS, KB, -ONE, A( IF+1, 1 ), LDA, + $ F( N+1, 1 ), LDF, ONE, A( IF+1, N+1 ), LDA ) + END IF +* +* There is no need to recompute the 2-norm of the +* difficult columns, since we stop the factorization. +* +* Set TAUs corresponding to the columns that were not +* factorized to ZERO, i.e. set TAU(KB+1:MINMNFACT) = ZERO, +* which is equivalent to seting TAU(K:MINMNFACT) = ZERO. +* + DO J = K, MINMNFACT + TAU( J ) = ZERO + END DO +* +* Return from the routine. +* + RETURN +* + END IF +* +* ============================================================ +* +* Check if the submatrix A(I:M,K:N) contains Inf, +* set INFO parameter to the column number, where +* the first Inf is found plus N, and continue +* the computation. +* We need to check the condition only if the +* column index (same as row index) of the original whole +* matrix is larger than 1, since the condition for whole +* original matrix is checked in the main routine. +* + IF( INFO.EQ.0 .AND. MAXC2NRMK.GT.HUGEVAL ) THEN + INFO = N + K - 1 + KP + END IF +* +* ============================================================ +* +* Test for the second and third tolerance stopping criteria. +* NOTE: There is no need to test for ABSTOL.GE.ZERO, since +* MAXC2NRMK is non-negative. Similarly, there is no need +* to test for RELTOL.GE.ZERO, since RELMAXC2NRMK is +* non-negative. +* We need to check the condition only if the +* column index (same as row index) of the original whole +* matrix is larger than 1, since the condition for whole +* original matrix is checked in the main routine. +* + RELMAXC2NRMK = MAXC2NRMK / MAXC2NRM +* + IF( MAXC2NRMK.LE.ABSTOL .OR. RELMAXC2NRMK.LE.RELTOL ) THEN +* + DONE = .TRUE. +* +* Set KB, the number of factorized partial columns +* that are non-zero in each step in the block, +* i.e. the rank of the factor R. +* Set IF, the number of processed rows in the block, which +* is the same as the number of processed rows in +* the original whole matrix A_orig; +* + KB = K - 1 + IF = I - 1 +* +* Apply the block reflector to the residual of the +* matrix A and the residual of the right hand sides B, if +* the residual matrix and and/or the residual of the right +* hand sides exist, i.e. if the submatrix +* A(I+1:M,KB+1:N+NRHS) exists. This occurs when +* KB < MINMNUPDT = min( M-IOFFSET, N+NRHS ): +* +* A(IF+1:M,K+1:N+NRHS) := A(IF+1:M,KB+1:N+NRHS) - +* A(IF+1:M,1:KB) * F(KB+1:N+NRHS,1:KB)**T. +* + IF( KB.LT.MINMNUPDT ) THEN + CALL DGEMM( 'No transpose', 'Transpose', + $ M-IF, N+NRHS-KB, KB,-ONE, A( IF+1, 1 ), LDA, + $ F( KB+1, 1 ), LDF, ONE, A( IF+1, KB+1 ), LDA ) + END IF +* +* There is no need to recompute the 2-norm of the +* difficult columns, since we stop the factorization. +* +* Set TAUs corresponding to the columns that were not +* factorized to ZERO, i.e. set TAU(KB+1:MINMNFACT) = ZERO, +* which is equivalent to seting TAU(K:MINMNFACT) = ZERO. +* + DO J = K, MINMNFACT + TAU( J ) = ZERO + END DO +* +* Return from the routine. +* + RETURN +* + END IF +* +* ============================================================ +* +* End ELSE of IF(I.EQ.1) +* + END IF +* +* =============================================================== +* +* If the pivot column is not the first column of the +* subblock A(1:M,K:N): +* 1) swap the K-th column and the KP-th pivot column +* in A(1:M,1:N); +* 2) swap the K-th row and the KP-th row in F(1:N,1:K-1) +* 3) copy the K-th element into the KP-th element of the partial +* and exact 2-norm vectors VN1 and VN2. (Swap is not needed +* for VN1 and VN2 since we use the element with the index +* larger than K in the next loop step.) +* 4) Save the pivot interchange with the indices relative to the +* the original matrix A_orig, not the block A(1:M,1:N). +* + IF( KP.NE.K ) THEN + CALL DSWAP( M, A( 1, KP ), 1, A( 1, K ), 1 ) + CALL DSWAP( K-1, F( KP, 1 ), LDF, F( K, 1 ), LDF ) + VN1( KP ) = VN1( K ) + VN2( KP ) = VN2( K ) + ITEMP = JPIV( KP ) + JPIV( KP ) = JPIV( K ) + JPIV( K ) = ITEMP + END IF +* +* Apply previous Householder reflectors to column K: +* A(I:M,K) := A(I:M,K) - A(I:M,1:K-1)*F(K,1:K-1)**T. +* + IF( K.GT.1 ) THEN + CALL DGEMV( 'No transpose', M-I+1, K-1, -ONE, A( I, 1 ), + $ LDA, F( K, 1 ), LDF, ONE, A( I, K ), 1 ) + END IF +* +* Generate elementary reflector H(k) using the column A(I:M,K). +* + IF( I.LT.M ) THEN + CALL DLARFG( M-I+1, A( I, K ), A( I+1, K ), 1, TAU( K ) ) + ELSE + TAU( K ) = ZERO + END IF +* +* Check if TAU(K) contains NaN, set INFO parameter +* to the column number where NaN is found and return from +* the routine. +* NOTE: There is no need to check TAU(K) for Inf, +* since DLARFG cannot produce TAU(K) or Householder vector +* below the diagonal containing Inf. Only BETA on the diagonal, +* returned by DLARFG can contain Inf, which requires +* TAU(K) to contain NaN. Therefore, this case of generating Inf +* by DLARFG is covered by checking TAU(K) for NaN. +* + IF( DISNAN( TAU(K) ) ) THEN +* + DONE = .TRUE. +* +* Set KB, the number of factorized partial columns +* that are non-zero in each step in the block, +* i.e. the rank of the factor R. +* Set IF, the number of processed rows in the block, which +* is the same as the number of processed rows in +* the original whole matrix A_orig. +* + KB = K - 1 + IF = I - 1 + INFO = K +* +* Set MAXC2NRMK and RELMAXC2NRMK to NaN. +* + MAXC2NRMK = TAU( K ) + RELMAXC2NRMK = TAU( K ) +* +* There is no need to apply the block reflector to the +* residual of the matrix A stored in A(KB+1:M,KB+1:N), +* since the submatrix contains NaN and we stop +* the computation. +* But, we need to apply the block reflector to the residual +* right hand sides stored in A(KB+1:M,N+1:N+NRHS), if the +* residual right hand sides exist. This occurs +* when ( NRHS != 0 AND KB <= (M-IOFFSET) ): +* +* A(I+1:M,N+1:N+NRHS) := A(I+1:M,N+1:N+NRHS) - +* A(I+1:M,1:KB) * F(N+1:N+NRHS,1:KB)**T. +* + IF( NRHS.GT.0 .AND. KB.LT.(M-IOFFSET) ) THEN + CALL DGEMM( 'No transpose', 'Transpose', + $ M-IF, NRHS, KB, -ONE, A( IF+1, 1 ), LDA, + $ F( N+1, 1 ), LDF, ONE, A( IF+1, N+1 ), LDA ) + END IF +* +* There is no need to recompute the 2-norm of the +* difficult columns, since we stop the factorization. +* +* Array TAU(KF+1:MINMNFACT) is not set and contains +* undefined elements. +* +* Return from the routine. +* + RETURN + END IF +* +* =============================================================== +* + AIK = A( I, K ) + A( I, K ) = ONE +* +* =============================================================== +* +* Compute the current K-th column of F: +* 1) F(K+1:N,K) := tau(K) * A(I:M,K+1:N)**T * A(I:M,K). +* + IF( K.LT.N+NRHS ) THEN + CALL DGEMV( 'Transpose', M-I+1, N+NRHS-K, + $ TAU( K ), A( I, K+1 ), LDA, A( I, K ), 1, + $ ZERO, F( K+1, K ), 1 ) + END IF +* +* 2) Zero out elements above and on the diagonal of the +* column K in matrix F, i.e elements F(1:K,K). +* + DO J = 1, K + F( J, K ) = ZERO + END DO +* +* 3) Incremental updating of the K-th column of F: +* F(1:N,K) := F(1:N,K) - tau(K) * F(1:N,1:K-1) * A(I:M,1:K-1)**T +* * A(I:M,K). +* + IF( K.GT.1 ) THEN + CALL DGEMV( 'Transpose', M-I+1, K-1, -TAU( K ), + $ A( I, 1 ), LDA, A( I, K ), 1, ZERO, + $ AUXV( 1 ), 1 ) +* + CALL DGEMV( 'No transpose', N+NRHS, K-1, ONE, + $ F( 1, 1 ), LDF, AUXV( 1 ), 1, ONE, + $ F( 1, K ), 1 ) + END IF +* +* =============================================================== +* +* Update the current I-th row of A: +* A(I,K+1:N+NRHS) := A(I,K+1:N+NRHS) +* - A(I,1:K)*F(K+1:N+NRHS,1:K)**T. +* + IF( K.LT.N+NRHS ) THEN + CALL DGEMV( 'No transpose', N+NRHS-K, K, -ONE, + $ F( K+1, 1 ), LDF, A( I, 1 ), LDA, ONE, + $ A( I, K+1 ), LDA ) + END IF +* + A( I, K ) = AIK +* +* Update the partial column 2-norms for the residual matrix, +* only if the residual matrix A(I+1:M,K+1:N) exists, i.e. +* when K < MINMNFACT = min( M-IOFFSET, N ). +* + IF( K.LT.MINMNFACT ) THEN +* + DO J = K + 1, N + IF( VN1( J ).NE.ZERO ) THEN +* +* NOTE: The following lines follow from the analysis in +* Lapack Working Note 176. +* + TEMP = ABS( A( I, J ) ) / VN1( J ) + TEMP = MAX( ZERO, ( ONE+TEMP )*( ONE-TEMP ) ) + TEMP2 = TEMP*( VN1( J ) / VN2( J ) )**2 + IF( TEMP2.LE.TOL3Z ) THEN +* +* At J-index, we have a difficult column for the +* update of the 2-norm. Save the index of the previous +* difficult column in IWORK(J-1). +* NOTE: ILSTCC > 1, threfore we can use IWORK only +* with N-1 elements, where the elements are +* shifted by 1 to the left. +* + IWORK( J-1 ) = LSTICC +* +* Set the index of the last difficult column LSTICC. +* + LSTICC = J +* + ELSE + VN1( J ) = VN1( J )*SQRT( TEMP ) + END IF + END IF + END DO +* + END IF +* +* End of while loop. +* + END DO +* +* Now, afler the loop: +* Set KB, the number of factorized columns in the block; +* Set IF, the number of processed rows in the block, which +* is the same as the number of processed rows in +* the original whole matrix A_orig, IF = IOFFSET + KB. +* + KB = K + IF = I +* +* Apply the block reflector to the residual of the matrix A +* and the residual of the right hand sides B, if the residual +* matrix and and/or the residual of the right hand sides +* exist, i.e. if the submatrix A(I+1:M,KB+1:N+NRHS) exists. +* This occurs when KB < MINMNUPDT = min( M-IOFFSET, N+NRHS ): +* +* A(IF+1:M,K+1:N+NRHS) := A(IF+1:M,KB+1:N+NRHS) - +* A(IF+1:M,1:KB) * F(KB+1:N+NRHS,1:KB)**T. +* + IF( KB.LT.MINMNUPDT ) THEN + CALL DGEMM( 'No transpose', 'Transpose', + $ M-IF, N+NRHS-KB, KB, -ONE, A( IF+1, 1 ), LDA, + $ F( KB+1, 1 ), LDF, ONE, A( IF+1, KB+1 ), LDA ) + END IF +* +* Recompute the 2-norm of the difficult columns. +* Loop over the index of the difficult columns from the largest +* to the smallest index. +* + DO WHILE( LSTICC.GT.0 ) +* +* LSTICC is the index of the last difficult column is greater +* than 1. +* ITEMP is the index of the previous difficult column. +* + ITEMP = IWORK( LSTICC-1 ) +* +* Compute the 2-norm explicilty for the last difficult column and +* save it in the partial and exact 2-norm vectors VN1 and VN2. +* +* NOTE: The computation of VN1( LSTICC ) relies on the fact that +* DNRM2 does not fail on vectors with norm below the value of +* SQRT(DLAMCH('S')) +* + VN1( LSTICC ) = DNRM2( M-IF, A( IF+1, LSTICC ), 1 ) + VN2( LSTICC ) = VN1( LSTICC ) +* +* Downdate the index of the last difficult column to +* the index of the previous difficult column. +* + LSTICC = ITEMP +* + END DO +* + RETURN +* +* End of DLAQP3RK +* + END diff --git a/SRC/ilaenv.f b/SRC/ilaenv.f index a03d0abe97..e74a2b35ec 100644 --- a/SRC/ilaenv.f +++ b/SRC/ilaenv.f @@ -355,6 +355,12 @@ INTEGER FUNCTION ILAENV( ISPEC, NAME, OPTS, N1, N2, N3, N4 ) ELSE NB = 64 END IF + ELSE IF( SUBNAM( 4: 7 ).EQ.'QP3RK' ) THEN + IF( SNAME ) THEN + NB = 32 + ELSE + NB = 32 + END IF END IF ELSE IF( C2.EQ.'PO' ) THEN IF( C3.EQ.'TRF' ) THEN @@ -541,7 +547,14 @@ INTEGER FUNCTION ILAENV( ISPEC, NAME, OPTS, N1, N2, N3, N4 ) ELSE NBMIN = 2 END IF + ELSE IF( SUBNAM( 4: 7 ).EQ.'QP3RK' ) THEN + IF( SNAME ) THEN + NBMIN = 2 + ELSE + NBMIN = 2 + END IF END IF + ELSE IF( C2.EQ.'SY' ) THEN IF( C3.EQ.'TRF' ) THEN IF( SNAME ) THEN @@ -618,6 +631,12 @@ INTEGER FUNCTION ILAENV( ISPEC, NAME, OPTS, N1, N2, N3, N4 ) ELSE NX = 128 END IF + ELSE IF( SUBNAM( 4: 7 ).EQ.'QP3RK' ) THEN + IF( SNAME ) THEN + NX = 128 + ELSE + NX = 128 + END IF END IF ELSE IF( C2.EQ.'SY' ) THEN IF( SNAME .AND. C3.EQ.'TRD' ) THEN diff --git a/SRC/sgeqp3rk.f b/SRC/sgeqp3rk.f new file mode 100755 index 0000000000..17559c7f44 --- /dev/null +++ b/SRC/sgeqp3rk.f @@ -0,0 +1,1081 @@ +*> \brief \b SGEQP3RK computes a truncated Householder QR factorization with column pivoting of a real m-by-n matrix A by using Level 3 BLAS and overwrites a real m-by-nrhs matrix B with Q**T * B. +* +* =========== DOCUMENTATION =========== +* +* Online html documentation available at +* http://www.netlib.org/lapack/explore-html/ +* +*> \htmlonly +*> Download SGEQP3RK + dependencies +*> +*> [TGZ] +*> +*> [ZIP] +*> +*> [TXT] +*> \endhtmlonly +* +* Definition: +* =========== +* +* SUBROUTINE SGEQP3RK( M, N, NRHS, KMAX, ABSTOL, RELTOL, A, LDA, +* $ K, MAXC2NRMK, RELMAXC2NRMK, JPIV, TAU, +* $ WORK, LWORK, IWORK, INFO ) +* IMPLICIT NONE +* +* .. Scalar Arguments .. +* INTEGER INFO, K, KMAX, LDA, LWORK, M, N, NRHS +* REAL ABSTOL, MAXC2NRMK, RELMAXC2NRMK, RELTOL +* .. +* .. Array Arguments .. +* INTEGER IWORK( * ), JPIV( * ) +* REAL A( LDA, * ), TAU( * ), WORK( * ) +* .. +* +* +*> \par Purpose: +* ============= +*> +*> \verbatim +*> +*> SGEQP3RK performs two tasks simultaneously: +*> +*> Task 1: The routine computes a truncated (rank K) or full rank +*> Householder QR factorization with column pivoting of a real +*> M-by-N matrix A using Level 3 BLAS. K is the number of columns +*> that were factorized, i.e. factorization rank of the +*> factor R, K <= min(M,N). +*> +*> A * P(K) = Q(K) * R(K) = +*> +*> = Q(K) * ( R11(K) R12(K) ) = Q(K) * ( R(K)_approx ) +*> ( 0 R22(K) ) ( 0 R(K)_residual ), +*> +*> where: +*> +*> P(K) is an N-by-N permutation matrix; +*> Q(K) is an M-by-M orthogonal matrix; +*> R(K)_approx = ( R11(K), R12(K) ) is a rank K approximation of the +*> full rank factor R with K-by-K upper-triangular +*> R11(K) and K-by-N rectangular R12(K). The diagonal +*> entries of R11(K) appear in non-increasing order +*> of absolute value, and absolute values of all of +*> them exceed the maximum column 2-norm of R22(K) +*> up to roundoff error. +*> R(K)_residual = R22(K) is the residual of a rank K approximation +*> of the full rank factor R. It is a +*> an (M-K)-by-(N-K) rectangular matrix; +*> 0 is a an (M-K)-by-K zero matrix. +*> +*> Task 2: At the same time, the routine overwrites a real M-by-NRHS +*> matrix B with Q(K)**T * B using Level 3 BLAS. +*> +*> ===================================================================== +*> +*> The matrices A and B are stored on input in the array A as +*> the left and right blocks A(1:M,1:N) and A(1:M, N+1:N+NRHS) +*> respectively. +*> +*> N NRHS +*> array_A = M [ mat_A, mat_B ] +*> +*> The truncation criteria (i.e. when to stop the factorization) +*> can be any of the following: +*> +*> 1) The input parameter KMAX, the maximum number of columns +*> KMAX to factorize, i.e. the factorization rank is limited +*> to KMAX. If KMAX >= min(M,N), the criterion is not used. +*> +*> 2) The input parameter ABSTOL, the absolute tolerance for +*> the maximum column 2-norm of the residual matrix R22(K). This +*> means that the factorization stops if this norm is less or +*> equal to ABSTOL. If ABSTOL < 0.0, the criterion is not used. +*> +*> 3) The input parameter RELTOL, the tolerance for the maximum +*> column 2-norm matrix of the residual matrix R22(K) divided +*> by the maximum column 2-norm of the original matrix A, which +*> is equal to abs(R(1,1)). This means that the factorization stops +*> when the ratio of the maximum column 2-norm of R22(K) to +*> the maximum column 2-norm of A is less than or equal to RELTOL. +*> If RELTOL < 0.0, the criterion is not used. +*> +*> 4) In case both stopping criteria ABSTOL or RELTOL are not used, +*> and when the residual matrix R22(K) is a zero matrix in some +*> factorization step K. ( This stopping criterion is implicit. ) +*> +*> The algorithm stops when any of these conditions is first +*> satisfied, otherwise the whole matrix A is factorized. +*> +*> To factorize the whole matrix A, use the values +*> KMAX >= min(M,N), ABSTOL < 0.0 and RELTOL < 0.0. +*> +*> The routine returns: +*> a) Q(K), R(K)_approx = ( R11(K), R12(K) ), +*> R(K)_residual = R22(K), P(K), i.e. the resulting matrices +*> of the factorization; P(K) is represented by JPIV, +*> ( if K = min(M,N), R(K)_approx is the full factor R, +*> and there is no residual matrix R(K)_residual); +*> b) K, the number of columns that were factorized, +*> i.e. factorization rank; +*> c) MAXC2NRMK, the maximum column 2-norm of the residual +*> matrix R(K)_residual = R22(K), +*> ( if K = min(M,N), MAXC2NRMK = 0.0 ); +*> d) RELMAXC2NRMK equals MAXC2NRMK divided by MAXC2NRM, the maximum +*> column 2-norm of the original matrix A, which is equal +*> to abs(R(1,1)), ( if K = min(M,N), RELMAXC2NRMK = 0.0 ); +*> e) Q(K)**T * B, the matrix B with the orthogonal +*> transformation Q(K)**T applied on the left. +*> +*> The N-by-N permutation matrix P(K) is stored in a compact form in +*> the integer array JPIV. For 1 <= j <= N, column j +*> of the matrix A was interchanged with column JPIV(j). +*> +*> The M-by-M orthogonal matrix Q is represented as a product +*> of elementary Householder reflectors +*> +*> Q(K) = H(1) * H(2) * . . . * H(K), +*> +*> where K is the number of columns that were factorized. +*> +*> Each H(j) has the form +*> +*> H(j) = I - tau * v * v**T, +*> +*> where 1 <= j <= K and +*> I is an M-by-M identity matrix, +*> tau is a real scalar, +*> v is a real vector with v(1:j-1) = 0 and v(j) = 1. +*> +*> v(j+1:M) is stored on exit in A(j+1:M,j) and tau in TAU(j). +*> +*> See the Further Details section for more information. +*> \endverbatim +* +* Arguments: +* ========== +* +*> \param[in] M +*> \verbatim +*> M is INTEGER +*> The number of rows of the matrix A. M >= 0. +*> \endverbatim +*> +*> \param[in] N +*> \verbatim +*> N is INTEGER +*> The number of columns of the matrix A. N >= 0. +*> \endverbatim +*> +*> \param[in] NRHS +*> \verbatim +*> NRHS is INTEGER +*> The number of right hand sides, i.e. the number of +*> columns of the matrix B. NRHS >= 0. +*> \endverbatim +*> +*> \param[in] KMAX +*> \verbatim +*> KMAX is INTEGER +*> +*> The first factorization stopping criterion. KMAX >= 0. +*> +*> The maximum number of columns of the matrix A to factorize, +*> i.e. the maximum factorization rank. +*> +*> a) If KMAX >= min(M,N), then this stopping criterion +*> is not used, the routine factorizes columns +*> depending on ABSTOL and RELTOL. +*> +*> b) If KMAX = 0, then this stopping criterion is +*> satisfied on input and the routine exits immediately. +*> This means that the factorization is not performed, +*> the matrices A and B are not modified, and +*> the matrix A is itself the residual. +*> \endverbatim +*> +*> \param[in] ABSTOL +*> \verbatim +*> ABSTOL is REAL +*> +*> The second factorization stopping criterion, cannot be NaN. +*> +*> The absolute tolerance (stopping threshold) for +*> maximum column 2-norm of the residual matrix R22(K). +*> The algorithm converges (stops the factorization) when +*> the maximum column 2-norm of the residual matrix R22(K) +*> is less than or equal to ABSTOL. Let SAFMIN = SLAMCH('S'). +*> +*> a) If ABSTOL is NaN, then no computation is performed +*> and an error message ( INFO = -5 ) is issued +*> by XERBLA. +*> +*> b) If ABSTOL < 0.0, then this stopping criterion is not +*> used, the routine factorizes columns depending +*> on KMAX and RELTOL. +*> This includes the case ABSTOL = -Inf. +*> +*> c) If 0.0 <= ABSTOL < 2*SAFMIN, then ABSTOL = 2*SAFMIN +*> is used. This includes the case ABSTOL = -0.0. +*> +*> d) If 2*SAFMIN <= ABSTOL then the input value +*> of ABSTOL is used. +*> +*> Let MAXC2NRM be the maximum column 2-norm of the +*> whole original matrix A. +*> If ABSTOL chosen above is >= MAXC2NRM, then this +*> stopping criterion is satisfied on input and routine exits +*> immediately after MAXC2NRM is computed. The routine +*> returns MAXC2NRM in MAXC2NORMK, +*> and 1.0 in RELMAXC2NORMK. +*> This includes the case ABSTOL = +Inf. This means that the +*> factorization is not performed, the matrices A and B are not +*> modified, and the matrix A is itself the residual. +*> \endverbatim +*> +*> \param[in] RELTOL +*> \verbatim +*> RELTOL is REAL +*> +*> The third factorization stopping criterion, cannot be NaN. +*> +*> The tolerance (stopping threshold) for the ratio +*> abs(R(K+1,K+1))/abs(R(1,1)) of the maximum column 2-norm of +*> the residual matrix R22(K) to the maximum column 2-norm of +*> the original matrix A. The algorithm converges (stops the +*> factorization), when abs(R(K+1,K+1))/abs(R(1,1)) A is less +*> than or equal to RELTOL. Let EPS = SLAMCH('E'). +*> +*> a) If RELTOL is NaN, then no computation is performed +*> and an error message ( INFO = -6 ) is issued +*> by XERBLA. +*> +*> b) If RELTOL < 0.0, then this stopping criterion is not +*> used, the routine factorizes columns depending +*> on KMAX and ABSTOL. +*> This includes the case RELTOL = -Inf. +*> +*> c) If 0.0 <= RELTOL < EPS, then RELTOL = EPS is used. +*> This includes the case RELTOL = -0.0. +*> +*> d) If EPS <= RELTOL then the input value of RELTOL +*> is used. +*> +*> Let MAXC2NRM be the maximum column 2-norm of the +*> whole original matrix A. +*> If RELTOL chosen above is >= 1.0, then this stopping +*> criterion is satisfied on input and routine exits +*> immediately after MAXC2NRM is computed. +*> The routine returns MAXC2NRM in MAXC2NORMK, +*> and 1.0 in RELMAXC2NORMK. +*> This includes the case RELTOL = +Inf. This means that the +*> factorization is not performed, the matrices A and B are not +*> modified, and the matrix A is itself the residual. +*> +*> NOTE: We recommend that RELTOL satisfy +*> min( max(M,N)*EPS, sqrt(EPS) ) <= RELTOL +*> \endverbatim +*> +*> \param[in,out] A +*> \verbatim +*> A is REAL array, dimension (LDA,N+NRHS) +*> +*> On entry: +*> +*> a) The subarray A(1:M,1:N) contains the M-by-N matrix A. +*> b) The subarray A(1:M,N+1:N+NRHS) contains the M-by-NRHS +*> matrix B. +*> +*> N NRHS +*> array_A = M [ mat_A, mat_B ] +*> +*> On exit: +*> +*> a) The subarray A(1:M,1:N) contains parts of the factors +*> of the matrix A: +*> +*> 1) If K = 0, A(1:M,1:N) contains the original matrix A. +*> 2) If K > 0, A(1:M,1:N) contains parts of the +*> factors: +*> +*> 1. The elements below the diagonal of the subarray +*> A(1:M,1:K) together with TAU(1:K) represent the +*> orthogonal matrix Q(K) as a product of K Householder +*> elementary reflectors. +*> +*> 2. The elements on and above the diagonal of +*> the subarray A(1:K,1:N) contain K-by-N +*> upper-trapezoidal matrix +*> R(K)_approx = ( R11(K), R12(K) ). +*> NOTE: If K=min(M,N), i.e. full rank factorization, +*> then R_approx(K) is the full factor R which +*> is upper-trapezoidal. If, in addition, M>=N, +*> then R is upper-triangular. +*> +*> 3. The subarray A(K+1:M,K+1:N) contains (M-K)-by-(N-K) +*> rectangular matrix R(K)_residual = R22(K). +*> +*> b) If NRHS > 0, the subarray A(1:M,N+1:N+NRHS) contains +*> the M-by-NRHS product Q(K)**T * B. +*> \endverbatim +*> +*> \param[in] LDA +*> \verbatim +*> LDA is INTEGER +*> The leading dimension of the array A. LDA >= max(1,M). +*> This is the leading dimension for both matrices, A and B. +*> \endverbatim +*> +*> \param[out] K +*> \verbatim +*> K is INTEGER +*> Factorization rank of the matrix A, i.e. the rank of +*> the factor R, which is the same as the number of non-zero +*> rows of the factor R. 0 <= K <= min(M,KMAX,N). +*> +*> K also represents the number of non-zero Householder +*> vectors. +*> +*> NOTE: If K = 0, a) the arrays A and B are not modified; +*> b) the array TAU(1:min(M,N)) is set to ZERO, +*> if the matrix A does not contain NaN, +*> otherwise the elements TAU(1:min(M,N)) +*> are undefined; +*> c) the elements of the array JPIV are set +*> as follows: for j = 1:N, JPIV(j) = j. +*> \endverbatim +*> +*> \param[out] MAXC2NRMK +*> \verbatim +*> MAXC2NRMK is REAL +*> The maximum column 2-norm of the residual matrix R22(K), +*> when the factorization stopped at rank K. MAXC2NRMK >= 0. +*> +*> a) If K = 0, i.e. the factorization was not performed, +*> the matrix A was not modified and is itself a residual +*> matrix, then MAXC2NRMK equals the maximum column 2-norm +*> of the original matrix A. +*> +*> b) If 0 < K < min(M,N), then MAXC2NRMK is returned. +*> +*> c) If K = min(M,N), i.e. the whole matrix A was +*> factorized and there is no residual matrix, +*> then MAXC2NRMK = 0.0. +*> +*> NOTE: MAXC2NRMK in the factorization step K would equal +*> R(K+1,K+1) in the next factorization step K+1. +*> \endverbatim +*> +*> \param[out] RELMAXC2NRMK +*> \verbatim +*> RELMAXC2NRMK is REAL +*> The ratio MAXC2NRMK / MAXC2NRM of the maximum column +*> 2-norm of the residual matrix R22(K) (when the factorization +*> stopped at rank K) to the maximum column 2-norm of the +*> whole original matrix A. RELMAXC2NRMK >= 0. +*> +*> a) If K = 0, i.e. the factorization was not performed, +*> the matrix A was not modified and is itself a residual +*> matrix, then RELMAXC2NRMK = 1.0. +*> +*> b) If 0 < K < min(M,N), then +*> RELMAXC2NRMK = MAXC2NRMK / MAXC2NRM is returned. +*> +*> c) If K = min(M,N), i.e. the whole matrix A was +*> factorized and there is no residual matrix, +*> then RELMAXC2NRMK = 0.0. +*> +*> NOTE: RELMAXC2NRMK in the factorization step K would equal +*> abs(R(K+1,K+1))/abs(R(1,1)) in the next factorization +*> step K+1. +*> \endverbatim +*> +*> \param[out] JPIV +*> \verbatim +*> JPIV is INTEGER array, dimension (N) +*> Column pivot indices. For 1 <= j <= N, column j +*> of the matrix A was interchanged with column JPIV(j). +*> +*> The elements of the array JPIV(1:N) are always set +*> by the routine, for example, even when no columns +*> were factorized, i.e. when K = 0, the elements are +*> set as JPIV(j) = j for j = 1:N. +*> \endverbatim +*> +*> \param[out] TAU +*> \verbatim +*> TAU is REAL array, dimension (min(M,N)) +*> The scalar factors of the elementary reflectors. +*> +*> If 0 < K <= min(M,N), only the elements TAU(1:K) of +*> the array TAU are modified by the factorization. +*> After the factorization computed, if no NaN was found +*> during the factorization, the remaining elements +*> TAU(K+1:min(M,N)) are set to zero, otherwise the +*> elements TAU(K+1:min(M,N)) are not set and therefore +*> undefined. +*> ( If K = 0, all elements of TAU are set to zero, if +*> the matrix A does not contain NaN. ) +*> \endverbatim +*> +*> \param[out] WORK +*> \verbatim +*> WORK is REAL array, dimension (MAX(1,LWORK)) +*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. +*> \endverbatim +*> +*> \param[in] LWORK +*> \verbatim +*> LWORK is INTEGER +*> The dimension of the array WORK. +*. LWORK >= (3*N + NRHS - 1) +*> For optimal performance LWORK >= (2*N + NB*( N+NRHS+1 )), +*> where NB is the optimal block size for SGEQP3RK returned +*> by ILAENV. Minimal block size MINNB=2. +*> +*> NOTE: The decision, whether to use unblocked BLAS 2 +*> or blocked BLAS 3 code is based not only on the dimension +*> LWORK of the availbale workspace WORK, but also also on the +*> matrix A dimension N via crossover point NX returned +*> by ILAENV. (For N less than NX, unblocked code should be +*> used.) +*> +*> If LWORK = -1, then a workspace query is assumed; +*> the routine only calculates the optimal size of the WORK +*> array, returns this value as the first entry of the WORK +*> array, and no error message related to LWORK is issued +*> by XERBLA. +*> \endverbatim +*> +*> \param[out] IWORK +*> \verbatim +*> IWORK is INTEGER array, dimension (N-1). +*> Is a work array. ( IWORK is used to store indices +*> of "bad" columns for norm downdating in the residual +*> matrix in the blocked step auxiliary subroutine SLAQP3RK ). +*> \endverbatim +*> +*> \param[out] INFO +*> \verbatim +*> INFO is INTEGER +*> 1) INFO = 0: successful exit. +*> 2) INFO < 0: if INFO = -i, the i-th argument had an +*> illegal value. +*> 3) If INFO = j_1, where 1 <= j_1 <= N, then NaN was +*> detected and the routine stops the computation. +*> The j_1-th column of the matrix A or the j_1-th +*> element of array TAU contains the first occurrence +*> of NaN in the factorization step K+1 ( when K columns +*> have been factorized ). +*> +*> On exit: +*> K is set to the number of +*> factorized columns without +*> exception. +*> MAXC2NRMK is set to NaN. +*> RELMAXC2NRMK is set to NaN. +*> TAU(K+1:min(M,N)) is not set and contains undefined +*> elements. If j_1=K+1, TAU(K+1) +*> may contain NaN. +*> 4) If INFO = j_2, where N+1 <= j_2 <= 2*N, then no NaN +*> was detected, but +Inf (or -Inf) was detected and +*> the routine continues the computation until completion. +*> The (j_2-N)-th column of the matrix A contains the first +*> occurrence of +Inf (or -Inf) in the factorization +*> step K+1 ( when K columns have been factorized ). +*> \endverbatim +* +* Authors: +* ======== +* +*> \author Univ. of Tennessee +*> \author Univ. of California Berkeley +*> \author Univ. of Colorado Denver +*> \author NAG Ltd. +* +*> \ingroup geqp3rk +* +*> \par Further Details: +* ===================== +* +*> \verbatim +*> SGEQP3RK is based on the same BLAS3 Householder QR factorization +*> algorithm with column pivoting as in SGEQP3 routine which uses +*> SLARFG routine to generate Householder reflectors +*> for QR factorization. +*> +*> We can also write: +*> +*> A = A_approx(K) + A_residual(K) +*> +*> The low rank approximation matrix A(K)_approx from +*> the truncated QR factorization of rank K of the matrix A is: +*> +*> A(K)_approx = Q(K) * ( R(K)_approx ) * P(K)**T +*> ( 0 0 ) +*> +*> = Q(K) * ( R11(K) R12(K) ) * P(K)**T +*> ( 0 0 ) +*> +*> The residual A_residual(K) of the matrix A is: +*> +*> A_residual(K) = Q(K) * ( 0 0 ) * P(K)**T = +*> ( 0 R(K)_residual ) +*> +*> = Q(K) * ( 0 0 ) * P(K)**T +*> ( 0 R22(K) ) +*> +*> The truncated (rank K) factorization guarantees that +*> the maximum column 2-norm of A_residual(K) is less than +*> or equal to MAXC2NRMK up to roundoff error. +*> +*> NOTE: An approximation of the null vectors +*> of A can be easily computed from R11(K) +*> and R12(K): +*> +*> Null( A(K) )_approx = P * ( inv(R11(K)) * R12(K) ) +*> ( -I ) +*> +*> \endverbatim +* +*> \par References: +* ================ +*> [1] A Level 3 BLAS QR factorization algorithm with column pivoting developed in 1996. +*> G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain. +*> X. Sun, Computer Science Dept., Duke University, USA. +*> C. H. Bischof, Math. and Comp. Sci. Div., Argonne National Lab, USA. +*> A BLAS-3 version of the QR factorization with column pivoting. +*> LAPACK Working Note 114 +*> \htmlonly +*> https://www.netlib.org/lapack/lawnspdf/lawn114.pdf +*> \endhtmlonly +*> and in +*> SIAM J. Sci. Comput., 19(5):1486-1494, Sept. 1998. +*> \htmlonly +*> https://doi.org/10.1137/S1064827595296732 +*> \endhtmlonly +*> +*> [2] A partial column norm updating strategy developed in 2006. +*> Z. Drmac and Z. Bujanovic, Dept. of Math., University of Zagreb, Croatia. +*> On the failure of rank revealing QR factorization software – a case study. +*> LAPACK Working Note 176. +*> \htmlonly +*> http://www.netlib.org/lapack/lawnspdf/lawn176.pdf +*> \endhtmlonly +*> and in +*> ACM Trans. Math. Softw. 35, 2, Article 12 (July 2008), 28 pages. +*> \htmlonly +*> https://doi.org/10.1145/1377612.1377616 +*> \endhtmlonly +* +*> \par Contributors: +* ================== +*> +*> \verbatim +*> +*> November 2023, Igor Kozachenko, James Demmel, +*> Computer Science Division, +*> University of California, Berkeley +*> +*> \endverbatim +* +* ===================================================================== + SUBROUTINE SGEQP3RK( M, N, NRHS, KMAX, ABSTOL, RELTOL, A, LDA, + $ K, MAXC2NRMK, RELMAXC2NRMK, JPIV, TAU, + $ WORK, LWORK, IWORK, INFO ) + IMPLICIT NONE +* +* -- LAPACK computational routine -- +* -- LAPACK is a software package provided by Univ. of Tennessee, -- +* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- +* +* .. Scalar Arguments .. + INTEGER INFO, K, KF, KMAX, LDA, LWORK, M, N, NRHS + REAL ABSTOL, MAXC2NRMK, RELMAXC2NRMK, RELTOL +* .. +* .. Array Arguments .. + INTEGER IWORK( * ), JPIV( * ) + REAL A( LDA, * ), TAU( * ), WORK( * ) +* .. +* +* ===================================================================== +* +* .. Parameters .. + INTEGER INB, INBMIN, IXOVER + PARAMETER ( INB = 1, INBMIN = 2, IXOVER = 3 ) + REAL ZERO, ONE, TWO + PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0, TWO = 2.0E+0 ) +* .. +* .. Local Scalars .. + LOGICAL LQUERY, DONE + INTEGER IINFO, IOFFSET, IWS, J, JB, JBF, JMAXB, JMAX, + $ JMAXC2NRM, KP1, LWKOPT, MINMN, N_SUB, NB, + $ NBMIN, NX + REAL EPS, HUGEVAL, MAXC2NRM, SAFMIN +* .. +* .. External Subroutines .. + EXTERNAL SLAQP2RK, SLAQP3RK, XERBLA +* .. +* .. External Functions .. + LOGICAL SISNAN + INTEGER ISAMAX, ILAENV + REAL SLAMCH, SNRM2 + EXTERNAL SISNAN, SLAMCH, SNRM2, ISAMAX, ILAENV +* .. +* .. Intrinsic Functions .. + INTRINSIC REAL, MAX, MIN +* .. +* .. Executable Statements .. +* +* Test input arguments +* ==================== +* + INFO = 0 + LQUERY = ( LWORK.EQ.-1 ) + IF( M.LT.0 ) THEN + INFO = -1 + ELSE IF( N.LT.0 ) THEN + INFO = -2 + ELSE IF( NRHS.LT.0 ) THEN + INFO = -3 + ELSE IF( KMAX.LT.0 ) THEN + INFO = -4 + ELSE IF( SISNAN( ABSTOL ) ) THEN + INFO = -5 + ELSE IF( SISNAN( RELTOL ) ) THEN + INFO = -6 + ELSE IF( LDA.LT.MAX( 1, M ) ) THEN + INFO = -8 + END IF +* +* If the input parameters M, N, NRHS, KMAX, LDA are valid: +* a) Test the input workspace size LWORK for the minimum +* size requirement IWS. +* b) Determine the optimal block size NB and optimal +* workspace size LWKOPT to be returned in WORK(1) +* in case of (1) LWORK < IWS, (2) LQUERY = .TRUE., +* (3) when routine exits. +* Here, IWS is the miminum workspace required for unblocked +* code. +* + IF( INFO.EQ.0 ) THEN + MINMN = MIN( M, N ) + IF( MINMN.EQ.0 ) THEN + IWS = 1 + LWKOPT = 1 + ELSE +* +* Minimal workspace size in case of using only unblocked +* BLAS 2 code in SLAQP2RK. +* 1) SGEQP3RK and SLAQP2RK: 2*N to store full and partial +* column 2-norms. +* 2) SLAQP2RK: N+NRHS-1 to use in WORK array that is used +* in SLARF subroutine inside SLAQP2RK to apply an +* elementary reflector from the left. +* TOTAL_WORK_SIZE = 3*N + NRHS - 1 +* + IWS = 3*N + NRHS - 1 +* +* Assign to NB optimal block size. +* + NB = ILAENV( INB, 'SGEQP3RK', ' ', M, N, -1, -1 ) +* +* A formula for the optimal workspace size in case of using +* both unblocked BLAS 2 in SLAQP2RK and blocked BLAS 3 code +* in SLAQP3RK. +* 1) SGEQP3RK, SLAQP2RK, SLAQP3RK: 2*N to store full and +* partial column 2-norms. +* 2) SLAQP2RK: N+NRHS-1 to use in WORK array that is used +* in SLARF subroutine to apply an elementary reflector +* from the left. +* 3) SLAQP3RK: NB*(N+NRHS) to use in the work array F that +* is used to apply a block reflector from +* the left. +* 4) SLAQP3RK: NB to use in the auxilixary array AUX. +* Sizes (2) and ((3) + (4)) should intersect, therefore +* TOTAL_WORK_SIZE = 2*N + NB*( N+NRHS+1 ), given NBMIN=2. +* + LWKOPT = 2*N + NB*( N+NRHS+1 ) + END IF + WORK( 1 ) = REAL( LWKOPT ) +* + IF( ( LWORK.LT.IWS ) .AND. .NOT.LQUERY ) THEN + INFO = -15 + END IF + END IF +* +* NOTE: The optimal workspace size is returned in WORK(1), if +* the input parameters M, N, NRHS, KMAX, LDA are valid. +* + IF( INFO.NE.0 ) THEN + CALL XERBLA( 'SGEQP3RK', -INFO ) + RETURN + ELSE IF( LQUERY ) THEN + RETURN + END IF +* +* Quick return if possible for M=0 or N=0. +* + IF( MINMN.EQ.0 ) THEN + K = 0 + MAXC2NRMK = ZERO + RELMAXC2NRMK = ZERO + WORK( 1 ) = REAL( LWKOPT ) + RETURN + END IF +* +* ================================================================== +* +* Initialize column pivot array JPIV. +* + DO J = 1, N + JPIV( J ) = J + END DO +* +* ================================================================== +* +* Initialize storage for partial and exact column 2-norms. +* a) The elements WORK(1:N) are used to store partial column +* 2-norms of the matrix A, and may decrease in each computation +* step; initialize to the values of complete columns 2-norms. +* b) The elements WORK(N+1:2*N) are used to store complete column +* 2-norms of the matrix A, they are not changed during the +* computation; initialize the values of complete columns 2-norms. +* + DO J = 1, N + WORK( J ) = SNRM2( M, A( 1, J ), 1 ) + WORK( N+J ) = WORK( J ) + END DO +* +* ================================================================== +* +* Compute the pivot column index and the maximum column 2-norm +* for the whole original matrix stored in A(1:M,1:N). +* + KP1 = ISAMAX( N, WORK( 1 ), 1 ) + MAXC2NRM = WORK( KP1 ) +* +* ==================================================================. +* + IF( SISNAN( MAXC2NRM ) ) THEN +* +* Check if the matrix A contains NaN, set INFO parameter +* to the column number where the first NaN is found and return +* from the routine. +* + K = 0 + INFO = KP1 +* +* Set MAXC2NRMK and RELMAXC2NRMK to NaN. +* + MAXC2NRMK = MAXC2NRM + RELMAXC2NRMK = MAXC2NRM +* +* Array TAU is not set and contains undefined elements. +* + WORK( 1 ) = REAL( LWKOPT ) + RETURN + END IF +* +* =================================================================== +* + IF( MAXC2NRM.EQ.ZERO ) THEN +* +* Check is the matrix A is a zero matrix, set array TAU and +* return from the routine. +* + K = 0 + MAXC2NRMK = ZERO + RELMAXC2NRMK = ZERO +* + DO J = 1, MINMN + TAU( J ) = ZERO + END DO +* + WORK( 1 ) = REAL( LWKOPT ) + RETURN +* + END IF +* +* =================================================================== +* + HUGEVAL = SLAMCH( 'Overflow' ) +* + IF( MAXC2NRM.GT.HUGEVAL ) THEN +* +* Check if the matrix A contains +Inf or -Inf, set INFO parameter +* to the column number, where the first +/-Inf is found plus N, +* and continue the computation. +* + INFO = N + KP1 +* + END IF +* +* ================================================================== +* +* Quick return if possible for the case when the first +* stopping criterion is satisfied, i.e. KMAX = 0. +* + IF( KMAX.EQ.0 ) THEN + K = 0 + MAXC2NRMK = MAXC2NRM + RELMAXC2NRMK = ONE + DO J = 1, MINMN + TAU( J ) = ZERO + END DO + WORK( 1 ) = REAL( LWKOPT ) + RETURN + END IF +* +* ================================================================== +* + EPS = SLAMCH('Epsilon') +* +* Adjust ABSTOL +* + IF( ABSTOL.GE.ZERO ) THEN + SAFMIN = SLAMCH('Safe minimum') + ABSTOL = MAX( ABSTOL, TWO*SAFMIN ) + END IF +* +* Adjust RELTOL +* + IF( RELTOL.GE.ZERO ) THEN + RELTOL = MAX( RELTOL, EPS ) + END IF +* +* =================================================================== +* +* JMAX is the maximum index of the column to be factorized, +* which is also limited by the first stopping criterion KMAX. +* + JMAX = MIN( KMAX, MINMN ) +* +* =================================================================== +* +* Quick return if possible for the case when the second or third +* stopping criterion for the whole original matrix is satified, +* i.e. MAXC2NRM <= ABSTOL or RELMAXC2NRM <= RELTOL +* (which is ONE <= RELTOL). +* + IF( MAXC2NRM.LE.ABSTOL .OR. ONE.LE.RELTOL ) THEN +* + K = 0 + MAXC2NRMK = MAXC2NRM + RELMAXC2NRMK = ONE +* + DO J = 1, MINMN + TAU( J ) = ZERO + END DO +* + WORK( 1 ) = REAL( LWKOPT ) + RETURN + END IF +* +* ================================================================== +* Factorize columns +* ================================================================== +* +* Determine the block size. +* + NBMIN = 2 + NX = 0 +* + IF( ( NB.GT.1 ) .AND. ( NB.LT.MINMN ) ) THEN +* +* Determine when to cross over from blocked to unblocked code. +* (for N less than NX, unblocked code should be used). +* + NX = MAX( 0, ILAENV( IXOVER, 'SGEQP3RK', ' ', M, N, -1, -1 )) +* + IF( NX.LT.MINMN ) THEN +* +* Determine if workspace is large enough for blocked code. +* + IF( LWORK.LT.LWKOPT ) THEN +* +* Not enough workspace to use optimal block size that +* is currently stored in NB. +* Reduce NB and determine the minimum value of NB. +* + NB = ( LWORK-2*N ) / ( N+1 ) + NBMIN = MAX( 2, ILAENV( INBMIN, 'SGEQP3RK', ' ', M, N, + $ -1, -1 ) ) +* + END IF + END IF + END IF +* +* ================================================================== +* +* DONE is the boolean flag to rerpresent the case when the +* factorization completed in the block factorization routine, +* before the end of the block. +* + DONE = .FALSE. +* +* J is the column index. +* + J = 1 +* +* (1) Use blocked code initially. +* +* JMAXB is the maximum column index of the block, when the +* blocked code is used, is also limited by the first stopping +* criterion KMAX. +* + JMAXB = MIN( KMAX, MINMN - NX ) +* + IF( NB.GE.NBMIN .AND. NB.LT.JMAX .AND. JMAXB.GT.0 ) THEN +* +* Loop over the column blocks of the matrix A(1:M,1:JMAXB). Here: +* J is the column index of a column block; +* JB is the column block size to pass to block factorization +* routine in a loop step; +* JBF is the number of columns that were actually factorized +* that was returned by the block factorization routine +* in a loop step, JBF <= JB; +* N_SUB is the number of columns in the submatrix; +* IOFFSET is the number of rows that should not be factorized. +* + DO WHILE( J.LE.JMAXB ) +* + JB = MIN( NB, JMAXB-J+1 ) + N_SUB = N-J+1 + IOFFSET = J-1 +* +* Factorize JB columns among the columns A(J:N). +* + CALL SLAQP3RK( M, N_SUB, NRHS, IOFFSET, JB, ABSTOL, + $ RELTOL, KP1, MAXC2NRM, A( 1, J ), LDA, + $ DONE, JBF, MAXC2NRMK, RELMAXC2NRMK, + $ JPIV( J ), TAU( J ), + $ WORK( J ), WORK( N+J ), + $ WORK( 2*N+1 ), WORK( 2*N+JB+1 ), + $ N+NRHS-J+1, IWORK, IINFO ) +* +* Set INFO on the first occurence of Inf. +* + IF( IINFO.GT.N_SUB .AND. INFO.EQ.0 ) THEN + INFO = 2*IOFFSET + IINFO + END IF +* + IF( DONE ) THEN +* +* Either the submatrix is zero before the end of the +* column block, or ABSTOL or RELTOL criterion is +* satisfied before the end of the column block, we can +* return from the routine. Perform the following before +* returning: +* a) Set the number of factorized columns K, +* K = IOFFSET + JBF from the last call of blocked +* routine. +* NOTE: 1) MAXC2NRMK and RELMAXC2NRMK are returned +* by the block factorization routine; +* 2) The remaining TAUs are set to ZERO by the +* block factorization routine. +* + K = IOFFSET + JBF +* +* Set INFO on the first occurrence of NaN, NaN takes +* prcedence over Inf. +* + IF( IINFO.LE.N_SUB .AND. IINFO.GT.0 ) THEN + INFO = IOFFSET + IINFO + END IF +* +* Return from the routine. +* + WORK( 1 ) = REAL( LWKOPT ) +* + RETURN +* + END IF +* + J = J + JBF +* + END DO +* + END IF +* +* Use unblocked code to factor the last or only block. +* J = JMAX+1 means we factorized the maximum possible number of +* columns, that is in ELSE clause we need to compute +* the MAXC2NORM and RELMAXC2NORM to return after we processed +* the blocks. +* + IF( J.LE.JMAX ) THEN +* +* N_SUB is the number of columns in the submatrix; +* IOFFSET is the number of rows that should not be factorized. +* + N_SUB = N-J+1 + IOFFSET = J-1 +* + CALL SLAQP2RK( M, N_SUB, NRHS, IOFFSET, JMAX-J+1, + $ ABSTOL, RELTOL, KP1, MAXC2NRM, A( 1, J ), LDA, + $ KF, MAXC2NRMK, RELMAXC2NRMK, JPIV( J ), + $ TAU( J ), WORK( J ), WORK( N+J ), + $ WORK( 2*N+1 ), IINFO ) +* +* ABSTOL or RELTOL criterion is satisfied when the number of +* the factorized columns KF is smaller then the number +* of columns JMAX-J+1 supplied to be factorized by the +* unblocked routine, we can return from +* the routine. Perform the following before returning: +* a) Set the number of factorized columns K, +* b) MAXC2NRMK and RELMAXC2NRMK are returned by the +* unblocked factorization routine above. +* + K = J - 1 + KF +* +* Set INFO on the first exception occurence. +* +* Set INFO on the first exception occurence of Inf or NaN, +* (NaN takes precedence over Inf). +* + IF( IINFO.GT.N_SUB .AND. INFO.EQ.0 ) THEN + INFO = 2*IOFFSET + IINFO + ELSE IF( IINFO.LE.N_SUB .AND. IINFO.GT.0 ) THEN + INFO = IOFFSET + IINFO + END IF +* + ELSE +* +* Compute the return values for blocked code. +* +* Set the number of factorized columns if the unblocked routine +* was not called. +* + K = JMAX +* +* If there exits a residual matrix after the blocked code: +* 1) compute the values of MAXC2NRMK, RELMAXC2NRMK of the +* residual matrix, otherwise set them to ZERO; +* 2) Set TAU(K+1:MINMN) to ZERO. +* + IF( K.LT.MINMN ) THEN + JMAXC2NRM = K + ISAMAX( N-K, WORK( K+1 ), 1 ) + MAXC2NRMK = WORK( JMAXC2NRM ) + IF( K.EQ.0 ) THEN + RELMAXC2NRMK = ONE + ELSE + RELMAXC2NRMK = MAXC2NRMK / MAXC2NRM + END IF +* + DO J = K + 1, MINMN + TAU( J ) = ZERO + END DO +* + END IF +* +* END IF( J.LE.JMAX ) THEN +* + END IF +* + WORK( 1 ) = REAL( LWKOPT ) +* + RETURN +* +* End of SGEQP3RK +* + END diff --git a/SRC/slaqp2rk.f b/SRC/slaqp2rk.f new file mode 100755 index 0000000000..d3dbb3d7c1 --- /dev/null +++ b/SRC/slaqp2rk.f @@ -0,0 +1,713 @@ +*> \brief \b SLAQP2RK computes truncated QR factorization with column pivoting of a real matrix block using Level 2 BLAS and overwrites a real m-by-nrhs matrix B with Q**T * B. +* +* =========== DOCUMENTATION =========== +* +* Online html documentation available at +* http://www.netlib.org/lapack/explore-html/ +* +*> \htmlonly +*> Download SLAQP2RK + dependencies +*> +*> [TGZ] +*> +*> [ZIP] +*> +*> [TXT] +*> \endhtmlonly +* +* Definition: +* =========== +* +* SUBROUTINE SLAQP2RK( M, N, NRHS, IOFFSET, KMAX, ABSTOL, RELTOL, +* $ KP1, MAXC2NRM, A, LDA, K, MAXC2NRMK, +* $ RELMAXC2NRMK, JPIV, TAU, VN1, VN2, WORK, +* $ INFO ) +* IMPLICIT NONE +* +* .. Scalar Arguments .. +* INTEGER INFO, IOFFSET, KP1, K, KMAX, LDA, M, N, NRHS +* REAL ABSTOL, MAXC2NRM, MAXC2NRMK, RELMAXC2NRMK, +* $ RELTOL +* .. +* .. Array Arguments .. +* INTEGER JPIV( * ) +* REAL A( LDA, * ), TAU( * ), VN1( * ), VN2( * ), +* $ WORK( * ) +* .. +* +* +*> \par Purpose: +* ============= +*> +*> \verbatim +*> +*> SLAQP2RK computes a truncated (rank K) or full rank Householder QR +*> factorization with column pivoting of a real matrix +*> block A(IOFFSET+1:M,1:N) as +*> +*> A * P(K) = Q(K) * R(K). +*> +*> The routine uses Level 2 BLAS. The block A(1:IOFFSET,1:N) +*> is accordingly pivoted, but not factorized. +*> +*> The routine also overwrites the right-hand-sides matrix block B +*> stored in A(IOFFSET+1:M,N+1:N+NRHS) with Q(K)**T * B. +*> \endverbatim +* +* Arguments: +* ========== +* +*> \param[in] M +*> \verbatim +*> M is INTEGER +*> The number of rows of the matrix A. M >= 0. +*> \endverbatim +*> +*> \param[in] N +*> \verbatim +*> N is INTEGER +*> The number of columns of the matrix A. N >= 0. +*> \endverbatim +*> +*> \param[in] NRHS +*> \verbatim +*> NRHS is INTEGER +*> The number of right hand sides, i.e., the number of +*> columns of the matrix B. NRHS >= 0. +*> \endverbatim +*> +*> \param[in] IOFFSET +*> \verbatim +*> IOFFSET is INTEGER +*> The number of rows of the matrix A that must be pivoted +*> but not factorized. IOFFSET >= 0. +*> +*> IOFFSET also represents the number of columns of the whole +*> original matrix A_orig that have been factorized +*> in the previous steps. +*> \endverbatim +*> +*> \param[in] KMAX +*> \verbatim +*> KMAX is INTEGER +*> +*> The first factorization stopping criterion. KMAX >= 0. +*> +*> The maximum number of columns of the matrix A to factorize, +*> i.e. the maximum factorization rank. +*> +*> a) If KMAX >= min(M-IOFFSET,N), then this stopping +*> criterion is not used, factorize columns +*> depending on ABSTOL and RELTOL. +*> +*> b) If KMAX = 0, then this stopping criterion is +*> satisfied on input and the routine exits immediately. +*> This means that the factorization is not performed, +*> the matrices A and B and the arrays TAU, IPIV +*> are not modified. +*> \endverbatim +*> +*> \param[in] ABSTOL +*> \verbatim +*> ABSTOL is DOUBLE PRECISION, cannot be NaN. +*> +*> The second factorization stopping criterion. +*> +*> The absolute tolerance (stopping threshold) for +*> maximum column 2-norm of the residual matrix. +*> The algorithm converges (stops the factorization) when +*> the maximum column 2-norm of the residual matrix +*> is less than or equal to ABSTOL. +*> +*> a) If ABSTOL < 0.0, then this stopping criterion is not +*> used, the routine factorizes columns depending +*> on KMAX and RELTOL. +*> This includes the case ABSTOL = -Inf. +*> +*> b) If 0.0 <= ABSTOL then the input value +*> of ABSTOL is used. +*> \endverbatim +*> +*> \param[in] RELTOL +*> \verbatim +*> RELTOL is DOUBLE PRECISION, cannot be NaN. +*> +*> The third factorization stopping criterion. +*> +*> The tolerance (stopping threshold) for the ratio of the +*> maximum column 2-norm of the residual matrix to the maximum +*> column 2-norm of the original matrix A_orig. The algorithm +*> converges (stops the factorization), when this ratio is +*> less than or equal to RELTOL. +*> +*> a) If RELTOL < 0.0, then this stopping criterion is not +*> used, the routine factorizes columns depending +*> on KMAX and ABSTOL. +*> This includes the case RELTOL = -Inf. +*> +*> d) If 0.0 <= RELTOL then the input value of RELTOL +*> is used. +*> \endverbatim +*> +*> \param[in] KP1 +*> \verbatim +*> KP1 is INTEGER +*> The index of the column with the maximum 2-norm in +*> the whole original matrix A_orig determined in the +*> main routine SGEQP3RK. 1 <= KP1 <= N_orig_mat. +*> \endverbatim +*> +*> \param[in] MAXC2NRM +*> \verbatim +*> MAXC2NRM is DOUBLE PRECISION +*> The maximum column 2-norm of the whole original +*> matrix A_orig computed in the main routine SGEQP3RK. +*> MAXC2NRM >= 0. +*> \endverbatim +*> +*> \param[in,out] A +*> \verbatim +*> A is REAL array, dimension (LDA,N+NRHS) +*> On entry: +*> the M-by-N matrix A and M-by-NRHS matrix B, as in +*> +*> N NRHS +*> array_A = M [ mat_A, mat_B ] +*> +*> On exit: +*> 1. The elements in block A(IOFFSET+1:M,1:K) below +*> the diagonal together with the array TAU represent +*> the orthogonal matrix Q(K) as a product of elementary +*> reflectors. +*> 2. The upper triangular block of the matrix A stored +*> in A(IOFFSET+1:M,1:K) is the triangular factor obtained. +*> 3. The block of the matrix A stored in A(1:IOFFSET,1:N) +*> has been accordingly pivoted, but not factorized. +*> 4. The rest of the array A, block A(IOFFSET+1:M,K+1:N+NRHS). +*> The left part A(IOFFSET+1:M,K+1:N) of this block +*> contains the residual of the matrix A, and, +*> if NRHS > 0, the right part of the block +*> A(IOFFSET+1:M,N+1:N+NRHS) contains the block of +*> the right-hand-side matrix B. Both these blocks have been +*> updated by multiplication from the left by Q(K)**T. +*> \endverbatim +*> +*> \param[in] LDA +*> \verbatim +*> LDA is INTEGER +*> The leading dimension of the array A. LDA >= max(1,M). +*> \endverbatim +*> +*> \param[out] K +*> \verbatim +*> K is INTEGER +*> Factorization rank of the matrix A, i.e. the rank of +*> the factor R, which is the same as the number of non-zero +*> rows of the factor R. 0 <= K <= min(M-IOFFSET,KMAX,N). +*> +*> K also represents the number of non-zero Householder +*> vectors. +*> \endverbatim +*> +*> \param[out] MAXC2NRMK +*> \verbatim +*> MAXC2NRMK is DOUBLE PRECISION +*> The maximum column 2-norm of the residual matrix, +*> when the factorization stopped at rank K. MAXC2NRMK >= 0. +*> \endverbatim +*> +*> \param[out] RELMAXC2NRMK +*> \verbatim +*> RELMAXC2NRMK is DOUBLE PRECISION +*> The ratio MAXC2NRMK / MAXC2NRM of the maximum column +*> 2-norm of the residual matrix (when the factorization +*> stopped at rank K) to the maximum column 2-norm of the +*> whole original matrix A. RELMAXC2NRMK >= 0. +*> \endverbatim +*> +*> \param[out] JPIV +*> \verbatim +*> JPIV is INTEGER array, dimension (N) +*> Column pivot indices, for 1 <= j <= N, column j +*> of the matrix A was interchanged with column JPIV(j). +*> \endverbatim +*> +*> \param[out] TAU +*> \verbatim +*> TAU is REAL array, dimension (min(M-IOFFSET,N)) +*> The scalar factors of the elementary reflectors. +*> \endverbatim +*> +*> \param[in,out] VN1 +*> \verbatim +*> VN1 is REAL array, dimension (N) +*> The vector with the partial column norms. +*> \endverbatim +*> +*> \param[in,out] VN2 +*> \verbatim +*> VN2 is REAL array, dimension (N) +*> The vector with the exact column norms. +*> \endverbatim +*> +*> \param[out] WORK +*> \verbatim +*> WORK is REAL array, dimension (N-1) +*> Used in SLARF subroutine to apply an elementary +*> reflector from the left. +*> \endverbatim +*> +*> \param[out] INFO +*> \verbatim +*> INFO is INTEGER +*> 1) INFO = 0: successful exit. +*> 2) If INFO = j_1, where 1 <= j_1 <= N, then NaN was +*> detected and the routine stops the computation. +*> The j_1-th column of the matrix A or the j_1-th +*> element of array TAU contains the first occurrence +*> of NaN in the factorization step K+1 ( when K columns +*> have been factorized ). +*> +*> On exit: +*> K is set to the number of +*> factorized columns without +*> exception. +*> MAXC2NRMK is set to NaN. +*> RELMAXC2NRMK is set to NaN. +*> TAU(K+1:min(M,N)) is not set and contains undefined +*> elements. If j_1=K+1, TAU(K+1) +*> may contain NaN. +*> 3) If INFO = j_2, where N+1 <= j_2 <= 2*N, then no NaN +*> was detected, but +Inf (or -Inf) was detected and +*> the routine continues the computation until completion. +*> The (j_2-N)-th column of the matrix A contains the first +*> occurrence of +Inf (or -Inf) in the factorization +*> step K+1 ( when K columns have been factorized ). +*> \endverbatim +* +* Authors: +* ======== +* +*> \author Univ. of Tennessee +*> \author Univ. of California Berkeley +*> \author Univ. of Colorado Denver +*> \author NAG Ltd. +* +*> \ingroup laqp2rk +* +*> \par References: +* ================ +*> [1] A Level 3 BLAS QR factorization algorithm with column pivoting developed in 1996. +*> G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain. +*> X. Sun, Computer Science Dept., Duke University, USA. +*> C. H. Bischof, Math. and Comp. Sci. Div., Argonne National Lab, USA. +*> A BLAS-3 version of the QR factorization with column pivoting. +*> LAPACK Working Note 114 +*> \htmlonly +*> https://www.netlib.org/lapack/lawnspdf/lawn114.pdf +*> \endhtmlonly +*> and in +*> SIAM J. Sci. Comput., 19(5):1486-1494, Sept. 1998. +*> \htmlonly +*> https://doi.org/10.1137/S1064827595296732 +*> \endhtmlonly +*> +*> [2] A partial column norm updating strategy developed in 2006. +*> Z. Drmac and Z. Bujanovic, Dept. of Math., University of Zagreb, Croatia. +*> On the failure of rank revealing QR factorization software – a case study. +*> LAPACK Working Note 176. +*> \htmlonly +*> http://www.netlib.org/lapack/lawnspdf/lawn176.pdf +*> \endhtmlonly +*> and in +*> ACM Trans. Math. Softw. 35, 2, Article 12 (July 2008), 28 pages. +*> \htmlonly +*> https://doi.org/10.1145/1377612.1377616 +*> \endhtmlonly +* +*> \par Contributors: +* ================== +*> +*> \verbatim +*> +*> November 2023, Igor Kozachenko, James Demmel, +*> Computer Science Division, +*> University of California, Berkeley +*> +*> \endverbatim +* +* ===================================================================== + SUBROUTINE SLAQP2RK( M, N, NRHS, IOFFSET, KMAX, ABSTOL, RELTOL, + $ KP1, MAXC2NRM, A, LDA, K, MAXC2NRMK, + $ RELMAXC2NRMK, JPIV, TAU, VN1, VN2, WORK, + $ INFO ) + IMPLICIT NONE +* +* -- LAPACK auxiliary routine -- +* -- LAPACK is a software package provided by Univ. of Tennessee, -- +* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- +* +* .. Scalar Arguments .. + INTEGER INFO, IOFFSET, KP1, K, KMAX, LDA, M, N, NRHS + REAL ABSTOL, MAXC2NRM, MAXC2NRMK, RELMAXC2NRMK, + $ RELTOL +* .. +* .. Array Arguments .. + INTEGER JPIV( * ) + REAL A( LDA, * ), TAU( * ), VN1( * ), VN2( * ), + $ WORK( * ) +* .. +* +* ===================================================================== +* +* .. Parameters .. + REAL ZERO, ONE + PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) +* .. +* .. Local Scalars .. + INTEGER I, ITEMP, J, JMAXC2NRM, KK, KP, MINMNFACT, + $ MINMNUPDT + REAL AIKK, HUGEVAL, TEMP, TEMP2, TOL3Z +* .. +* .. External Subroutines .. + EXTERNAL SLARF, SLARFG, SSWAP +* .. +* .. Intrinsic Functions .. + INTRINSIC ABS, MAX, MIN, SQRT +* .. +* .. External Functions .. + LOGICAL SISNAN + INTEGER ISAMAX + REAL SLAMCH, SNRM2 + EXTERNAL SISNAN, SLAMCH, ISAMAX, SNRM2 +* .. +* .. Executable Statements .. +* +* Initialize INFO +* + INFO = 0 +* +* MINMNFACT in the smallest dimension of the submatrix +* A(IOFFSET+1:M,1:N) to be factorized. +* +* MINMNUPDT is the smallest dimension +* of the subarray A(IOFFSET+1:M,1:N+NRHS) to be udated, which +* contains the submatrices A(IOFFSET+1:M,1:N) and +* B(IOFFSET+1:M,1:NRHS) as column blocks. +* + MINMNFACT = MIN( M-IOFFSET, N ) + MINMNUPDT = MIN( M-IOFFSET, N+NRHS ) + KMAX = MIN( KMAX, MINMNFACT ) + TOL3Z = SQRT( SLAMCH( 'Epsilon' ) ) + HUGEVAL = SLAMCH( 'Overflow' ) +* +* Compute the factorization, KK is the lomn loop index. +* + DO KK = 1, KMAX +* + I = IOFFSET + KK +* + IF( I.EQ.1 ) THEN +* +* ============================================================ +* +* We are at the first column of the original whole matrix A, +* therefore we use the computed KP1 and MAXC2NRM from the +* main routine. +* + + KP = KP1 +* +* ============================================================ +* + ELSE +* +* ============================================================ +* +* Determine the pivot column in KK-th step, i.e. the index +* of the column with the maximum 2-norm in the +* submatrix A(I:M,K:N). +* + KP = ( KK-1 ) + ISAMAX( N-KK+1, VN1( KK ), 1 ) +* +* Determine the maximum column 2-norm and the relative maximum +* column 2-norm of the submatrix A(I:M,KK:N) in step KK. +* RELMAXC2NRMK will be computed later, after somecondition +* checks on MAXC2NRMK. +* + MAXC2NRMK = VN1( KP ) +* +* ============================================================ +* +* Check if the submatrix A(I:M,KK:N) contains NaN, and set +* INFO parameter to the column number, where the first NaN +* is found and return from the routine. +* We need to check the condition only if the +* column index (same as row index) of the original whole +* matrix is larger than 1, since the condition for whole +* original matrix is checked in the main routine. +* + IF( SISNAN( MAXC2NRMK ) ) THEN +* +* Set K, the number of factorized columns. +* that are not zero. +* + K = KK - 1 + INFO = K + KP +* +* Set RELMAXC2NRMK to NaN. +* + RELMAXC2NRMK = MAXC2NRMK +* +* Array TAU(K+1:MINMNFACT) is not set and contains +* undefined elements. +* + RETURN + END IF +* +* ============================================================ +* +* Quick return, if the submatrix A(I:M,KK:N) is +* a zero matrix. +* We need to check the condition only if the +* column index (same as row index) of the original whole +* matrix is larger than 1, since the condition for whole +* original matrix is checked in the main routine. +* + IF( MAXC2NRMK.EQ.ZERO ) THEN +* +* Set K, the number of factorized columns. +* that are not zero. +* + K = KK - 1 + RELMAXC2NRMK = ZERO +* +* Set TAUs corresponding to the columns that were not +* factorized to ZERO, i.e. set TAU(KK:MINMNFACT) to ZERO. +* + DO J = KK, MINMNFACT + TAU( J ) = ZERO + END DO +* +* Return from the routine. +* + RETURN +* + END IF +* +* ============================================================ +* +* Check if the submatrix A(I:M,KK:N) contains Inf, +* set INFO parameter to the column number, where +* the first Inf is found plus N, and continue +* the computation. +* We need to check the condition only if the +* column index (same as row index) of the original whole +* matrix is larger than 1, since the condition for whole +* original matrix is checked in the main routine. +* + IF( INFO.EQ.0 .AND. MAXC2NRMK.GT.HUGEVAL ) THEN + INFO = N + KK - 1 + KP + END IF +* +* ============================================================ +* +* Test for the second and third stopping criteria. +* NOTE: There is no need to test for ABSTOL >= ZERO, since +* MAXC2NRMK is non-negative. Similarly, there is no need +* to test for RELTOL >= ZERO, since RELMAXC2NRMK is +* non-negative. +* We need to check the condition only if the +* column index (same as row index) of the original whole +* matrix is larger than 1, since the condition for whole +* original matrix is checked in the main routine. + + RELMAXC2NRMK = MAXC2NRMK / MAXC2NRM +* + IF( MAXC2NRMK.LE.ABSTOL .OR. RELMAXC2NRMK.LE.RELTOL ) THEN +* +* Set K, the number of factorized columns. +* + K = KK - 1 +* +* Set TAUs corresponding to the columns that were not +* factorized to ZERO, i.e. set TAU(KK:MINMNFACT) to ZERO. +* + DO J = KK, MINMNFACT + TAU( J ) = ZERO + END DO +* +* Return from the routine. +* + RETURN +* + END IF +* +* ============================================================ +* +* End ELSE of IF(I.EQ.1) +* + END IF +* +* =============================================================== +* +* If the pivot column is not the first column of the +* subblock A(1:M,KK:N): +* 1) swap the KK-th column and the KP-th pivot column +* in A(1:M,1:N); +* 2) copy the KK-th element into the KP-th element of the partial +* and exact 2-norm vectors VN1 and VN2. ( Swap is not needed +* for VN1 and VN2 since we use the element with the index +* larger than KK in the next loop step.) +* 3) Save the pivot interchange with the indices relative to the +* the original matrix A, not the block A(1:M,1:N). +* + IF( KP.NE.KK ) THEN + CALL SSWAP( M, A( 1, KP ), 1, A( 1, KK ), 1 ) + VN1( KP ) = VN1( KK ) + VN2( KP ) = VN2( KK ) + ITEMP = JPIV( KP ) + JPIV( KP ) = JPIV( KK ) + JPIV( KK ) = ITEMP + END IF +* +* Generate elementary reflector H(KK) using the column A(I:M,KK), +* if the column has more than one element, otherwise +* the elementary reflector would be an identity matrix, +* and TAU(KK) = ZERO. +* + IF( I.LT.M ) THEN + CALL SLARFG( M-I+1, A( I, KK ), A( I+1, KK ), 1, + $ TAU( KK ) ) + ELSE + TAU( KK ) = ZERO + END IF +* +* Check if TAU(KK) contains NaN, set INFO parameter +* to the column number where NaN is found and return from +* the routine. +* NOTE: There is no need to check TAU(KK) for Inf, +* since SLARFG cannot produce TAU(KK) or Householder vector +* below the diagonal containing Inf. Only BETA on the diagonal, +* returned by SLARFG can contain Inf, which requires +* TAU(KK) to contain NaN. Therefore, this case of generating Inf +* by SLARFG is covered by checking TAU(KK) for NaN. +* + IF( SISNAN( TAU(KK) ) ) THEN + K = KK - 1 + INFO = KK +* +* Set MAXC2NRMK and RELMAXC2NRMK to NaN. +* + MAXC2NRMK = TAU( KK ) + RELMAXC2NRMK = TAU( KK ) +* +* Array TAU(KK:MINMNFACT) is not set and contains +* undefined elements, except the first element TAU(KK) = NaN. +* + RETURN + END IF +* +* Apply H(KK)**T to A(I:M,KK+1:N+NRHS) from the left. +* ( If M >= N, then at KK = N there is no residual matrix, +* i.e. no columns of A to update, only columns of B. +* If M < N, then at KK = M-IOFFSET, I = M and we have a +* one-row residual matrix in A and the elementary +* reflector is a unit matrix, TAU(KK) = ZERO, i.e. no update +* is needed for the residual matrix in A and the +* right-hand-side-matrix in B. +* Therefore, we update only if +* KK < MINMNUPDT = min(M-IOFFSET, N+NRHS) +* condition is satisfied, not only KK < N+NRHS ) +* + IF( KK.LT.MINMNUPDT ) THEN + AIKK = A( I, KK ) + A( I, KK ) = ONE + CALL SLARF( 'Left', M-I+1, N+NRHS-KK, A( I, KK ), 1, + $ TAU( KK ), A( I, KK+1 ), LDA, WORK( 1 ) ) + A( I, KK ) = AIKK + END IF +* + IF( KK.LT.MINMNFACT ) THEN +* +* Update the partial column 2-norms for the residual matrix, +* only if the residual matrix A(I+1:M,KK+1:N) exists, i.e. +* when KK < min(M-IOFFSET, N). +* + DO J = KK + 1, N + IF( VN1( J ).NE.ZERO ) THEN +* +* NOTE: The following lines follow from the analysis in +* Lapack Working Note 176. +* + TEMP = ONE - ( ABS( A( I, J ) ) / VN1( J ) )**2 + TEMP = MAX( TEMP, ZERO ) + TEMP2 = TEMP*( VN1( J ) / VN2( J ) )**2 + IF( TEMP2 .LE. TOL3Z ) THEN +* +* Compute the column 2-norm for the partial +* column A(I+1:M,J) by explicitly computing it, +* and store it in both partial 2-norm vector VN1 +* and exact column 2-norm vector VN2. +* + VN1( J ) = SNRM2( M-I, A( I+1, J ), 1 ) + VN2( J ) = VN1( J ) +* + ELSE +* +* Update the column 2-norm for the partial +* column A(I+1:M,J) by removing one +* element A(I,J) and store it in partial +* 2-norm vector VN1. +* + VN1( J ) = VN1( J )*SQRT( TEMP ) +* + END IF + END IF + END DO +* + END IF +* +* End factorization loop +* + END DO +* +* If we reached this point, all colunms have been factorized, +* i.e. no condition was triggered to exit the routine. +* Set the number of factorized columns. +* + K = KMAX +* +* We reached the end of the loop, i.e. all KMAX columns were +* factorized, we need to set MAXC2NRMK and RELMAXC2NRMK before +* we return. +* + IF( K.LT.MINMNFACT ) THEN +* + JMAXC2NRM = K + ISAMAX( N-K, VN1( K+1 ), 1 ) + MAXC2NRMK = VN1( JMAXC2NRM ) +* + IF( K.EQ.0 ) THEN + RELMAXC2NRMK = ONE + ELSE + RELMAXC2NRMK = MAXC2NRMK / MAXC2NRM + END IF +* + ELSE + MAXC2NRMK = ZERO + RELMAXC2NRMK = ZERO + END IF +* +* We reached the end of the loop, i.e. all KMAX columns were +* factorized, set TAUs corresponding to the columns that were +* not factorized to ZERO, i.e. TAU(K+1:MINMNFACT) set to ZERO. +* + DO J = K + 1, MINMNFACT + TAU( J ) = ZERO + END DO +* + RETURN +* +* End of SLAQP2RK +* + END diff --git a/SRC/slaqp3rk.f b/SRC/slaqp3rk.f new file mode 100755 index 0000000000..fa735bb9d7 --- /dev/null +++ b/SRC/slaqp3rk.f @@ -0,0 +1,935 @@ +*> \brief \b SLAQP3RK computes a step of truncated QR factorization with column pivoting of a real m-by-n matrix A using Level 3 BLAS and overwrites a real m-by-nrhs matrix B with Q**T * B. +* +* =========== DOCUMENTATION =========== +* +* Online html documentation available at +* http://www.netlib.org/lapack/explore-html/ +* +*> \htmlonly +*> Download SLAQP3RK + dependencies +*> +*> [TGZ] +*> +*> [ZIP] +*> +*> [TXT] +*> \endhtmlonly +* +* Definition: +* =========== +* +* SUBROUTINE SLAQP3RK( M, N, NRHS, IOFFSET, NB, ABSTOL, +* $ RELTOL, KP1, MAXC2NRM, A, LDA, DONE, KB, +* $ MAXC2NRMK, RELMAXC2NRMK, JPIV, TAU, +* $ VN1, VN2, AUXV, F, LDF, IWORK, INFO ) +* IMPLICIT NONE +* LOGICAL DONE +* INTEGER INFO, IOFFSET, KB, KP1, LDA, LDF, M, N, +* $ NB, NRHS +* REAL ABSTOL, MAXC2NRM, MAXC2NRMK, RELMAXC2NRMK, +* $ RELTOL +* +* .. Scalar Arguments .. +* LOGICAL DONE +* INTEGER KB, LDA, LDF, M, N, NB, NRHS, IOFFSET +* REAL ABSTOL, MAXC2NRM, MAXC2NRMK, RELMAXC2NRMK, +* $ RELTOL +* .. +* .. Array Arguments .. +* INTEGER IWORK( * ), JPIV( * ) +* REAL A( LDA, * ), AUXV( * ), F( LDF, * ), TAU( * ), +* $ VN1( * ), VN2( * ) +* .. +* +* +*> \par Purpose: +* ============= +*> +*> \verbatim +*> +*> SLAQP3RK computes a step of truncated QR factorization with column +*> pivoting of a real M-by-N matrix A block A(IOFFSET+1:M,1:N) +*> by using Level 3 BLAS as +*> +*> A * P(KB) = Q(KB) * R(KB). +*> +*> The routine tries to factorize NB columns from A starting from +*> the row IOFFSET+1 and updates the residual matrix with BLAS 3 +*> xGEMM. The number of actually factorized columns is returned +*> is smaller than NB. +*> +*> Block A(1:IOFFSET,1:N) is accordingly pivoted, but not factorized. +*> +*> The routine also overwrites the right-hand-sides B matrix stored +*> in A(IOFFSET+1:M,1:N+1:N+NRHS) with Q(KB)**T * B. +*> +*> Cases when the number of factorized columns KB < NB: +*> +*> (1) In some cases, due to catastrophic cancellations, it cannot +*> factorize all NB columns and need to update the residual matrix. +*> Hence, the actual number of factorized columns in the block returned +*> in KB is smaller than NB. The logical DONE is returned as FALSE. +*> The factorization of the whole original matrix A_orig must proceed +*> with the next block. +*> +*> (2) Whenever the stopping criterion ABSTOL or RELTOL is satisfied, +*> the factorization of the whole original matrix A_orig is stopped, +*> the logical DONE is returned as TRUE. The number of factorized +*> columns which is smaller than NB is returned in KB. +*> +*> (3) In case both stopping criteria ABSTOL or RELTOL are not used, +*> and when the residual matrix is a zero matrix in some factorization +*> step KB, the factorization of the whole original matrix A_orig is +*> stopped, the logical DONE is returned as TRUE. The number of +*> factorized columns which is smaller than NB is returned in KB. +*> +*> (4) Whenever NaN is detected in the matrix A or in the array TAU, +*> the factorization of the whole original matrix A_orig is stopped, +*> the logical DONE is returned as TRUE. The number of factorized +*> columns which is smaller than NB is returned in KB. The INFO +*> parameter is set to the column index of the first NaN occurrence. +*> +*> \endverbatim +* +* Arguments: +* ========== +* +*> \param[in] M +*> \verbatim +*> M is INTEGER +*> The number of rows of the matrix A. M >= 0. +*> \endverbatim +*> +*> \param[in] N +*> \verbatim +*> N is INTEGER +*> The number of columns of the matrix A. N >= 0 +*> \endverbatim +*> +*> \param[in] NRHS +*> \verbatim +*> NRHS is INTEGER +*> The number of right hand sides, i.e., the number of +*> columns of the matrix B. NRHS >= 0. +*> \endverbatim +*> +*> \param[in] IOFFSET +*> \verbatim +*> IOFFSET is INTEGER +*> The number of rows of the matrix A that must be pivoted +*> but not factorized. IOFFSET >= 0. +*> +*> IOFFSET also represents the number of columns of the whole +*> original matrix A_orig that have been factorized +*> in the previous steps. +*> \endverbatim +*> +*> \param[in] NB +*> \verbatim +*> NB is INTEGER +*> Factorization block size, i.e the number of columns +*> to factorize in the matrix A. 0 <= NB +*> +*> If NB = 0, then the routine exits immediately. +*> This means that the factorization is not performed, +*> the matrices A and B and the arrays TAU, IPIV +*> are not modified. +*> \endverbatim +*> +*> \param[in] ABSTOL +*> \verbatim +*> ABSTOL is REAL, cannot be NaN. +*> +*> The absolute tolerance (stopping threshold) for +*> maximum column 2-norm of the residual matrix. +*> The algorithm converges (stops the factorization) when +*> the maximum column 2-norm of the residual matrix +*> is less than or equal to ABSTOL. +*> +*> a) If ABSTOL < 0.0, then this stopping criterion is not +*> used, the routine factorizes columns depending +*> on NB and RELTOL. +*> This includes the case ABSTOL = -Inf. +*> +*> b) If 0.0 <= ABSTOL then the input value +*> of ABSTOL is used. +*> \endverbatim +*> +*> \param[in] RELTOL +*> \verbatim +*> RELTOL is REAL, cannot be NaN. +*> +*> The tolerance (stopping threshold) for the ratio of the +*> maximum column 2-norm of the residual matrix to the maximum +*> column 2-norm of the original matrix A_orig. The algorithm +*> converges (stops the factorization), when this ratio is +*> less than or equal to RELTOL. +*> +*> a) If RELTOL < 0.0, then this stopping criterion is not +*> used, the routine factorizes columns depending +*> on NB and ABSTOL. +*> This includes the case RELTOL = -Inf. +*> +*> d) If 0.0 <= RELTOL then the input value of RELTOL +*> is used. +*> \endverbatim +*> +*> \param[in] KP1 +*> \verbatim +*> KP1 is INTEGER +*> The index of the column with the maximum 2-norm in +*> the whole original matrix A_orig determined in the +*> main routine SGEQP3RK. 1 <= KP1 <= N_orig. +*> \endverbatim +*> +*> \param[in] MAXC2NRM +*> \verbatim +*> MAXC2NRM is REAL +*> The maximum column 2-norm of the whole original +*> matrix A_orig computed in the main routine SGEQP3RK. +*> MAXC2NRM >= 0. +*> \endverbatim +*> +*> \param[in,out] A +*> \verbatim +*> A is REAL array, dimension (LDA,N+NRHS) +*> On entry: +*> the M-by-N matrix A and M-by-NRHS matrix B, as in +*> +*> N NRHS +*> array_A = M [ mat_A, mat_B ] +*> +*> On exit: +*> 1. The elements in block A(IOFFSET+1:M,1:KB) below +*> the diagonal together with the array TAU represent +*> the orthogonal matrix Q(KB) as a product of elementary +*> reflectors. +*> 2. The upper triangular block of the matrix A stored +*> in A(IOFFSET+1:M,1:KB) is the triangular factor obtained. +*> 3. The block of the matrix A stored in A(1:IOFFSET,1:N) +*> has been accordingly pivoted, but not factorized. +*> 4. The rest of the array A, block A(IOFFSET+1:M,KB+1:N+NRHS). +*> The left part A(IOFFSET+1:M,KB+1:N) of this block +*> contains the residual of the matrix A, and, +*> if NRHS > 0, the right part of the block +*> A(IOFFSET+1:M,N+1:N+NRHS) contains the block of +*> the right-hand-side matrix B. Both these blocks have been +*> updated by multiplication from the left by Q(KB)**T. +*> \endverbatim +*> +*> \param[in] LDA +*> \verbatim +*> LDA is INTEGER +*> The leading dimension of the array A. LDA >= max(1,M). +*> \endverbatim +*> +*> \param[out] +*> \verbatim +*> DONE is LOGICAL +*> TRUE: a) if the factorization completed before processing +*> all min(M-IOFFSET,NB,N) columns due to ABSTOL +*> or RELTOL criterion, +*> b) if the factorization completed before processing +*> all min(M-IOFFSET,NB,N) columns due to the +*> residual matrix being a ZERO matrix. +*> c) when NaN was detected in the matrix A +*> or in the array TAU. +*> FALSE: otherwise. +*> \endverbatim +*> +*> \param[out] KB +*> \verbatim +*> KB is INTEGER +*> Factorization rank of the matrix A, i.e. the rank of +*> the factor R, which is the same as the number of non-zero +*> rows of the factor R. 0 <= KB <= min(M-IOFFSET,NB,N). +*> +*> KB also represents the number of non-zero Householder +*> vectors. +*> \endverbatim +*> +*> \param[out] MAXC2NRMK +*> \verbatim +*> MAXC2NRMK is REAL +*> The maximum column 2-norm of the residual matrix, +*> when the factorization stopped at rank KB. MAXC2NRMK >= 0. +*> \endverbatim +*> +*> \param[out] RELMAXC2NRMK +*> \verbatim +*> RELMAXC2NRMK is REAL +*> The ratio MAXC2NRMK / MAXC2NRM of the maximum column +*> 2-norm of the residual matrix (when the factorization +*> stopped at rank KB) to the maximum column 2-norm of the +*> original matrix A_orig. RELMAXC2NRMK >= 0. +*> \endverbatim +*> +*> \param[out] JPIV +*> \verbatim +*> JPIV is INTEGER array, dimension (N) +*> Column pivot indices, for 1 <= j <= N, column j +*> of the matrix A was interchanged with column JPIV(j). +*> \endverbatim +*> +*> \param[out] TAU +*> \verbatim +*> TAU is REAL array, dimension (min(M-IOFFSET,N)) +*> The scalar factors of the elementary reflectors. +*> \endverbatim +*> +*> \param[in,out] VN1 +*> \verbatim +*> VN1 is REAL array, dimension (N) +*> The vector with the partial column norms. +*> \endverbatim +*> +*> \param[in,out] VN2 +*> \verbatim +*> VN2 is REAL array, dimension (N) +*> The vector with the exact column norms. +*> \endverbatim +*> +*> \param[out] AUXV +*> \verbatim +*> AUXV is REAL array, dimension (NB) +*> Auxiliary vector. +*> \endverbatim +*> +*> \param[out] F +*> \verbatim +*> F is REAL array, dimension (LDF,NB) +*> Matrix F**T = L*(Y**T)*A. +*> \endverbatim +*> +*> \param[in] LDF +*> \verbatim +*> LDF is INTEGER +*> The leading dimension of the array F. LDF >= max(1,N+NRHS). +*> \endverbatim +*> +*> \param[out] IWORK +*> \verbatim +*> IWORK is INTEGER array, dimension (N-1). +*> Is a work array. ( IWORK is used to store indices +*> of "bad" columns for norm downdating in the residual +*> matrix ). +*> \endverbatim +*> +*> \param[out] INFO +*> \verbatim +*> INFO is INTEGER +*> 1) INFO = 0: successful exit. +*> 2) If INFO = j_1, where 1 <= j_1 <= N, then NaN was +*> detected and the routine stops the computation. +*> The j_1-th column of the matrix A or the j_1-th +*> element of array TAU contains the first occurrence +*> of NaN in the factorization step KB+1 ( when KB columns +*> have been factorized ). +*> +*> On exit: +*> KB is set to the number of +*> factorized columns without +*> exception. +*> MAXC2NRMK is set to NaN. +*> RELMAXC2NRMK is set to NaN. +*> TAU(KB+1:min(M,N)) is not set and contains undefined +*> elements. If j_1=KB+1, TAU(KB+1) +*> may contain NaN. +*> 3) If INFO = j_2, where N+1 <= j_2 <= 2*N, then no NaN +*> was detected, but +Inf (or -Inf) was detected and +*> the routine continues the computation until completion. +*> The (j_2-N)-th column of the matrix A contains the first +*> occurrence of +Inf (or -Inf) in the actorization +*> step KB+1 ( when KB columns have been factorized ). +*> \endverbatim +* +* Authors: +* ======== +* +*> \author Univ. of Tennessee +*> \author Univ. of California Berkeley +*> \author Univ. of Colorado Denver +*> \author NAG Ltd. +* +*> \ingroup laqp3rk +* +*> \par References: +* ================ +*> [1] A Level 3 BLAS QR factorization algorithm with column pivoting developed in 1996. +*> G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain. +*> X. Sun, Computer Science Dept., Duke University, USA. +*> C. H. Bischof, Math. and Comp. Sci. Div., Argonne National Lab, USA. +*> A BLAS-3 version of the QR factorization with column pivoting. +*> LAPACK Working Note 114 +*> \htmlonly +*> https://www.netlib.org/lapack/lawnspdf/lawn114.pdf +*> \endhtmlonly +*> and in +*> SIAM J. Sci. Comput., 19(5):1486-1494, Sept. 1998. +*> \htmlonly +*> https://doi.org/10.1137/S1064827595296732 +*> \endhtmlonly +*> +*> [2] A partial column norm updating strategy developed in 2006. +*> Z. Drmac and Z. Bujanovic, Dept. of Math., University of Zagreb, Croatia. +*> On the failure of rank revealing QR factorization software – a case study. +*> LAPACK Working Note 176. +*> \htmlonly +*> http://www.netlib.org/lapack/lawnspdf/lawn176.pdf +*> \endhtmlonly +*> and in +*> ACM Trans. Math. Softw. 35, 2, Article 12 (July 2008), 28 pages. +*> \htmlonly +*> https://doi.org/10.1145/1377612.1377616 +*> \endhtmlonly +* +*> \par Contributors: +* ================== +*> +*> \verbatim +*> +*> November 2023, Igor Kozachenko, James Demmel, +*> Computer Science Division, +*> University of California, Berkeley +*> +*> \endverbatim +* +* ===================================================================== + SUBROUTINE SLAQP3RK( M, N, NRHS, IOFFSET, NB, ABSTOL, + $ RELTOL, KP1, MAXC2NRM, A, LDA, DONE, KB, + $ MAXC2NRMK, RELMAXC2NRMK, JPIV, TAU, + $ VN1, VN2, AUXV, F, LDF, IWORK, INFO ) + IMPLICIT NONE +* +* -- LAPACK auxiliary routine -- +* -- LAPACK is a software package provided by Univ. of Tennessee, -- +* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- +* +* .. Scalar Arguments .. + LOGICAL DONE + INTEGER INFO, IOFFSET, KB, KP1, LDA, LDF, M, N, + $ NB, NRHS + REAL ABSTOL, MAXC2NRM, MAXC2NRMK, RELMAXC2NRMK, + $ RELTOL +* .. +* .. Array Arguments .. + INTEGER IWORK( * ), JPIV( * ) + REAL A( LDA, * ), AUXV( * ), F( LDF, * ), TAU( * ), + $ VN1( * ), VN2( * ) +* .. +* +* ===================================================================== +* +* .. Parameters .. + REAL ZERO, ONE + PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) +* .. +* .. Local Scalars .. + INTEGER ITEMP, J, K, MINMNFACT, MINMNUPDT, + $ LSTICC, KP, I, IF + REAL AIK, HUGEVAL, TEMP, TEMP2, TOL3Z +* .. +* .. External Subroutines .. + EXTERNAL SGEMM, SGEMV, SLARFG, SSWAP +* .. +* .. Intrinsic Functions .. + INTRINSIC ABS, MAX, MIN, SQRT +* .. +* .. External Functions .. + LOGICAL SISNAN + INTEGER ISAMAX + REAL SLAMCH, SNRM2 + EXTERNAL SISNAN, SLAMCH, ISAMAX, SNRM2 +* .. +* .. Executable Statements .. +* +* Initialize INFO +* + INFO = 0 +* +* MINMNFACT in the smallest dimension of the submatrix +* A(IOFFSET+1:M,1:N) to be factorized. +* + MINMNFACT = MIN( M-IOFFSET, N ) + MINMNUPDT = MIN( M-IOFFSET, N+NRHS ) + NB = MIN( NB, MINMNFACT ) + TOL3Z = SQRT( SLAMCH( 'Epsilon' ) ) + HUGEVAL = SLAMCH( 'Overflow' ) +* +* Compute factorization in a while loop over NB columns, +* K is the column index in the block A(1:M,1:N). +* + K = 0 + LSTICC = 0 + DONE = .FALSE. +* + DO WHILE ( K.LT.NB .AND. LSTICC.EQ.0 ) + K = K + 1 + I = IOFFSET + K +* + IF( I.EQ.1 ) THEN +* +* We are at the first column of the original whole matrix A_orig, +* therefore we use the computed KP1 and MAXC2NRM from the +* main routine. +* + KP = KP1 +* + ELSE +* +* Determine the pivot column in K-th step, i.e. the index +* of the column with the maximum 2-norm in the +* submatrix A(I:M,K:N). +* + KP = ( K-1 ) + ISAMAX( N-K+1, VN1( K ), 1 ) +* +* Determine the maximum column 2-norm and the relative maximum +* column 2-norm of the submatrix A(I:M,K:N) in step K. +* + MAXC2NRMK = VN1( KP ) +* +* ============================================================ +* +* Check if the submatrix A(I:M,K:N) contains NaN, set +* INFO parameter to the column number, where the first NaN +* is found and return from the routine. +* We need to check the condition only if the +* column index (same as row index) of the original whole +* matrix is larger than 1, since the condition for whole +* original matrix is checked in the main routine. +* + IF( SISNAN( MAXC2NRMK ) ) THEN +* + DONE = .TRUE. +* +* Set KB, the number of factorized partial columns +* that are non-zero in each step in the block, +* i.e. the rank of the factor R. +* Set IF, the number of processed rows in the block, which +* is the same as the number of processed rows in +* the original whole matrix A_orig. +* + KB = K - 1 + IF = I - 1 + INFO = KB + KP +* +* Set RELMAXC2NRMK to NaN. +* + RELMAXC2NRMK = MAXC2NRMK +* +* There is no need to apply the block reflector to the +* residual of the matrix A stored in A(KB+1:M,KB+1:N), +* since the submatrix contains NaN and we stop +* the computation. +* But, we need to apply the block reflector to the residual +* right hand sides stored in A(KB+1:M,N+1:N+NRHS), if the +* residual right hand sides exist. This occurs +* when ( NRHS != 0 AND KB <= (M-IOFFSET) ): +* +* A(I+1:M,N+1:N+NRHS) := A(I+1:M,N+1:N+NRHS) - +* A(I+1:M,1:KB) * F(N+1:N+NRHS,1:KB)**T. + + IF( NRHS.GT.0 .AND. KB.LT.(M-IOFFSET) ) THEN + CALL SGEMM( 'No transpose', 'Transpose', + $ M-IF, NRHS, KB, -ONE, A( IF+1, 1 ), LDA, + $ F( N+1, 1 ), LDF, ONE, A( IF+1, N+1 ), LDA ) + END IF +* +* There is no need to recompute the 2-norm of the +* difficult columns, since we stop the factorization. +* +* Array TAU(KF+1:MINMNFACT) is not set and contains +* undefined elements. +* +* Return from the routine. +* + RETURN + END IF +* +* Quick return, if the submatrix A(I:M,K:N) is +* a zero matrix. We need to check it only if the column index +* (same as row index) is larger than 1, since the condition +* for the whole original matrix A_orig is checked in the main +* routine. +* + IF( MAXC2NRMK.EQ.ZERO ) THEN +* + DONE = .TRUE. +* +* Set KB, the number of factorized partial columns +* that are non-zero in each step in the block, +* i.e. the rank of the factor R. +* Set IF, the number of processed rows in the block, which +* is the same as the number of processed rows in +* the original whole matrix A_orig. +* + KB = K - 1 + IF = I - 1 + RELMAXC2NRMK = ZERO +* +* There is no need to apply the block reflector to the +* residual of the matrix A stored in A(KB+1:M,KB+1:N), +* since the submatrix is zero and we stop the computation. +* But, we need to apply the block reflector to the residual +* right hand sides stored in A(KB+1:M,N+1:N+NRHS), if the +* residual right hand sides exist. This occurs +* when ( NRHS != 0 AND KB <= (M-IOFFSET) ): +* +* A(I+1:M,N+1:N+NRHS) := A(I+1:M,N+1:N+NRHS) - +* A(I+1:M,1:KB) * F(N+1:N+NRHS,1:KB)**T. +* + IF( NRHS.GT.0 .AND. KB.LT.(M-IOFFSET) ) THEN + CALL SGEMM( 'No transpose', 'Transpose', + $ M-IF, NRHS, KB, -ONE, A( IF+1, 1 ), LDA, + $ F( N+1, 1 ), LDF, ONE, A( IF+1, N+1 ), LDA ) + END IF +* +* There is no need to recompute the 2-norm of the +* difficult columns, since we stop the factorization. +* +* Set TAUs corresponding to the columns that were not +* factorized to ZERO, i.e. set TAU(KB+1:MINMNFACT) = ZERO, +* which is equivalent to seting TAU(K:MINMNFACT) = ZERO. +* + DO J = K, MINMNFACT + TAU( J ) = ZERO + END DO +* +* Return from the routine. +* + RETURN +* + END IF +* +* ============================================================ +* +* Check if the submatrix A(I:M,K:N) contains Inf, +* set INFO parameter to the column number, where +* the first Inf is found plus N, and continue +* the computation. +* We need to check the condition only if the +* column index (same as row index) of the original whole +* matrix is larger than 1, since the condition for whole +* original matrix is checked in the main routine. +* + IF( INFO.EQ.0 .AND. MAXC2NRMK.GT.HUGEVAL ) THEN + INFO = N + K - 1 + KP + END IF +* +* ============================================================ +* +* Test for the second and third tolerance stopping criteria. +* NOTE: There is no need to test for ABSTOL.GE.ZERO, since +* MAXC2NRMK is non-negative. Similarly, there is no need +* to test for RELTOL.GE.ZERO, since RELMAXC2NRMK is +* non-negative. +* We need to check the condition only if the +* column index (same as row index) of the original whole +* matrix is larger than 1, since the condition for whole +* original matrix is checked in the main routine. +* + RELMAXC2NRMK = MAXC2NRMK / MAXC2NRM +* + IF( MAXC2NRMK.LE.ABSTOL .OR. RELMAXC2NRMK.LE.RELTOL ) THEN +* + DONE = .TRUE. +* +* Set KB, the number of factorized partial columns +* that are non-zero in each step in the block, +* i.e. the rank of the factor R. +* Set IF, the number of processed rows in the block, which +* is the same as the number of processed rows in +* the original whole matrix A_orig; +* + KB = K - 1 + IF = I - 1 +* +* Apply the block reflector to the residual of the +* matrix A and the residual of the right hand sides B, if +* the residual matrix and and/or the residual of the right +* hand sides exist, i.e. if the submatrix +* A(I+1:M,KB+1:N+NRHS) exists. This occurs when +* KB < MINMNUPDT = min( M-IOFFSET, N+NRHS ): +* +* A(IF+1:M,K+1:N+NRHS) := A(IF+1:M,KB+1:N+NRHS) - +* A(IF+1:M,1:KB) * F(KB+1:N+NRHS,1:KB)**T. +* + IF( KB.LT.MINMNUPDT ) THEN + CALL SGEMM( 'No transpose', 'Transpose', + $ M-IF, N+NRHS-KB, KB,-ONE, A( IF+1, 1 ), LDA, + $ F( KB+1, 1 ), LDF, ONE, A( IF+1, KB+1 ), LDA ) + END IF +* +* There is no need to recompute the 2-norm of the +* difficult columns, since we stop the factorization. +* +* Set TAUs corresponding to the columns that were not +* factorized to ZERO, i.e. set TAU(KB+1:MINMNFACT) = ZERO, +* which is equivalent to seting TAU(K:MINMNFACT) = ZERO. +* + DO J = K, MINMNFACT + TAU( J ) = ZERO + END DO +* +* Return from the routine. +* + RETURN +* + END IF +* +* ============================================================ +* +* End ELSE of IF(I.EQ.1) +* + END IF +* +* =============================================================== +* +* If the pivot column is not the first column of the +* subblock A(1:M,K:N): +* 1) swap the K-th column and the KP-th pivot column +* in A(1:M,1:N); +* 2) swap the K-th row and the KP-th row in F(1:N,1:K-1) +* 3) copy the K-th element into the KP-th element of the partial +* and exact 2-norm vectors VN1 and VN2. (Swap is not needed +* for VN1 and VN2 since we use the element with the index +* larger than K in the next loop step.) +* 4) Save the pivot interchange with the indices relative to the +* the original matrix A_orig, not the block A(1:M,1:N). +* + IF( KP.NE.K ) THEN + CALL SSWAP( M, A( 1, KP ), 1, A( 1, K ), 1 ) + CALL SSWAP( K-1, F( KP, 1 ), LDF, F( K, 1 ), LDF ) + VN1( KP ) = VN1( K ) + VN2( KP ) = VN2( K ) + ITEMP = JPIV( KP ) + JPIV( KP ) = JPIV( K ) + JPIV( K ) = ITEMP + END IF +* +* Apply previous Householder reflectors to column K: +* A(I:M,K) := A(I:M,K) - A(I:M,1:K-1)*F(K,1:K-1)**T. +* + IF( K.GT.1 ) THEN + CALL SGEMV( 'No transpose', M-I+1, K-1, -ONE, A( I, 1 ), + $ LDA, F( K, 1 ), LDF, ONE, A( I, K ), 1 ) + END IF +* +* Generate elementary reflector H(k) using the column A(I:M,K). +* + IF( I.LT.M ) THEN + CALL SLARFG( M-I+1, A( I, K ), A( I+1, K ), 1, TAU( K ) ) + ELSE + TAU( K ) = ZERO + END IF +* +* Check if TAU(K) contains NaN, set INFO parameter +* to the column number where NaN is found and return from +* the routine. +* NOTE: There is no need to check TAU(K) for Inf, +* since SLARFG cannot produce TAU(K) or Householder vector +* below the diagonal containing Inf. Only BETA on the diagonal, +* returned by SLARFG can contain Inf, which requires +* TAU(K) to contain NaN. Therefore, this case of generating Inf +* by SLARFG is covered by checking TAU(K) for NaN. +* + IF( SISNAN( TAU(K) ) ) THEN +* + DONE = .TRUE. +* +* Set KB, the number of factorized partial columns +* that are non-zero in each step in the block, +* i.e. the rank of the factor R. +* Set IF, the number of processed rows in the block, which +* is the same as the number of processed rows in +* the original whole matrix A_orig. +* + KB = K - 1 + IF = I - 1 + INFO = K +* +* Set MAXC2NRMK and RELMAXC2NRMK to NaN. +* + MAXC2NRMK = TAU( K ) + RELMAXC2NRMK = TAU( K ) +* +* There is no need to apply the block reflector to the +* residual of the matrix A stored in A(KB+1:M,KB+1:N), +* since the submatrix contains NaN and we stop +* the computation. +* But, we need to apply the block reflector to the residual +* right hand sides stored in A(KB+1:M,N+1:N+NRHS), if the +* residual right hand sides exist. This occurs +* when ( NRHS != 0 AND KB <= (M-IOFFSET) ): +* +* A(I+1:M,N+1:N+NRHS) := A(I+1:M,N+1:N+NRHS) - +* A(I+1:M,1:KB) * F(N+1:N+NRHS,1:KB)**T. +* + IF( NRHS.GT.0 .AND. KB.LT.(M-IOFFSET) ) THEN + CALL SGEMM( 'No transpose', 'Transpose', + $ M-IF, NRHS, KB, -ONE, A( IF+1, 1 ), LDA, + $ F( N+1, 1 ), LDF, ONE, A( IF+1, N+1 ), LDA ) + END IF +* +* There is no need to recompute the 2-norm of the +* difficult columns, since we stop the factorization. +* +* Array TAU(KF+1:MINMNFACT) is not set and contains +* undefined elements. +* +* Return from the routine. +* + RETURN + END IF +* +* =============================================================== +* + AIK = A( I, K ) + A( I, K ) = ONE +* +* =============================================================== +* +* Compute the current K-th column of F: +* 1) F(K+1:N,K) := tau(K) * A(I:M,K+1:N)**T * A(I:M,K). +* + IF( K.LT.N+NRHS ) THEN + CALL SGEMV( 'Transpose', M-I+1, N+NRHS-K, + $ TAU( K ), A( I, K+1 ), LDA, A( I, K ), 1, + $ ZERO, F( K+1, K ), 1 ) + END IF +* +* 2) Zero out elements above and on the diagonal of the +* column K in matrix F, i.e elements F(1:K,K). +* + DO J = 1, K + F( J, K ) = ZERO + END DO +* +* 3) Incremental updating of the K-th column of F: +* F(1:N,K) := F(1:N,K) - tau(K) * F(1:N,1:K-1) * A(I:M,1:K-1)**T +* * A(I:M,K). +* + IF( K.GT.1 ) THEN + CALL SGEMV( 'Transpose', M-I+1, K-1, -TAU( K ), + $ A( I, 1 ), LDA, A( I, K ), 1, ZERO, + $ AUXV( 1 ), 1 ) +* + CALL SGEMV( 'No transpose', N+NRHS, K-1, ONE, + $ F( 1, 1 ), LDF, AUXV( 1 ), 1, ONE, + $ F( 1, K ), 1 ) + END IF +* +* =============================================================== +* +* Update the current I-th row of A: +* A(I,K+1:N+NRHS) := A(I,K+1:N+NRHS) +* - A(I,1:K)*F(K+1:N+NRHS,1:K)**T. +* + IF( K.LT.N+NRHS ) THEN + CALL SGEMV( 'No transpose', N+NRHS-K, K, -ONE, + $ F( K+1, 1 ), LDF, A( I, 1 ), LDA, ONE, + $ A( I, K+1 ), LDA ) + END IF +* + A( I, K ) = AIK +* +* Update the partial column 2-norms for the residual matrix, +* only if the residual matrix A(I+1:M,K+1:N) exists, i.e. +* when K < MINMNFACT = min( M-IOFFSET, N ). +* + IF( K.LT.MINMNFACT ) THEN +* + DO J = K + 1, N + IF( VN1( J ).NE.ZERO ) THEN +* +* NOTE: The following lines follow from the analysis in +* Lapack Working Note 176. +* + TEMP = ABS( A( I, J ) ) / VN1( J ) + TEMP = MAX( ZERO, ( ONE+TEMP )*( ONE-TEMP ) ) + TEMP2 = TEMP*( VN1( J ) / VN2( J ) )**2 + IF( TEMP2.LE.TOL3Z ) THEN +* +* At J-index, we have a difficult column for the +* update of the 2-norm. Save the index of the previous +* difficult column in IWORK(J-1). +* NOTE: ILSTCC > 1, threfore we can use IWORK only +* with N-1 elements, where the elements are +* shifted by 1 to the left. +* + IWORK( J-1 ) = LSTICC +* +* Set the index of the last difficult column LSTICC. +* + LSTICC = J +* + ELSE + VN1( J ) = VN1( J )*SQRT( TEMP ) + END IF + END IF + END DO +* + END IF +* +* End of while loop. +* + END DO +* +* Now, afler the loop: +* Set KB, the number of factorized columns in the block; +* Set IF, the number of processed rows in the block, which +* is the same as the number of processed rows in +* the original whole matrix A_orig, IF = IOFFSET + KB. +* + KB = K + IF = I +* +* Apply the block reflector to the residual of the matrix A +* and the residual of the right hand sides B, if the residual +* matrix and and/or the residual of the right hand sides +* exist, i.e. if the submatrix A(I+1:M,KB+1:N+NRHS) exists. +* This occurs when KB < MINMNUPDT = min( M-IOFFSET, N+NRHS ): +* +* A(IF+1:M,K+1:N+NRHS) := A(IF+1:M,KB+1:N+NRHS) - +* A(IF+1:M,1:KB) * F(KB+1:N+NRHS,1:KB)**T. +* + IF( KB.LT.MINMNUPDT ) THEN + CALL SGEMM( 'No transpose', 'Transpose', + $ M-IF, N+NRHS-KB, KB, -ONE, A( IF+1, 1 ), LDA, + $ F( KB+1, 1 ), LDF, ONE, A( IF+1, KB+1 ), LDA ) + END IF +* +* Recompute the 2-norm of the difficult columns. +* Loop over the index of the difficult columns from the largest +* to the smallest index. +* + DO WHILE( LSTICC.GT.0 ) +* +* LSTICC is the index of the last difficult column is greater +* than 1. +* ITEMP is the index of the previous difficult column. +* + ITEMP = IWORK( LSTICC-1 ) +* +* Compute the 2-norm explicilty for the last difficult column and +* save it in the partial and exact 2-norm vectors VN1 and VN2. +* +* NOTE: The computation of VN1( LSTICC ) relies on the fact that +* SNRM2 does not fail on vectors with norm below the value of +* SQRT(SLAMCH('S')) +* + VN1( LSTICC ) = SNRM2( M-IF, A( IF+1, LSTICC ), 1 ) + VN2( LSTICC ) = VN1( LSTICC ) +* +* Downdate the index of the last difficult column to +* the index of the previous difficult column. +* + LSTICC = ITEMP +* + END DO +* + RETURN +* +* End of SLAQP3RK +* + END diff --git a/SRC/zgeqp3rk.f b/SRC/zgeqp3rk.f new file mode 100755 index 0000000000..f8ef986c70 --- /dev/null +++ b/SRC/zgeqp3rk.f @@ -0,0 +1,1091 @@ +*> \brief \b ZGEQP3RK computes a truncated Householder QR factorization with column pivoting of a complex m-by-n matrix A by using Level 3 BLAS and overwrites m-by-nrhs matrix B with Q**H * B. +* +* =========== DOCUMENTATION =========== +* +* Online html documentation available at +* http://www.netlib.org/lapack/explore-html/ +* +*> \htmlonly +*> Download ZGEQP3RK + dependencies +*> +*> [TGZ] +*> +*> [ZIP] +*> +*> [TXT] +*> \endhtmlonly +* +* Definition: +* =========== +* +* SUBROUTINE ZGEQP3RK( M, N, NRHS, KMAX, ABSTOL, RELTOL, A, LDA, +* $ K, MAXC2NRMK, RELMAXC2NRMK, JPIV, TAU, +* $ WORK, LWORK, RWORK, IWORK, INFO ) +* IMPLICIT NONE +* +* .. Scalar Arguments .. +* INTEGER INFO, K, KMAX, LDA, LWORK, M, N, NRHS +* DOUBLE PRECISION ABSTOL, MAXC2NRMK, RELMAXC2NRMK, RELTOL +* .. +* .. Array Arguments .. +* INTEGER IWORK( * ), JPIV( * ) +* DOUBLE PRECISION RWORK( * ) +* COMPLEX*16 A( LDA, * ), TAU( * ), WORK( * ) +* .. +* +* +*> \par Purpose: +* ============= +*> +*> \verbatim +*> +*> ZGEQP3RK performs two tasks simultaneously: +*> +*> Task 1: The routine computes a truncated (rank K) or full rank +*> Householder QR factorization with column pivoting of a complex +*> M-by-N matrix A using Level 3 BLAS. K is the number of columns +*> that were factorized, i.e. factorization rank of the +*> factor R, K <= min(M,N). +*> +*> A * P(K) = Q(K) * R(K) = +*> +*> = Q(K) * ( R11(K) R12(K) ) = Q(K) * ( R(K)_approx ) +*> ( 0 R22(K) ) ( 0 R(K)_residual ), +*> +*> where: +*> +*> P(K) is an N-by-N permutation matrix; +*> Q(K) is an M-by-M orthogonal matrix; +*> R(K)_approx = ( R11(K), R12(K) ) is a rank K approximation of the +*> full rank factor R with K-by-K upper-triangular +*> R11(K) and K-by-N rectangular R12(K). The diagonal +*> entries of R11(K) appear in non-increasing order +*> of absolute value, and absolute values of all of +*> them exceed the maximum column 2-norm of R22(K) +*> up to roundoff error. +*> R(K)_residual = R22(K) is the residual of a rank K approximation +*> of the full rank factor R. It is a +*> an (M-K)-by-(N-K) rectangular matrix; +*> 0 is a an (M-K)-by-K zero matrix. +*> +*> Task 2: At the same time, the routine overwrites a complex M-by-NRHS +*> matrix B with Q(K)**H * B using Level 3 BLAS. +*> +*> ===================================================================== +*> +*> The matrices A and B are stored on input in the array A as +*> the left and right blocks A(1:M,1:N) and A(1:M, N+1:N+NRHS) +*> respectively. +*> +*> N NRHS +*> array_A = M [ mat_A, mat_B ] +*> +*> The truncation criteria (i.e. when to stop the factorization) +*> can be any of the following: +*> +*> 1) The input parameter KMAX, the maximum number of columns +*> KMAX to factorize, i.e. the factorization rank is limited +*> to KMAX. If KMAX >= min(M,N), the criterion is not used. +*> +*> 2) The input parameter ABSTOL, the absolute tolerance for +*> the maximum column 2-norm of the residual matrix R22(K). This +*> means that the factorization stops if this norm is less or +*> equal to ABSTOL. If ABSTOL < 0.0, the criterion is not used. +*> +*> 3) The input parameter RELTOL, the tolerance for the maximum +*> column 2-norm matrix of the residual matrix R22(K) divided +*> by the maximum column 2-norm of the original matrix A, which +*> is equal to abs(R(1,1)). This means that the factorization stops +*> when the ratio of the maximum column 2-norm of R22(K) to +*> the maximum column 2-norm of A is less than or equal to RELTOL. +*> If RELTOL < 0.0, the criterion is not used. +*> +*> 4) In case both stopping criteria ABSTOL or RELTOL are not used, +*> and when the residual matrix R22(K) is a zero matrix in some +*> factorization step K. ( This stopping criterion is implicit. ) +*> +*> The algorithm stops when any of these conditions is first +*> satisfied, otherwise the whole matrix A is factorized. +*> +*> To factorize the whole matrix A, use the values +*> KMAX >= min(M,N), ABSTOL < 0.0 and RELTOL < 0.0. +*> +*> The routine returns: +*> a) Q(K), R(K)_approx = ( R11(K), R12(K) ), +*> R(K)_residual = R22(K), P(K), i.e. the resulting matrices +*> of the factorization; P(K) is represented by JPIV, +*> ( if K = min(M,N), R(K)_approx is the full factor R, +*> and there is no residual matrix R(K)_residual); +*> b) K, the number of columns that were factorized, +*> i.e. factorization rank; +*> c) MAXC2NRMK, the maximum column 2-norm of the residual +*> matrix R(K)_residual = R22(K), +*> ( if K = min(M,N), MAXC2NRMK = 0.0 ); +*> d) RELMAXC2NRMK equals MAXC2NRMK divided by MAXC2NRM, the maximum +*> column 2-norm of the original matrix A, which is equal +*> to abs(R(1,1)), ( if K = min(M,N), RELMAXC2NRMK = 0.0 ); +*> e) Q(K)**H * B, the matrix B with the orthogonal +*> transformation Q(K)**H applied on the left. +*> +*> The N-by-N permutation matrix P(K) is stored in a compact form in +*> the integer array JPIV. For 1 <= j <= N, column j +*> of the matrix A was interchanged with column JPIV(j). +*> +*> The M-by-M orthogonal matrix Q is represented as a product +*> of elementary Householder reflectors +*> +*> Q(K) = H(1) * H(2) * . . . * H(K), +*> +*> where K is the number of columns that were factorized. +*> +*> Each H(j) has the form +*> +*> H(j) = I - tau * v * v**H, +*> +*> where 1 <= j <= K and +*> I is an M-by-M identity matrix, +*> tau is a complex scalar, +*> v is a complex vector with v(1:j-1) = 0 and v(j) = 1. +*> +*> v(j+1:M) is stored on exit in A(j+1:M,j) and tau in TAU(j). +*> +*> See the Further Details section for more information. +*> \endverbatim +* +* Arguments: +* ========== +* +*> \param[in] M +*> \verbatim +*> M is INTEGER +*> The number of rows of the matrix A. M >= 0. +*> \endverbatim +*> +*> \param[in] N +*> \verbatim +*> N is INTEGER +*> The number of columns of the matrix A. N >= 0. +*> \endverbatim +*> +*> \param[in] NRHS +*> \verbatim +*> NRHS is INTEGER +*> The number of right hand sides, i.e. the number of +*> columns of the matrix B. NRHS >= 0. +*> \endverbatim +*> +*> \param[in] KMAX +*> \verbatim +*> KMAX is INTEGER +*> +*> The first factorization stopping criterion. KMAX >= 0. +*> +*> The maximum number of columns of the matrix A to factorize, +*> i.e. the maximum factorization rank. +*> +*> a) If KMAX >= min(M,N), then this stopping criterion +*> is not used, the routine factorizes columns +*> depending on ABSTOL and RELTOL. +*> +*> b) If KMAX = 0, then this stopping criterion is +*> satisfied on input and the routine exits immediately. +*> This means that the factorization is not performed, +*> the matrices A and B are not modified, and +*> the matrix A is itself the residual. +*> \endverbatim +*> +*> \param[in] ABSTOL +*> \verbatim +*> ABSTOL is DOUBLE PRECISION +*> +*> The second factorization stopping criterion, cannot be NaN. +*> +*> The absolute tolerance (stopping threshold) for +*> maximum column 2-norm of the residual matrix R22(K). +*> The algorithm converges (stops the factorization) when +*> the maximum column 2-norm of the residual matrix R22(K) +*> is less than or equal to ABSTOL. Let SAFMIN = DLAMCH('S'). +*> +*> a) If ABSTOL is NaN, then no computation is performed +*> and an error message ( INFO = -5 ) is issued +*> by XERBLA. +*> +*> b) If ABSTOL < 0.0, then this stopping criterion is not +*> used, the routine factorizes columns depending +*> on KMAX and RELTOL. +*> This includes the case ABSTOL = -Inf. +*> +*> c) If 0.0 <= ABSTOL < 2*SAFMIN, then ABSTOL = 2*SAFMIN +*> is used. This includes the case ABSTOL = -0.0. +*> +*> d) If 2*SAFMIN <= ABSTOL then the input value +*> of ABSTOL is used. +*> +*> Let MAXC2NRM be the maximum column 2-norm of the +*> whole original matrix A. +*> If ABSTOL chosen above is >= MAXC2NRM, then this +*> stopping criterion is satisfied on input and routine exits +*> immediately after MAXC2NRM is computed. The routine +*> returns MAXC2NRM in MAXC2NORMK, +*> and 1.0 in RELMAXC2NORMK. +*> This includes the case ABSTOL = +Inf. This means that the +*> factorization is not performed, the matrices A and B are not +*> modified, and the matrix A is itself the residual. +*> \endverbatim +*> +*> \param[in] RELTOL +*> \verbatim +*> RELTOL is DOUBLE PRECISION +*> +*> The third factorization stopping criterion, cannot be NaN. +*> +*> The tolerance (stopping threshold) for the ratio +*> abs(R(K+1,K+1))/abs(R(1,1)) of the maximum column 2-norm of +*> the residual matrix R22(K) to the maximum column 2-norm of +*> the original matrix A. The algorithm converges (stops the +*> factorization), when abs(R(K+1,K+1))/abs(R(1,1)) A is less +*> than or equal to RELTOL. Let EPS = DLAMCH('E'). +*> +*> a) If RELTOL is NaN, then no computation is performed +*> and an error message ( INFO = -6 ) is issued +*> by XERBLA. +*> +*> b) If RELTOL < 0.0, then this stopping criterion is not +*> used, the routine factorizes columns depending +*> on KMAX and ABSTOL. +*> This includes the case RELTOL = -Inf. +*> +*> c) If 0.0 <= RELTOL < EPS, then RELTOL = EPS is used. +*> This includes the case RELTOL = -0.0. +*> +*> d) If EPS <= RELTOL then the input value of RELTOL +*> is used. +*> +*> Let MAXC2NRM be the maximum column 2-norm of the +*> whole original matrix A. +*> If RELTOL chosen above is >= 1.0, then this stopping +*> criterion is satisfied on input and routine exits +*> immediately after MAXC2NRM is computed. +*> The routine returns MAXC2NRM in MAXC2NORMK, +*> and 1.0 in RELMAXC2NORMK. +*> This includes the case RELTOL = +Inf. This means that the +*> factorization is not performed, the matrices A and B are not +*> modified, and the matrix A is itself the residual. +*> +*> NOTE: We recommend that RELTOL satisfy +*> min( 10*max(M,N)*EPS, sqrt(EPS) ) <= RELTOL +*> \endverbatim +*> +*> \param[in,out] A +*> \verbatim +*> A is COMPLEX*16 array, dimension (LDA,N+NRHS) +*> +*> On entry: +*> +*> a) The subarray A(1:M,1:N) contains the M-by-N matrix A. +*> b) The subarray A(1:M,N+1:N+NRHS) contains the M-by-NRHS +*> matrix B. +*> +*> N NRHS +*> array_A = M [ mat_A, mat_B ] +*> +*> On exit: +*> +*> a) The subarray A(1:M,1:N) contains parts of the factors +*> of the matrix A: +*> +*> 1) If K = 0, A(1:M,1:N) contains the original matrix A. +*> 2) If K > 0, A(1:M,1:N) contains parts of the +*> factors: +*> +*> 1. The elements below the diagonal of the subarray +*> A(1:M,1:K) together with TAU(1:K) represent the +*> orthogonal matrix Q(K) as a product of K Householder +*> elementary reflectors. +*> +*> 2. The elements on and above the diagonal of +*> the subarray A(1:K,1:N) contain K-by-N +*> upper-trapezoidal matrix +*> R(K)_approx = ( R11(K), R12(K) ). +*> NOTE: If K=min(M,N), i.e. full rank factorization, +*> then R_approx(K) is the full factor R which +*> is upper-trapezoidal. If, in addition, M>=N, +*> then R is upper-triangular. +*> +*> 3. The subarray A(K+1:M,K+1:N) contains (M-K)-by-(N-K) +*> rectangular matrix R(K)_residual = R22(K). +*> +*> b) If NRHS > 0, the subarray A(1:M,N+1:N+NRHS) contains +*> the M-by-NRHS product Q(K)**H * B. +*> \endverbatim +*> +*> \param[in] LDA +*> \verbatim +*> LDA is INTEGER +*> The leading dimension of the array A. LDA >= max(1,M). +*> This is the leading dimension for both matrices, A and B. +*> \endverbatim +*> +*> \param[out] K +*> \verbatim +*> K is INTEGER +*> Factorization rank of the matrix A, i.e. the rank of +*> the factor R, which is the same as the number of non-zero +*> rows of the factor R. 0 <= K <= min(M,KMAX,N). +*> +*> K also represents the number of non-zero Householder +*> vectors. +*> +*> NOTE: If K = 0, a) the arrays A and B are not modified; +*> b) the array TAU(1:min(M,N)) is set to ZERO, +*> if the matrix A does not contain NaN, +*> otherwise the elements TAU(1:min(M,N)) +*> are undefined; +*> c) the elements of the array JPIV are set +*> as follows: for j = 1:N, JPIV(j) = j. +*> \endverbatim +*> +*> \param[out] MAXC2NRMK +*> \verbatim +*> MAXC2NRMK is DOUBLE PRECISION +*> The maximum column 2-norm of the residual matrix R22(K), +*> when the factorization stopped at rank K. MAXC2NRMK >= 0. +*> +*> a) If K = 0, i.e. the factorization was not performed, +*> the matrix A was not modified and is itself a residual +*> matrix, then MAXC2NRMK equals the maximum column 2-norm +*> of the original matrix A. +*> +*> b) If 0 < K < min(M,N), then MAXC2NRMK is returned. +*> +*> c) If K = min(M,N), i.e. the whole matrix A was +*> factorized and there is no residual matrix, +*> then MAXC2NRMK = 0.0. +*> +*> NOTE: MAXC2NRMK in the factorization step K would equal +*> R(K+1,K+1) in the next factorization step K+1. +*> \endverbatim +*> +*> \param[out] RELMAXC2NRMK +*> \verbatim +*> RELMAXC2NRMK is DOUBLE PRECISION +*> The ratio MAXC2NRMK / MAXC2NRM of the maximum column +*> 2-norm of the residual matrix R22(K) (when the factorization +*> stopped at rank K) to the maximum column 2-norm of the +*> whole original matrix A. RELMAXC2NRMK >= 0. +*> +*> a) If K = 0, i.e. the factorization was not performed, +*> the matrix A was not modified and is itself a residual +*> matrix, then RELMAXC2NRMK = 1.0. +*> +*> b) If 0 < K < min(M,N), then +*> RELMAXC2NRMK = MAXC2NRMK / MAXC2NRM is returned. +*> +*> c) If K = min(M,N), i.e. the whole matrix A was +*> factorized and there is no residual matrix, +*> then RELMAXC2NRMK = 0.0. +*> +*> NOTE: RELMAXC2NRMK in the factorization step K would equal +*> abs(R(K+1,K+1))/abs(R(1,1)) in the next factorization +*> step K+1. +*> \endverbatim +*> +*> \param[out] JPIV +*> \verbatim +*> JPIV is INTEGER array, dimension (N) +*> Column pivot indices. For 1 <= j <= N, column j +*> of the matrix A was interchanged with column JPIV(j). +*> +*> The elements of the array JPIV(1:N) are always set +*> by the routine, for example, even when no columns +*> were factorized, i.e. when K = 0, the elements are +*> set as JPIV(j) = j for j = 1:N. +*> \endverbatim +*> +*> \param[out] TAU +*> \verbatim +*> TAU is COMPLEX*16 array, dimension (min(M,N)) +*> The scalar factors of the elementary reflectors. +*> +*> If 0 < K <= min(M,N), only the elements TAU(1:K) of +*> the array TAU are modified by the factorization. +*> After the factorization computed, if no NaN was found +*> during the factorization, the remaining elements +*> TAU(K+1:min(M,N)) are set to zero, otherwise the +*> elements TAU(K+1:min(M,N)) are not set and therefore +*> undefined. +*> ( If K = 0, all elements of TAU are set to zero, if +*> the matrix A does not contain NaN. ) +*> \endverbatim +*> +*> \param[out] WORK +*> \verbatim +*> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) +*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. +*> \endverbatim +*> +*> \param[in] LWORK +*> \verbatim +*> LWORK is INTEGER +*> The dimension of the array WORK. +*. LWORK >= N+NRHS-1 +*> For optimal performance LWORK >= NB*( N+NRHS+1 ), +*> where NB is the optimal block size for ZGEQP3RK returned +*> by ILAENV. Minimal block size MINNB=2. +*> +*> NOTE: The decision, whether to use unblocked BLAS 2 +*> or blocked BLAS 3 code is based not only on the dimension +*> LWORK of the availbale workspace WORK, but also also on the +*> matrix A dimension N via crossover point NX returned +*> by ILAENV. (For N less than NX, unblocked code should be +*> used.) +*> +*> If LWORK = -1, then a workspace query is assumed; +*> the routine only calculates the optimal size of the WORK +*> array, returns this value as the first entry of the WORK +*> array, and no error message related to LWORK is issued +*> by XERBLA. +*> \endverbatim +*> +*> \param[out] RWORK +*> \verbatim +*> RWORK is DOUBLE PRECISION array, dimension (2*N) +*> \endverbatim +*> +*> \param[out] IWORK +*> \verbatim +*> IWORK is INTEGER array, dimension (N-1). +*> Is a work array. ( IWORK is used to store indices +*> of "bad" columns for norm downdating in the residual +*> matrix in the blocked step auxiliary subroutine ZLAQP3RK ). +*> \endverbatim +*> +*> \param[out] INFO +*> \verbatim +*> INFO is INTEGER +*> 1) INFO = 0: successful exit. +*> 2) INFO < 0: if INFO = -i, the i-th argument had an +*> illegal value. +*> 3) If INFO = j_1, where 1 <= j_1 <= N, then NaN was +*> detected and the routine stops the computation. +*> The j_1-th column of the matrix A or the j_1-th +*> element of array TAU contains the first occurrence +*> of NaN in the factorization step K+1 ( when K columns +*> have been factorized ). +*> +*> On exit: +*> K is set to the number of +*> factorized columns without +*> exception. +*> MAXC2NRMK is set to NaN. +*> RELMAXC2NRMK is set to NaN. +*> TAU(K+1:min(M,N)) is not set and contains undefined +*> elements. If j_1=K+1, TAU(K+1) +*> may contain NaN. +*> 4) If INFO = j_2, where N+1 <= j_2 <= 2*N, then no NaN +*> was detected, but +Inf (or -Inf) was detected and +*> the routine continues the computation until completion. +*> The (j_2-N)-th column of the matrix A contains the first +*> occurrence of +Inf (or -Inf) in the factorization +*> step K+1 ( when K columns have been factorized ). +*> \endverbatim +* +* Authors: +* ======== +* +*> \author Univ. of Tennessee +*> \author Univ. of California Berkeley +*> \author Univ. of Colorado Denver +*> \author NAG Ltd. +* +*> \ingroup geqp3rk +* +*> \par Further Details: +* ===================== +* +*> \verbatim +*> ZGEQP3RK is based on the same BLAS3 Householder QR factorization +*> algorithm with column pivoting as in ZGEQP3 routine which uses +*> ZLARFG routine to generate Householder reflectors +*> for QR factorization. +*> +*> We can also write: +*> +*> A = A_approx(K) + A_residual(K) +*> +*> The low rank approximation matrix A(K)_approx from +*> the truncated QR factorization of rank K of the matrix A is: +*> +*> A(K)_approx = Q(K) * ( R(K)_approx ) * P(K)**T +*> ( 0 0 ) +*> +*> = Q(K) * ( R11(K) R12(K) ) * P(K)**T +*> ( 0 0 ) +*> +*> The residual A_residual(K) of the matrix A is: +*> +*> A_residual(K) = Q(K) * ( 0 0 ) * P(K)**T = +*> ( 0 R(K)_residual ) +*> +*> = Q(K) * ( 0 0 ) * P(K)**T +*> ( 0 R22(K) ) +*> +*> The truncated (rank K) factorization guarantees that +*> the maximum column 2-norm of A_residual(K) is less than +*> or equal to MAXC2NRMK up to roundoff error. +*> +*> NOTE: An approximation of the null vectors +*> of A can be easily computed from R11(K) +*> and R12(K): +*> +*> Null( A(K) )_approx = P * ( inv(R11(K)) * R12(K) ) +*> ( -I ) +*> +*> \endverbatim +* +*> \par References: +* ================ +*> [1] A Level 3 BLAS QR factorization algorithm with column pivoting developed in 1996. +*> G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain. +*> X. Sun, Computer Science Dept., Duke University, USA. +*> C. H. Bischof, Math. and Comp. Sci. Div., Argonne National Lab, USA. +*> A BLAS-3 version of the QR factorization with column pivoting. +*> LAPACK Working Note 114 +*> \htmlonly +*> https://www.netlib.org/lapack/lawnspdf/lawn114.pdf +*> \endhtmlonly +*> and in +*> SIAM J. Sci. Comput., 19(5):1486-1494, Sept. 1998. +*> \htmlonly +*> https://doi.org/10.1137/S1064827595296732 +*> \endhtmlonly +*> +*> [2] A partial column norm updating strategy developed in 2006. +*> Z. Drmac and Z. Bujanovic, Dept. of Math., University of Zagreb, Croatia. +*> On the failure of rank revealing QR factorization software – a case study. +*> LAPACK Working Note 176. +*> \htmlonly +*> http://www.netlib.org/lapack/lawnspdf/lawn176.pdf +*> \endhtmlonly +*> and in +*> ACM Trans. Math. Softw. 35, 2, Article 12 (July 2008), 28 pages. +*> \htmlonly +*> https://doi.org/10.1145/1377612.1377616 +*> \endhtmlonly +* +*> \par Contributors: +* ================== +*> +*> \verbatim +*> +*> November 2023, Igor Kozachenko, James Demmel, +*> Computer Science Division, +*> University of California, Berkeley +*> +*> \endverbatim +* +* ===================================================================== + SUBROUTINE ZGEQP3RK( M, N, NRHS, KMAX, ABSTOL, RELTOL, A, LDA, + $ K, MAXC2NRMK, RELMAXC2NRMK, JPIV, TAU, + $ WORK, LWORK, RWORK, IWORK, INFO ) + IMPLICIT NONE +* +* -- LAPACK computational routine -- +* -- LAPACK is a software package provided by Univ. of Tennessee, -- +* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- +* +* .. Scalar Arguments .. + INTEGER INFO, K, KF, KMAX, LDA, LWORK, M, N, NRHS + DOUBLE PRECISION ABSTOL, MAXC2NRMK, RELMAXC2NRMK, RELTOL +* .. +* .. Array Arguments .. + INTEGER IWORK( * ), JPIV( * ) + DOUBLE PRECISION RWORK( * ) + COMPLEX*16 A( LDA, * ), TAU( * ), WORK( * ) +* .. +* +* ===================================================================== +* +* .. Parameters .. + INTEGER INB, INBMIN, IXOVER + PARAMETER ( INB = 1, INBMIN = 2, IXOVER = 3 ) + DOUBLE PRECISION ZERO, ONE, TWO + PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 ) + COMPLEX*16 CZERO + PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ) ) +* .. +* .. Local Scalars .. + LOGICAL LQUERY, DONE + INTEGER IINFO, IOFFSET, IWS, J, JB, JBF, JMAXB, JMAX, + $ JMAXC2NRM, KP1, LWKOPT, MINMN, N_SUB, NB, + $ NBMIN, NX + DOUBLE PRECISION EPS, HUGEVAL, MAXC2NRM, SAFMIN +* .. +* .. External Subroutines .. + EXTERNAL ZLAQP2RK, ZLAQP3RK, XERBLA +* .. +* .. External Functions .. + LOGICAL DISNAN + INTEGER IDAMAX, ILAENV + DOUBLE PRECISION DLAMCH, DZNRM2 + EXTERNAL DISNAN, DLAMCH, DZNRM2, IDAMAX, ILAENV +* .. +* .. Intrinsic Functions .. + INTRINSIC DCMPLX, MAX, MIN +* .. +* .. Executable Statements .. +* +* Test input arguments +* ==================== +* + INFO = 0 + LQUERY = ( LWORK.EQ.-1 ) + IF( M.LT.0 ) THEN + INFO = -1 + ELSE IF( N.LT.0 ) THEN + INFO = -2 + ELSE IF( NRHS.LT.0 ) THEN + INFO = -3 + ELSE IF( KMAX.LT.0 ) THEN + INFO = -4 + ELSE IF( DISNAN( ABSTOL ) ) THEN + INFO = -5 + ELSE IF( DISNAN( RELTOL ) ) THEN + INFO = -6 + ELSE IF( LDA.LT.MAX( 1, M ) ) THEN + INFO = -8 + END IF +* +* If the input parameters M, N, NRHS, KMAX, LDA are valid: +* a) Test the input workspace size LWORK for the minimum +* size requirement IWS. +* b) Determine the optimal block size NB and optimal +* workspace size LWKOPT to be returned in WORK(1) +* in case of (1) LWORK < IWS, (2) LQUERY = .TRUE., +* (3) when routine exits. +* Here, IWS is the miminum workspace required for unblocked +* code. +* + IF( INFO.EQ.0 ) THEN + MINMN = MIN( M, N ) + IF( MINMN.EQ.0 ) THEN + IWS = 1 + LWKOPT = 1 + ELSE +* +* Minimal workspace size in case of using only unblocked +* BLAS 2 code in ZLAQP2RK. +* 1) ZLAQP2RK: N+NRHS-1 to use in WORK array that is used +* in ZLARF subroutine inside ZLAQP2RK to apply an +* elementary reflector from the left. +* TOTAL_WORK_SIZE = 3*N + NRHS - 1 +* + IWS = N + NRHS - 1 +* +* Assign to NB optimal block size. +* + NB = ILAENV( INB, 'ZGEQP3RK', ' ', M, N, -1, -1 ) +* +* A formula for the optimal workspace size in case of using +* both unblocked BLAS 2 in ZLAQP2RK and blocked BLAS 3 code +* in ZLAQP3RK. +* 1) ZGEQP3RK, ZLAQP2RK, ZLAQP3RK: 2*N to store full and +* partial column 2-norms. +* 2) ZLAQP2RK: N+NRHS-1 to use in WORK array that is used +* in ZLARF subroutine to apply an elementary reflector +* from the left. +* 3) ZLAQP3RK: NB*(N+NRHS) to use in the work array F that +* is used to apply a block reflector from +* the left. +* 4) ZLAQP3RK: NB to use in the auxilixary array AUX. +* Sizes (2) and ((3) + (4)) should intersect, therefore +* TOTAL_WORK_SIZE = 2*N + NB*( N+NRHS+1 ), given NBMIN=2. +* + LWKOPT = 2*N + NB*( N+NRHS+1 ) + END IF + WORK( 1 ) = DCMPLX( LWKOPT ) +* + IF( ( LWORK.LT.IWS ) .AND. .NOT.LQUERY ) THEN + INFO = -15 + END IF + END IF +* +* NOTE: The optimal workspace size is returned in WORK(1), if +* the input parameters M, N, NRHS, KMAX, LDA are valid. +* + IF( INFO.NE.0 ) THEN + CALL XERBLA( 'ZGEQP3RK', -INFO ) + RETURN + ELSE IF( LQUERY ) THEN + RETURN + END IF +* +* Quick return if possible for M=0 or N=0. +* + IF( MINMN.EQ.0 ) THEN + K = 0 + MAXC2NRMK = ZERO + RELMAXC2NRMK = ZERO + WORK( 1 ) = DCMPLX( LWKOPT ) + RETURN + END IF +* +* ================================================================== +* +* Initialize column pivot array JPIV. +* + DO J = 1, N + JPIV( J ) = J + END DO +* +* ================================================================== +* +* Initialize storage for partial and exact column 2-norms. +* a) The elements WORK(1:N) are used to store partial column +* 2-norms of the matrix A, and may decrease in each computation +* step; initialize to the values of complete columns 2-norms. +* b) The elements WORK(N+1:2*N) are used to store complete column +* 2-norms of the matrix A, they are not changed during the +* computation; initialize the values of complete columns 2-norms. +* + DO J = 1, N + RWORK( J ) = DZNRM2( M, A( 1, J ), 1 ) + RWORK( N+J ) = RWORK( J ) + END DO +* +* ================================================================== +* +* Compute the pivot column index and the maximum column 2-norm +* for the whole original matrix stored in A(1:M,1:N). +* + KP1 = IDAMAX( N, RWORK( 1 ), 1 ) +* +* ==================================================================. +* + IF( DISNAN( MAXC2NRM ) ) THEN +* +* Check if the matrix A contains NaN, set INFO parameter +* to the column number where the first NaN is found and return +* from the routine. +* + K = 0 + INFO = KP1 +* +* Set MAXC2NRMK and RELMAXC2NRMK to NaN. +* + MAXC2NRMK = MAXC2NRM + RELMAXC2NRMK = MAXC2NRM +* +* Array TAU is not set and contains undefined elements. +* + WORK( 1 ) = DCMPLX( LWKOPT ) + RETURN + END IF +* +* =================================================================== +* + IF( MAXC2NRM.EQ.ZERO ) THEN +* +* Check is the matrix A is a zero matrix, set array TAU and +* return from the routine. +* + K = 0 + MAXC2NRMK = ZERO + RELMAXC2NRMK = ZERO +* + DO J = 1, MINMN + TAU( J ) = CZERO + END DO +* + WORK( 1 ) = DCMPLX( LWKOPT ) + RETURN +* + END IF +* +* =================================================================== +* + HUGEVAL = DLAMCH( 'Overflow' ) +* + IF( MAXC2NRM.GT.HUGEVAL ) THEN +* +* Check if the matrix A contains +Inf or -Inf, set INFO parameter +* to the column number, where the first +/-Inf is found plus N, +* and continue the computation. +* + INFO = N + KP1 +* + END IF +* +* ================================================================== +* +* Quick return if possible for the case when the first +* stopping criterion is satisfied, i.e. KMAX = 0. +* + IF( KMAX.EQ.0 ) THEN + K = 0 + MAXC2NRMK = MAXC2NRM + RELMAXC2NRMK = ONE + DO J = 1, MINMN + TAU( J ) = CZERO + END DO + WORK( 1 ) = DCMPLX( LWKOPT ) + RETURN + END IF +* +* ================================================================== +* + EPS = DLAMCH('Epsilon') +* +* Adjust ABSTOL +* + IF( ABSTOL.GE.ZERO ) THEN + SAFMIN = DLAMCH('Safe minimum') + ABSTOL = MAX( ABSTOL, TWO*SAFMIN ) + END IF +* +* Adjust RELTOL +* + IF( RELTOL.GE.ZERO ) THEN + RELTOL = MAX( RELTOL, EPS ) + END IF +* +* =================================================================== +* +* JMAX is the maximum index of the column to be factorized, +* which is also limited by the first stopping criterion KMAX. +* + JMAX = MIN( KMAX, MINMN ) +* +* =================================================================== +* +* Quick return if possible for the case when the second or third +* stopping criterion for the whole original matrix is satified, +* i.e. MAXC2NRM <= ABSTOL or RELMAXC2NRM <= RELTOL +* (which is ONE <= RELTOL). +* + IF( MAXC2NRM.LE.ABSTOL .OR. ONE.LE.RELTOL ) THEN +* + K = 0 + MAXC2NRMK = MAXC2NRM + RELMAXC2NRMK = ONE +* + DO J = 1, MINMN + TAU( J ) = CZERO + END DO +* + WORK( 1 ) = DCMPLX( LWKOPT ) + RETURN + END IF +* +* ================================================================== +* Factorize columns +* ================================================================== +* +* Determine the block size. +* + NBMIN = 2 + NX = 0 +* + IF( ( NB.GT.1 ) .AND. ( NB.LT.MINMN ) ) THEN +* +* Determine when to cross over from blocked to unblocked code. +* (for N less than NX, unblocked code should be used). +* + NX = MAX( 0, ILAENV( IXOVER, 'ZGEQP3RK', ' ', M, N, -1, -1 ) ) +* + IF( NX.LT.MINMN ) THEN +* +* Determine if workspace is large enough for blocked code. +* + IF( LWORK.LT.LWKOPT ) THEN +* +* Not enough workspace to use optimal block size that +* is currently stored in NB. +* Reduce NB and determine the minimum value of NB. +* + NB = ( LWORK-2*N ) / ( N+1 ) + NBMIN = MAX( 2, ILAENV( INBMIN, 'ZGEQP3RK', ' ', M, N, + $ -1, -1 ) ) +* + END IF + END IF + END IF +* +* ================================================================== +* +* DONE is the boolean flag to rerpresent the case when the +* factorization completed in the block factorization routine, +* before the end of the block. +* + DONE = .FALSE. +* +* J is the column index. +* + J = 1 +* +* (1) Use blocked code initially. +* +* JMAXB is the maximum column index of the block, when the +* blocked code is used, is also limited by the first stopping +* criterion KMAX. +* + JMAXB = MIN( KMAX, MINMN - NX ) +* + IF( NB.GE.NBMIN .AND. NB.LT.JMAX .AND. JMAXB.GT.0 ) THEN +* +* Loop over the column blocks of the matrix A(1:M,1:JMAXB). Here: +* J is the column index of a column block; +* JB is the column block size to pass to block factorization +* routine in a loop step; +* JBF is the number of columns that were actually factorized +* that was returned by the block factorization routine +* in a loop step, JBF <= JB; +* N_SUB is the number of columns in the submatrix; +* IOFFSET is the number of rows that should not be factorized. +* + DO WHILE( J.LE.JMAXB ) +* + JB = MIN( NB, JMAXB-J+1 ) + N_SUB = N-J+1 + IOFFSET = J-1 +* +* Factorize JB columns among the columns A(J:N). +* + CALL ZLAQP3RK( M, N_SUB, NRHS, IOFFSET, JB, ABSTOL, + $ RELTOL, KP1, MAXC2NRM, A( 1, J ), LDA, + $ DONE, JBF, MAXC2NRMK, RELMAXC2NRMK, + $ JPIV( J ), TAU( J ), + $ RWORK( J ), RWORK( N+J ), + $ WORK( 1 ), WORK( JB+1 ), + $ N+NRHS-J+1, IWORK, IINFO ) +* +* Set INFO on the first occurence of Inf. +* + IF( IINFO.GT.N_SUB .AND. INFO.EQ.0 ) THEN + INFO = 2*IOFFSET + IINFO + END IF +* + IF( DONE ) THEN +* +* Either the submatrix is zero before the end of the +* column block, or ABSTOL or RELTOL criterion is +* satisfied before the end of the column block, we can +* return from the routine. Perform the following before +* returning: +* a) Set the number of factorized columns K, +* K = IOFFSET + JBF from the last call of blocked +* routine. +* NOTE: 1) MAXC2NRMK and RELMAXC2NRMK are returned +* by the block factorization routine; +* 2) The remaining TAUs are set to ZERO by the +* block factorization routine. +* + K = IOFFSET + JBF +* +* Set INFO on the first occurrence of NaN, NaN takes +* prcedence over Inf. +* + IF( IINFO.LE.N_SUB .AND. IINFO.GT.0 ) THEN + INFO = IOFFSET + IINFO + END IF +* +* Return from the routine. +* + WORK( 1 ) = DCMPLX( LWKOPT ) +* + RETURN +* + END IF +* + J = J + JBF +* + END DO +* + END IF +* +* Use unblocked code to factor the last or only block. +* J = JMAX+1 means we factorized the maximum possible number of +* columns, that is in ELSE clause we need to compute +* the MAXC2NORM and RELMAXC2NORM to return after we processed +* the blocks. +* + IF( J.LE.JMAX ) THEN +* +* N_SUB is the number of columns in the submatrix; +* IOFFSET is the number of rows that should not be factorized. +* + N_SUB = N-J+1 + IOFFSET = J-1 +* + CALL ZLAQP2RK( M, N_SUB, NRHS, IOFFSET, JMAX-J+1, + $ ABSTOL, RELTOL, KP1, MAXC2NRM, A( 1, J ), LDA, + $ KF, MAXC2NRMK, RELMAXC2NRMK, JPIV( J ), + $ TAU( J ), RWORK( J ), RWORK( N+J ), + $ WORK( 1 ), IINFO ) +* +* ABSTOL or RELTOL criterion is satisfied when the number of +* the factorized columns KF is smaller then the number +* of columns JMAX-J+1 supplied to be factorized by the +* unblocked routine, we can return from +* the routine. Perform the following before returning: +* a) Set the number of factorized columns K, +* b) MAXC2NRMK and RELMAXC2NRMK are returned by the +* unblocked factorization routine above. +* + K = J - 1 + KF +* +* Set INFO on the first exception occurence. +* +* Set INFO on the first exception occurence of Inf or NaN, +* (NaN takes precedence over Inf). +* + IF( IINFO.GT.N_SUB .AND. INFO.EQ.0 ) THEN + INFO = 2*IOFFSET + IINFO + ELSE IF( IINFO.LE.N_SUB .AND. IINFO.GT.0 ) THEN + INFO = IOFFSET + IINFO + END IF +* + ELSE +* +* Compute the return values for blocked code. +* +* Set the number of factorized columns if the unblocked routine +* was not called. +* + K = JMAX +* +* If there exits a residual matrix after the blocked code: +* 1) compute the values of MAXC2NRMK, RELMAXC2NRMK of the +* residual matrix, otherwise set them to ZERO; +* 2) Set TAU(K+1:MINMN) to ZERO. +* + IF( K.LT.MINMN ) THEN + JMAXC2NRM = K + IDAMAX( N-K, RWORK( K+1 ), 1 ) + MAXC2NRMK = RWORK( JMAXC2NRM ) + IF( K.EQ.0 ) THEN + RELMAXC2NRMK = ONE + ELSE + RELMAXC2NRMK = MAXC2NRMK / MAXC2NRM + END IF +* + DO J = K + 1, MINMN + TAU( J ) = CZERO + END DO +* + ELSE + MAXC2NRMK = ZERO + RELMAXC2NRMK = ZERO +* + END IF +* +* END IF( J.LE.JMAX ) THEN +* + END IF +* + WORK( 1 ) = DCMPLX( LWKOPT ) +* + RETURN +* +* End of ZGEQP3RK +* + END diff --git a/SRC/zlaqp2rk.f b/SRC/zlaqp2rk.f new file mode 100755 index 0000000000..f1e9f48993 --- /dev/null +++ b/SRC/zlaqp2rk.f @@ -0,0 +1,726 @@ +*> \brief \b ZLAQP2RK computes truncated QR factorization with column pivoting of a complex matrix block using Level 2 BLAS and overwrites a complex m-by-nrhs matrix B with Q**H * B. +* +* =========== DOCUMENTATION =========== +* +* Online html documentation available at +* http://www.netlib.org/lapack/explore-html/ +* +*> \htmlonly +*> Download ZLAQP2RK + dependencies +*> +*> [TGZ] +*> +*> [ZIP] +*> +*> [TXT] +*> \endhtmlonly +* +* Definition: +* =========== +* +* SUBROUTINE ZLAQP2RK( M, N, NRHS, IOFFSET, KMAX, ABSTOL, RELTOL, +* $ KP1, MAXC2NRM, A, LDA, K, MAXC2NRMK, +* $ RELMAXC2NRMK, JPIV, TAU, VN1, VN2, WORK, +* $ INFO ) +* IMPLICIT NONE +* +* .. Scalar Arguments .. +* INTEGER INFO, IOFFSET, KP1, K, KMAX, LDA, M, N, NRHS +* DOUBLE PRECISION ABSTOL, MAXC2NRM, MAXC2NRMK, RELMAXC2NRMK, +* $ RELTOL +* .. +* .. Array Arguments .. +* INTEGER JPIV( * ) +* DOUBLE PRECISION VN1( * ), VN2( * ) +* COMPLEX*16 A( LDA, * ), TAU( * ), WORK( * ) +* $ +* .. +* +* +*> \par Purpose: +* ============= +*> +*> \verbatim +*> +*> ZLAQP2RK computes a truncated (rank K) or full rank Householder QR +*> factorization with column pivoting of the complex matrix +*> block A(IOFFSET+1:M,1:N) as +*> +*> A * P(K) = Q(K) * R(K). +*> +*> The routine uses Level 2 BLAS. The block A(1:IOFFSET,1:N) +*> is accordingly pivoted, but not factorized. +*> +*> The routine also overwrites the right-hand-sides matrix block B +*> stored in A(IOFFSET+1:M,N+1:N+NRHS) with Q(K)**H * B. +*> \endverbatim +* +* Arguments: +* ========== +* +*> \param[in] M +*> \verbatim +*> M is INTEGER +*> The number of rows of the matrix A. M >= 0. +*> \endverbatim +*> +*> \param[in] N +*> \verbatim +*> N is INTEGER +*> The number of columns of the matrix A. N >= 0. +*> \endverbatim +*> +*> \param[in] NRHS +*> \verbatim +*> NRHS is INTEGER +*> The number of right hand sides, i.e., the number of +*> columns of the matrix B. NRHS >= 0. +*> \endverbatim +*> +*> \param[in] IOFFSET +*> \verbatim +*> IOFFSET is INTEGER +*> The number of rows of the matrix A that must be pivoted +*> but not factorized. IOFFSET >= 0. +*> +*> IOFFSET also represents the number of columns of the whole +*> original matrix A_orig that have been factorized +*> in the previous steps. +*> \endverbatim +*> +*> \param[in] KMAX +*> \verbatim +*> KMAX is INTEGER +*> +*> The first factorization stopping criterion. KMAX >= 0. +*> +*> The maximum number of columns of the matrix A to factorize, +*> i.e. the maximum factorization rank. +*> +*> a) If KMAX >= min(M-IOFFSET,N), then this stopping +*> criterion is not used, factorize columns +*> depending on ABSTOL and RELTOL. +*> +*> b) If KMAX = 0, then this stopping criterion is +*> satisfied on input and the routine exits immediately. +*> This means that the factorization is not performed, +*> the matrices A and B and the arrays TAU, IPIV +*> are not modified. +*> \endverbatim +*> +*> \param[in] ABSTOL +*> \verbatim +*> ABSTOL is DOUBLE PRECISION, cannot be NaN. +*> +*> The second factorization stopping criterion. +*> +*> The absolute tolerance (stopping threshold) for +*> maximum column 2-norm of the residual matrix. +*> The algorithm converges (stops the factorization) when +*> the maximum column 2-norm of the residual matrix +*> is less than or equal to ABSTOL. +*> +*> a) If ABSTOL < 0.0, then this stopping criterion is not +*> used, the routine factorizes columns depending +*> on KMAX and RELTOL. +*> This includes the case ABSTOL = -Inf. +*> +*> b) If 0.0 <= ABSTOL then the input value +*> of ABSTOL is used. +*> \endverbatim +*> +*> \param[in] RELTOL +*> \verbatim +*> RELTOL is DOUBLE PRECISION, cannot be NaN. +*> +*> The third factorization stopping criterion. +*> +*> The tolerance (stopping threshold) for the ratio of the +*> maximum column 2-norm of the residual matrix to the maximum +*> column 2-norm of the original matrix A_orig. The algorithm +*> converges (stops the factorization), when this ratio is +*> less than or equal to RELTOL. +*> +*> a) If RELTOL < 0.0, then this stopping criterion is not +*> used, the routine factorizes columns depending +*> on KMAX and ABSTOL. +*> This includes the case RELTOL = -Inf. +*> +*> d) If 0.0 <= RELTOL then the input value of RELTOL +*> is used. +*> \endverbatim +*> +*> \param[in] KP1 +*> \verbatim +*> KP1 is INTEGER +*> The index of the column with the maximum 2-norm in +*> the whole original matrix A_orig determined in the +*> main routine ZGEQP3RK. 1 <= KP1 <= N_orig_mat. +*> \endverbatim +*> +*> \param[in] MAXC2NRM +*> \verbatim +*> MAXC2NRM is DOUBLE PRECISION +*> The maximum column 2-norm of the whole original +*> matrix A_orig computed in the main routine ZGEQP3RK. +*> MAXC2NRM >= 0. +*> \endverbatim +*> +*> \param[in,out] A +*> \verbatim +*> A is COMPLEX*16 array, dimension (LDA,N+NRHS) +*> On entry: +*> the M-by-N matrix A and M-by-NRHS matrix B, as in +*> +*> N NRHS +*> array_A = M [ mat_A, mat_B ] +*> +*> On exit: +*> 1. The elements in block A(IOFFSET+1:M,1:K) below +*> the diagonal together with the array TAU represent +*> the orthogonal matrix Q(K) as a product of elementary +*> reflectors. +*> 2. The upper triangular block of the matrix A stored +*> in A(IOFFSET+1:M,1:K) is the triangular factor obtained. +*> 3. The block of the matrix A stored in A(1:IOFFSET,1:N) +*> has been accordingly pivoted, but not factorized. +*> 4. The rest of the array A, block A(IOFFSET+1:M,K+1:N+NRHS). +*> The left part A(IOFFSET+1:M,K+1:N) of this block +*> contains the residual of the matrix A, and, +*> if NRHS > 0, the right part of the block +*> A(IOFFSET+1:M,N+1:N+NRHS) contains the block of +*> the right-hand-side matrix B. Both these blocks have been +*> updated by multiplication from the left by Q(K)**H. +*> \endverbatim +*> +*> \param[in] LDA +*> \verbatim +*> LDA is INTEGER +*> The leading dimension of the array A. LDA >= max(1,M). +*> \endverbatim +*> +*> \param[out] K +*> \verbatim +*> K is INTEGER +*> Factorization rank of the matrix A, i.e. the rank of +*> the factor R, which is the same as the number of non-zero +*> rows of the factor R. 0 <= K <= min(M-IOFFSET,KMAX,N). +*> +*> K also represents the number of non-zero Householder +*> vectors. +*> \endverbatim +*> +*> \param[out] MAXC2NRMK +*> \verbatim +*> MAXC2NRMK is DOUBLE PRECISION +*> The maximum column 2-norm of the residual matrix, +*> when the factorization stopped at rank K. MAXC2NRMK >= 0. +*> \endverbatim +*> +*> \param[out] RELMAXC2NRMK +*> \verbatim +*> RELMAXC2NRMK is DOUBLE PRECISION +*> The ratio MAXC2NRMK / MAXC2NRM of the maximum column +*> 2-norm of the residual matrix (when the factorization +*> stopped at rank K) to the maximum column 2-norm of the +*> whole original matrix A. RELMAXC2NRMK >= 0. +*> \endverbatim +*> +*> \param[out] JPIV +*> \verbatim +*> JPIV is INTEGER array, dimension (N) +*> Column pivot indices, for 1 <= j <= N, column j +*> of the matrix A was interchanged with column JPIV(j). +*> \endverbatim +*> +*> \param[out] TAU +*> \verbatim +*> TAU is COMPLEX*16 array, dimension (min(M-IOFFSET,N)) +*> The scalar factors of the elementary reflectors. +*> \endverbatim +*> +*> \param[in,out] VN1 +*> \verbatim +*> VN1 is DOUBLE PRECISION array, dimension (N) +*> The vector with the partial column norms. +*> \endverbatim +*> +*> \param[in,out] VN2 +*> \verbatim +*> VN2 is DOUBLE PRECISION array, dimension (N) +*> The vector with the exact column norms. +*> \endverbatim +*> +*> \param[out] WORK +*> \verbatim +*> WORK is COMPLEX*16 array, dimension (N-1) +*> Used in ZLARF subroutine to apply an elementary +*> reflector from the left. +*> \endverbatim +*> +*> \param[out] INFO +*> \verbatim +*> INFO is INTEGER +*> 1) INFO = 0: successful exit. +*> 2) If INFO = j_1, where 1 <= j_1 <= N, then NaN was +*> detected and the routine stops the computation. +*> The j_1-th column of the matrix A or the j_1-th +*> element of array TAU contains the first occurrence +*> of NaN in the factorization step K+1 ( when K columns +*> have been factorized ). +*> +*> On exit: +*> K is set to the number of +*> factorized columns without +*> exception. +*> MAXC2NRMK is set to NaN. +*> RELMAXC2NRMK is set to NaN. +*> TAU(K+1:min(M,N)) is not set and contains undefined +*> elements. If j_1=K+1, TAU(K+1) +*> may contain NaN. +*> 3) If INFO = j_2, where N+1 <= j_2 <= 2*N, then no NaN +*> was detected, but +Inf (or -Inf) was detected and +*> the routine continues the computation until completion. +*> The (j_2-N)-th column of the matrix A contains the first +*> occurrence of +Inf (or -Inf) in the factorization +*> step K+1 ( when K columns have been factorized ). +*> \endverbatim +* +* Authors: +* ======== +* +*> \author Univ. of Tennessee +*> \author Univ. of California Berkeley +*> \author Univ. of Colorado Denver +*> \author NAG Ltd. +* +*> \ingroup laqp2rk +* +*> \par References: +* ================ +*> [1] A Level 3 BLAS QR factorization algorithm with column pivoting developed in 1996. +*> G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain. +*> X. Sun, Computer Science Dept., Duke University, USA. +*> C. H. Bischof, Math. and Comp. Sci. Div., Argonne National Lab, USA. +*> A BLAS-3 version of the QR factorization with column pivoting. +*> LAPACK Working Note 114 +*> \htmlonly +*> https://www.netlib.org/lapack/lawnspdf/lawn114.pdf +*> \endhtmlonly +*> and in +*> SIAM J. Sci. Comput., 19(5):1486-1494, Sept. 1998. +*> \htmlonly +*> https://doi.org/10.1137/S1064827595296732 +*> \endhtmlonly +*> +*> [2] A partial column norm updating strategy developed in 2006. +*> Z. Drmac and Z. Bujanovic, Dept. of Math., University of Zagreb, Croatia. +*> On the failure of rank revealing QR factorization software – a case study. +*> LAPACK Working Note 176. +*> \htmlonly +*> http://www.netlib.org/lapack/lawnspdf/lawn176.pdf +*> \endhtmlonly +*> and in +*> ACM Trans. Math. Softw. 35, 2, Article 12 (July 2008), 28 pages. +*> \htmlonly +*> https://doi.org/10.1145/1377612.1377616 +*> \endhtmlonly +* +*> \par Contributors: +* ================== +*> +*> \verbatim +*> +*> November 2023, Igor Kozachenko, James Demmel, +*> Computer Science Division, +*> University of California, Berkeley +*> +*> \endverbatim +* +* ===================================================================== + SUBROUTINE ZLAQP2RK( M, N, NRHS, IOFFSET, KMAX, ABSTOL, RELTOL, + $ KP1, MAXC2NRM, A, LDA, K, MAXC2NRMK, + $ RELMAXC2NRMK, JPIV, TAU, VN1, VN2, WORK, + $ INFO ) + IMPLICIT NONE +* +* -- LAPACK auxiliary routine -- +* -- LAPACK is a software package provided by Univ. of Tennessee, -- +* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- +* +* .. Scalar Arguments .. + INTEGER INFO, IOFFSET, KP1, K, KMAX, LDA, M, N, NRHS + DOUBLE PRECISION ABSTOL, MAXC2NRM, MAXC2NRMK, RELMAXC2NRMK, + $ RELTOL +* .. +* .. Array Arguments .. + INTEGER JPIV( * ) + DOUBLE PRECISION VN1( * ), VN2( * ) + COMPLEX*16 A( LDA, * ), TAU( * ), WORK( * ) +* .. +* +* ===================================================================== +* +* .. Parameters .. + DOUBLE PRECISION ZERO, ONE + PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) + COMPLEX*16 CZERO, CONE + PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ), + $ CONE = ( 1.0D+0, 0.0D+0 ) ) +* .. +* .. Local Scalars .. + INTEGER I, ITEMP, J, JMAXC2NRM, KK, KP, MINMNFACT, + $ MINMNUPDT + DOUBLE PRECISION HUGEVAL, TAUNAN, TEMP, TEMP2, TOL3Z + COMPLEX*16 AIKK +* .. +* .. External Subroutines .. + EXTERNAL ZLARF, ZLARFG, ZSWAP +* .. +* .. Intrinsic Functions .. + INTRINSIC ABS, DBLE, DCONJG, DIMAG, MAX, MIN, SQRT +* .. +* .. External Functions .. + LOGICAL DISNAN + INTEGER IDAMAX + DOUBLE PRECISION DLAMCH, DZNRM2 + EXTERNAL DISNAN, DLAMCH, IDAMAX, DZNRM2 +* .. +* .. Executable Statements .. +* +* Initialize INFO +* + INFO = 0 +* +* MINMNFACT in the smallest dimension of the submatrix +* A(IOFFSET+1:M,1:N) to be factorized. +* +* MINMNUPDT is the smallest dimension +* of the subarray A(IOFFSET+1:M,1:N+NRHS) to be udated, which +* contains the submatrices A(IOFFSET+1:M,1:N) and +* B(IOFFSET+1:M,1:NRHS) as column blocks. +* + MINMNFACT = MIN( M-IOFFSET, N ) + MINMNUPDT = MIN( M-IOFFSET, N+NRHS ) + KMAX = MIN( KMAX, MINMNFACT ) + TOL3Z = SQRT( DLAMCH( 'Epsilon' ) ) + HUGEVAL = DLAMCH( 'Overflow' ) +* +* Compute the factorization, KK is the lomn loop index. +* + DO KK = 1, KMAX +* + I = IOFFSET + KK +* + IF( I.EQ.1 ) THEN +* +* ============================================================ +* +* We are at the first column of the original whole matrix A, +* therefore we use the computed KP1 and MAXC2NRM from the +* main routine. +* + KP = KP1 +* +* ============================================================ +* + ELSE +* +* ============================================================ +* +* Determine the pivot column in KK-th step, i.e. the index +* of the column with the maximum 2-norm in the +* submatrix A(I:M,K:N). +* + KP = ( KK-1 ) + IDAMAX( N-KK+1, VN1( KK ), 1 ) +* +* Determine the maximum column 2-norm and the relative maximum +* column 2-norm of the submatrix A(I:M,KK:N) in step KK. +* RELMAXC2NRMK will be computed later, after somecondition +* checks on MAXC2NRMK. +* + MAXC2NRMK = VN1( KP ) +* +* ============================================================ +* +* Check if the submatrix A(I:M,KK:N) contains NaN, and set +* INFO parameter to the column number, where the first NaN +* is found and return from the routine. +* We need to check the condition only if the +* column index (same as row index) of the original whole +* matrix is larger than 1, since the condition for whole +* original matrix is checked in the main routine. +* + IF( DISNAN( MAXC2NRMK ) ) THEN +* +* Set K, the number of factorized columns. +* that are not zero. +* + K = KK - 1 + INFO = K + KP +* +* Set RELMAXC2NRMK to NaN. +* + RELMAXC2NRMK = MAXC2NRMK +* +* Array TAU(K+1:MINMNFACT) is not set and contains +* undefined elements. +* + RETURN + END IF +* +* ============================================================ +* +* Quick return, if the submatrix A(I:M,KK:N) is +* a zero matrix. +* We need to check the condition only if the +* column index (same as row index) of the original whole +* matrix is larger than 1, since the condition for whole +* original matrix is checked in the main routine. +* + IF( MAXC2NRMK.EQ.ZERO ) THEN +* +* Set K, the number of factorized columns. +* that are not zero. +* + K = KK - 1 + RELMAXC2NRMK = ZERO +* +* Set TAUs corresponding to the columns that were not +* factorized to ZERO, i.e. set TAU(KK:MINMNFACT) to CZERO. +* + DO J = KK, MINMNFACT + TAU( J ) = CZERO + END DO +* +* Return from the routine. +* + RETURN +* + END IF +* +* ============================================================ +* +* Check if the submatrix A(I:M,KK:N) contains Inf, +* set INFO parameter to the column number, where +* the first Inf is found plus N, and continue +* the computation. +* We need to check the condition only if the +* column index (same as row index) of the original whole +* matrix is larger than 1, since the condition for whole +* original matrix is checked in the main routine. +* + IF( INFO.EQ.0 .AND. MAXC2NRMK.GT.HUGEVAL ) THEN + INFO = N + KK - 1 + KP + END IF +* +* ============================================================ +* +* Test for the second and third stopping criteria. +* NOTE: There is no need to test for ABSTOL >= ZERO, since +* MAXC2NRMK is non-negative. Similarly, there is no need +* to test for RELTOL >= ZERO, since RELMAXC2NRMK is +* non-negative. +* We need to check the condition only if the +* column index (same as row index) of the original whole +* matrix is larger than 1, since the condition for whole +* original matrix is checked in the main routine. + + RELMAXC2NRMK = MAXC2NRMK / MAXC2NRM +* + IF( MAXC2NRMK.LE.ABSTOL .OR. RELMAXC2NRMK.LE.RELTOL ) THEN +* +* Set K, the number of factorized columns. +* + K = KK - 1 +* +* Set TAUs corresponding to the columns that were not +* factorized to ZERO, i.e. set TAU(KK:MINMNFACT) to CZERO. +* + DO J = KK, MINMNFACT + TAU( J ) = CZERO + END DO +* +* Return from the routine. +* + RETURN +* + END IF +* +* ============================================================ +* +* End ELSE of IF(I.EQ.1) +* + END IF +* +* =============================================================== +* +* If the pivot column is not the first column of the +* subblock A(1:M,KK:N): +* 1) swap the KK-th column and the KP-th pivot column +* in A(1:M,1:N); +* 2) copy the KK-th element into the KP-th element of the partial +* and exact 2-norm vectors VN1 and VN2. ( Swap is not needed +* for VN1 and VN2 since we use the element with the index +* larger than KK in the next loop step.) +* 3) Save the pivot interchange with the indices relative to the +* the original matrix A, not the block A(1:M,1:N). +* + IF( KP.NE.KK ) THEN + CALL ZSWAP( M, A( 1, KP ), 1, A( 1, KK ), 1 ) + VN1( KP ) = VN1( KK ) + VN2( KP ) = VN2( KK ) + ITEMP = JPIV( KP ) + JPIV( KP ) = JPIV( KK ) + JPIV( KK ) = ITEMP + END IF +* +* Generate elementary reflector H(KK) using the column A(I:M,KK), +* if the column has more than one element, otherwise +* the elementary reflector would be an identity matrix, +* and TAU(KK) = CZERO. +* + IF( I.LT.M ) THEN + CALL ZLARFG( M-I+1, A( I, KK ), A( I+1, KK ), 1, + $ TAU( KK ) ) + ELSE + TAU( KK ) = CZERO + END IF +* +* Check if TAU(KK) contains NaN, set INFO parameter +* to the column number where NaN is found and return from +* the routine. +* NOTE: There is no need to check TAU(KK) for Inf, +* since ZLARFG cannot produce TAU(KK) or Householder vector +* below the diagonal containing Inf. Only BETA on the diagonal, +* returned by ZLARFG can contain Inf, which requires +* TAU(KK) to contain NaN. Therefore, this case of generating Inf +* by ZLARFG is covered by checking TAU(KK) for NaN. +* + IF( DISNAN( DBLE( TAU(KK) ) ) ) THEN + TAUNAN = DBLE( TAU(KK) ) + ELSE IF( DISNAN( DIMAG( TAU(KK) ) ) ) THEN + TAUNAN = DIMAG( TAU(KK) ) + ELSE + TAUNAN = ZERO + END IF +* + IF( DISNAN( TAUNAN ) ) THEN + K = KK - 1 + INFO = KK +* +* Set MAXC2NRMK and RELMAXC2NRMK to NaN. +* + MAXC2NRMK = TAUNAN + RELMAXC2NRMK = TAUNAN +* +* Array TAU(KK:MINMNFACT) is not set and contains +* undefined elements, except the first element TAU(KK) = NaN. +* + RETURN + END IF +* +* Apply H(KK)**H to A(I:M,KK+1:N+NRHS) from the left. +* ( If M >= N, then at KK = N there is no residual matrix, +* i.e. no columns of A to update, only columns of B. +* If M < N, then at KK = M-IOFFSET, I = M and we have a +* one-row residual matrix in A and the elementary +* reflector is a unit matrix, TAU(KK) = CZERO, i.e. no update +* is needed for the residual matrix in A and the +* right-hand-side-matrix in B. +* Therefore, we update only if +* KK < MINMNUPDT = min(M-IOFFSET, N+NRHS) +* condition is satisfied, not only KK < N+NRHS ) +* + IF( KK.LT.MINMNUPDT ) THEN + AIKK = A( I, KK ) + A( I, KK ) = CONE + CALL ZLARF( 'Left', M-I+1, N+NRHS-KK, A( I, KK ), 1, + $ DCONJG( TAU( KK ) ), A( I, KK+1 ), LDA, + $ WORK( 1 ) ) + A( I, KK ) = AIKK + END IF +* + IF( KK.LT.MINMNFACT ) THEN +* +* Update the partial column 2-norms for the residual matrix, +* only if the residual matrix A(I+1:M,KK+1:N) exists, i.e. +* when KK < min(M-IOFFSET, N). +* + DO J = KK + 1, N + IF( VN1( J ).NE.ZERO ) THEN +* +* NOTE: The following lines follow from the analysis in +* Lapack Working Note 176. +* + TEMP = ONE - ( ABS( A( I, J ) ) / VN1( J ) )**2 + TEMP = MAX( TEMP, ZERO ) + TEMP2 = TEMP*( VN1( J ) / VN2( J ) )**2 + IF( TEMP2 .LE. TOL3Z ) THEN +* +* Compute the column 2-norm for the partial +* column A(I+1:M,J) by explicitly computing it, +* and store it in both partial 2-norm vector VN1 +* and exact column 2-norm vector VN2. +* + VN1( J ) = DZNRM2( M-I, A( I+1, J ), 1 ) + VN2( J ) = VN1( J ) +* + ELSE +* +* Update the column 2-norm for the partial +* column A(I+1:M,J) by removing one +* element A(I,J) and store it in partial +* 2-norm vector VN1. +* + VN1( J ) = VN1( J )*SQRT( TEMP ) +* + END IF + END IF + END DO +* + END IF +* +* End factorization loop +* + END DO +* +* If we reached this point, all colunms have been factorized, +* i.e. no condition was triggered to exit the routine. +* Set the number of factorized columns. +* + K = KMAX +* +* We reached the end of the loop, i.e. all KMAX columns were +* factorized, we need to set MAXC2NRMK and RELMAXC2NRMK before +* we return. +* + IF( K.LT.MINMNFACT ) THEN +* + JMAXC2NRM = K + IDAMAX( N-K, VN1( K+1 ), 1 ) + MAXC2NRMK = VN1( JMAXC2NRM ) +* + IF( K.EQ.0 ) THEN + RELMAXC2NRMK = ONE + ELSE + RELMAXC2NRMK = MAXC2NRMK / MAXC2NRM + END IF +* + ELSE + MAXC2NRMK = ZERO + RELMAXC2NRMK = ZERO + END IF +* +* We reached the end of the loop, i.e. all KMAX columns were +* factorized, set TAUs corresponding to the columns that were +* not factorized to ZERO, i.e. TAU(K+1:MINMNFACT) set to CZERO. +* + DO J = K + 1, MINMNFACT + TAU( J ) = CZERO + END DO +* + RETURN +* +* End of ZLAQP2RK +* + END diff --git a/SRC/zlaqp3rk.f b/SRC/zlaqp3rk.f new file mode 100755 index 0000000000..7a9fdfd95b --- /dev/null +++ b/SRC/zlaqp3rk.f @@ -0,0 +1,947 @@ +*> \brief \b ZLAQP3RK computes a step of truncated QR factorization with column pivoting of a complex m-by-n matrix A using Level 3 BLAS and overwrites a complex m-by-nrhs matrix B with Q**H * B. +* +* =========== DOCUMENTATION =========== +* +* Online html documentation available at +* http://www.netlib.org/lapack/explore-html/ +* +*> \htmlonly +*> Download ZLAQP3RK + dependencies +*> +*> [TGZ] +*> +*> [ZIP] +*> +*> [TXT] +*> \endhtmlonly +* +* Definition: +* =========== +* +* SUBROUTINE ZLAQP3RK( M, N, NRHS, IOFFSET, NB, ABSTOL, +* $ RELTOL, KP1, MAXC2NRM, A, LDA, DONE, KB, +* $ MAXC2NRMK, RELMAXC2NRMK, JPIV, TAU, +* $ VN1, VN2, AUXV, F, LDF, IWORK, INFO ) +* IMPLICIT NONE +* LOGICAL DONE +* INTEGER INFO, IOFFSET, KB, KP1, LDA, LDF, M, N, +* $ NB, NRHS +* DOUBLE PRECISION ABSTOL, MAXC2NRM, MAXC2NRMK, RELMAXC2NRMK, +* $ RELTOL +* .. +* .. Array Arguments .. +* INTEGER IWORK( * ), JPIV( * ) +* DOUBLE PRECISION VN1( * ), VN2( * ) +* COMPLEX*16 A( LDA, * ), AUXV( * ), F( LDF, * ), TAU( * ) +* .. +* +* +*> \par Purpose: +* ============= +*> +*> \verbatim +*> +*> ZLAQP3RK computes a step of truncated QR factorization with column +*> pivoting of a complex M-by-N matrix A block A(IOFFSET+1:M,1:N) +*> by using Level 3 BLAS as +*> +*> A * P(KB) = Q(KB) * R(KB). +*> +*> The routine tries to factorize NB columns from A starting from +*> the row IOFFSET+1 and updates the residual matrix with BLAS 3 +*> xGEMM. The number of actually factorized columns is returned +*> is smaller than NB. +*> +*> Block A(1:IOFFSET,1:N) is accordingly pivoted, but not factorized. +*> +*> The routine also overwrites the right-hand-sides B matrix stored +*> in A(IOFFSET+1:M,1:N+1:N+NRHS) with Q(KB)**H * B. +*> +*> Cases when the number of factorized columns KB < NB: +*> +*> (1) In some cases, due to catastrophic cancellations, it cannot +*> factorize all NB columns and need to update the residual matrix. +*> Hence, the actual number of factorized columns in the block returned +*> in KB is smaller than NB. The logical DONE is returned as FALSE. +*> The factorization of the whole original matrix A_orig must proceed +*> with the next block. +*> +*> (2) Whenever the stopping criterion ABSTOL or RELTOL is satisfied, +*> the factorization of the whole original matrix A_orig is stopped, +*> the logical DONE is returned as TRUE. The number of factorized +*> columns which is smaller than NB is returned in KB. +*> +*> (3) In case both stopping criteria ABSTOL or RELTOL are not used, +*> and when the residual matrix is a zero matrix in some factorization +*> step KB, the factorization of the whole original matrix A_orig is +*> stopped, the logical DONE is returned as TRUE. The number of +*> factorized columns which is smaller than NB is returned in KB. +*> +*> (4) Whenever NaN is detected in the matrix A or in the array TAU, +*> the factorization of the whole original matrix A_orig is stopped, +*> the logical DONE is returned as TRUE. The number of factorized +*> columns which is smaller than NB is returned in KB. The INFO +*> parameter is set to the column index of the first NaN occurrence. +*> +*> \endverbatim +* +* Arguments: +* ========== +* +*> \param[in] M +*> \verbatim +*> M is INTEGER +*> The number of rows of the matrix A. M >= 0. +*> \endverbatim +*> +*> \param[in] N +*> \verbatim +*> N is INTEGER +*> The number of columns of the matrix A. N >= 0 +*> \endverbatim +*> +*> \param[in] NRHS +*> \verbatim +*> NRHS is INTEGER +*> The number of right hand sides, i.e., the number of +*> columns of the matrix B. NRHS >= 0. +*> \endverbatim +*> +*> \param[in] IOFFSET +*> \verbatim +*> IOFFSET is INTEGER +*> The number of rows of the matrix A that must be pivoted +*> but not factorized. IOFFSET >= 0. +*> +*> IOFFSET also represents the number of columns of the whole +*> original matrix A_orig that have been factorized +*> in the previous steps. +*> \endverbatim +*> +*> \param[in] NB +*> \verbatim +*> NB is INTEGER +*> Factorization block size, i.e the number of columns +*> to factorize in the matrix A. 0 <= NB +*> +*> If NB = 0, then the routine exits immediately. +*> This means that the factorization is not performed, +*> the matrices A and B and the arrays TAU, IPIV +*> are not modified. +*> \endverbatim +*> +*> \param[in] ABSTOL +*> \verbatim +*> ABSTOL is DOUBLE PRECISION, cannot be NaN. +*> +*> The absolute tolerance (stopping threshold) for +*> maximum column 2-norm of the residual matrix. +*> The algorithm converges (stops the factorization) when +*> the maximum column 2-norm of the residual matrix +*> is less than or equal to ABSTOL. +*> +*> a) If ABSTOL < 0.0, then this stopping criterion is not +*> used, the routine factorizes columns depending +*> on NB and RELTOL. +*> This includes the case ABSTOL = -Inf. +*> +*> b) If 0.0 <= ABSTOL then the input value +*> of ABSTOL is used. +*> \endverbatim +*> +*> \param[in] RELTOL +*> \verbatim +*> RELTOL is DOUBLE PRECISION, cannot be NaN. +*> +*> The tolerance (stopping threshold) for the ratio of the +*> maximum column 2-norm of the residual matrix to the maximum +*> column 2-norm of the original matrix A_orig. The algorithm +*> converges (stops the factorization), when this ratio is +*> less than or equal to RELTOL. +*> +*> a) If RELTOL < 0.0, then this stopping criterion is not +*> used, the routine factorizes columns depending +*> on NB and ABSTOL. +*> This includes the case RELTOL = -Inf. +*> +*> d) If 0.0 <= RELTOL then the input value of RELTOL +*> is used. +*> \endverbatim +*> +*> \param[in] KP1 +*> \verbatim +*> KP1 is INTEGER +*> The index of the column with the maximum 2-norm in +*> the whole original matrix A_orig determined in the +*> main routine ZGEQP3RK. 1 <= KP1 <= N_orig. +*> \endverbatim +*> +*> \param[in] MAXC2NRM +*> \verbatim +*> MAXC2NRM is DOUBLE PRECISION +*> The maximum column 2-norm of the whole original +*> matrix A_orig computed in the main routine ZGEQP3RK. +*> MAXC2NRM >= 0. +*> \endverbatim +*> +*> \param[in,out] A +*> \verbatim +*> A is COMPLEX*16 array, dimension (LDA,N+NRHS) +*> On entry: +*> the M-by-N matrix A and M-by-NRHS matrix B, as in +*> +*> N NRHS +*> array_A = M [ mat_A, mat_B ] +*> +*> On exit: +*> 1. The elements in block A(IOFFSET+1:M,1:KB) below +*> the diagonal together with the array TAU represent +*> the orthogonal matrix Q(KB) as a product of elementary +*> reflectors. +*> 2. The upper triangular block of the matrix A stored +*> in A(IOFFSET+1:M,1:KB) is the triangular factor obtained. +*> 3. The block of the matrix A stored in A(1:IOFFSET,1:N) +*> has been accordingly pivoted, but not factorized. +*> 4. The rest of the array A, block A(IOFFSET+1:M,KB+1:N+NRHS). +*> The left part A(IOFFSET+1:M,KB+1:N) of this block +*> contains the residual of the matrix A, and, +*> if NRHS > 0, the right part of the block +*> A(IOFFSET+1:M,N+1:N+NRHS) contains the block of +*> the right-hand-side matrix B. Both these blocks have been +*> updated by multiplication from the left by Q(KB)**H. +*> \endverbatim +*> +*> \param[in] LDA +*> \verbatim +*> LDA is INTEGER +*> The leading dimension of the array A. LDA >= max(1,M). +*> \endverbatim +*> +*> \param[out] +*> \verbatim +*> DONE is LOGICAL +*> TRUE: a) if the factorization completed before processing +*> all min(M-IOFFSET,NB,N) columns due to ABSTOL +*> or RELTOL criterion, +*> b) if the factorization completed before processing +*> all min(M-IOFFSET,NB,N) columns due to the +*> residual matrix being a ZERO matrix. +*> c) when NaN was detected in the matrix A +*> or in the array TAU. +*> FALSE: otherwise. +*> \endverbatim +*> +*> \param[out] KB +*> \verbatim +*> KB is INTEGER +*> Factorization rank of the matrix A, i.e. the rank of +*> the factor R, which is the same as the number of non-zero +*> rows of the factor R. 0 <= KB <= min(M-IOFFSET,NB,N). +*> +*> KB also represents the number of non-zero Householder +*> vectors. +*> \endverbatim +*> +*> \param[out] MAXC2NRMK +*> \verbatim +*> MAXC2NRMK is DOUBLE PRECISION +*> The maximum column 2-norm of the residual matrix, +*> when the factorization stopped at rank KB. MAXC2NRMK >= 0. +*> \endverbatim +*> +*> \param[out] RELMAXC2NRMK +*> \verbatim +*> RELMAXC2NRMK is DOUBLE PRECISION +*> The ratio MAXC2NRMK / MAXC2NRM of the maximum column +*> 2-norm of the residual matrix (when the factorization +*> stopped at rank KB) to the maximum column 2-norm of the +*> original matrix A_orig. RELMAXC2NRMK >= 0. +*> \endverbatim +*> +*> \param[out] JPIV +*> \verbatim +*> JPIV is INTEGER array, dimension (N) +*> Column pivot indices, for 1 <= j <= N, column j +*> of the matrix A was interchanged with column JPIV(j). +*> \endverbatim +*> +*> \param[out] TAU +*> \verbatim +*> TAU is COMPLEX*16 array, dimension (min(M-IOFFSET,N)) +*> The scalar factors of the elementary reflectors. +*> \endverbatim +*> +*> \param[in,out] VN1 +*> \verbatim +*> VN1 is DOUBLE PRECISION array, dimension (N) +*> The vector with the partial column norms. +*> \endverbatim +*> +*> \param[in,out] VN2 +*> \verbatim +*> VN2 is DOUBLE PRECISION array, dimension (N) +*> The vector with the exact column norms. +*> \endverbatim +*> +*> \param[out] AUXV +*> \verbatim +*> AUXV is COMPLEX*16 array, dimension (NB) +*> Auxiliary vector. +*> \endverbatim +*> +*> \param[out] F +*> \verbatim +*> F is COMPLEX*16 array, dimension (LDF,NB) +*> Matrix F**H = L*(Y**H)*A. +*> \endverbatim +*> +*> \param[in] LDF +*> \verbatim +*> LDF is INTEGER +*> The leading dimension of the array F. LDF >= max(1,N+NRHS). +*> \endverbatim +*> +*> \param[out] IWORK +*> \verbatim +*> IWORK is INTEGER array, dimension (N-1). +*> Is a work array. ( IWORK is used to store indices +*> of "bad" columns for norm downdating in the residual +*> matrix ). +*> \endverbatim +*> +*> \param[out] INFO +*> \verbatim +*> INFO is INTEGER +*> 1) INFO = 0: successful exit. +*> 2) If INFO = j_1, where 1 <= j_1 <= N, then NaN was +*> detected and the routine stops the computation. +*> The j_1-th column of the matrix A or the j_1-th +*> element of array TAU contains the first occurrence +*> of NaN in the factorization step KB+1 ( when KB columns +*> have been factorized ). +*> +*> On exit: +*> KB is set to the number of +*> factorized columns without +*> exception. +*> MAXC2NRMK is set to NaN. +*> RELMAXC2NRMK is set to NaN. +*> TAU(KB+1:min(M,N)) is not set and contains undefined +*> elements. If j_1=KB+1, TAU(KB+1) +*> may contain NaN. +*> 3) If INFO = j_2, where N+1 <= j_2 <= 2*N, then no NaN +*> was detected, but +Inf (or -Inf) was detected and +*> the routine continues the computation until completion. +*> The (j_2-N)-th column of the matrix A contains the first +*> occurrence of +Inf (or -Inf) in the actorization +*> step KB+1 ( when KB columns have been factorized ). +*> \endverbatim +* +* Authors: +* ======== +* +*> \author Univ. of Tennessee +*> \author Univ. of California Berkeley +*> \author Univ. of Colorado Denver +*> \author NAG Ltd. +* +*> \ingroup laqp3rk +* +*> \par References: +* ================ +*> [1] A Level 3 BLAS QR factorization algorithm with column pivoting developed in 1996. +*> G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain. +*> X. Sun, Computer Science Dept., Duke University, USA. +*> C. H. Bischof, Math. and Comp. Sci. Div., Argonne National Lab, USA. +*> A BLAS-3 version of the QR factorization with column pivoting. +*> LAPACK Working Note 114 +*> \htmlonly +*> https://www.netlib.org/lapack/lawnspdf/lawn114.pdf +*> \endhtmlonly +*> and in +*> SIAM J. Sci. Comput., 19(5):1486-1494, Sept. 1998. +*> \htmlonly +*> https://doi.org/10.1137/S1064827595296732 +*> \endhtmlonly +*> +*> [2] A partial column norm updating strategy developed in 2006. +*> Z. Drmac and Z. Bujanovic, Dept. of Math., University of Zagreb, Croatia. +*> On the failure of rank revealing QR factorization software – a case study. +*> LAPACK Working Note 176. +*> \htmlonly +*> http://www.netlib.org/lapack/lawnspdf/lawn176.pdf +*> \endhtmlonly +*> and in +*> ACM Trans. Math. Softw. 35, 2, Article 12 (July 2008), 28 pages. +*> \htmlonly +*> https://doi.org/10.1145/1377612.1377616 +*> \endhtmlonly +* +*> \par Contributors: +* ================== +*> +*> \verbatim +*> +*> November 2023, Igor Kozachenko, James Demmel, +*> Computer Science Division, +*> University of California, Berkeley +*> +*> \endverbatim +* +* ===================================================================== + SUBROUTINE ZLAQP3RK( M, N, NRHS, IOFFSET, NB, ABSTOL, + $ RELTOL, KP1, MAXC2NRM, A, LDA, DONE, KB, + $ MAXC2NRMK, RELMAXC2NRMK, JPIV, TAU, + $ VN1, VN2, AUXV, F, LDF, IWORK, INFO ) + IMPLICIT NONE +* +* -- LAPACK auxiliary routine -- +* -- LAPACK is a software package provided by Univ. of Tennessee, -- +* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- +* +* .. Scalar Arguments .. + LOGICAL DONE + INTEGER INFO, IOFFSET, KB, KP1, LDA, LDF, M, N, + $ NB, NRHS + DOUBLE PRECISION ABSTOL, MAXC2NRM, MAXC2NRMK, RELMAXC2NRMK, + $ RELTOL +* .. +* .. Array Arguments .. + INTEGER IWORK( * ), JPIV( * ) + DOUBLE PRECISION VN1( * ), VN2( * ) + COMPLEX*16 A( LDA, * ), AUXV( * ), F( LDF, * ), TAU( * ) +* .. +* +* ===================================================================== +* +* .. Parameters .. + DOUBLE PRECISION ZERO, ONE + PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) + COMPLEX*16 CZERO, CONE + PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ), + $ CONE = ( 1.0D+0, 0.0D+0 ) ) +* .. +* .. Local Scalars .. + INTEGER ITEMP, J, K, MINMNFACT, MINMNUPDT, + $ LSTICC, KP, I, IF + DOUBLE PRECISION HUGEVAL, TAUNAN, TEMP, TEMP2, TOL3Z + COMPLEX*16 AIK +* .. +* .. External Subroutines .. + EXTERNAL ZGEMM, ZGEMV, ZLARFG, ZSWAP +* .. +* .. Intrinsic Functions .. + INTRINSIC ABS, DBLE, DCONJG, DIMAG, MAX, MIN, SQRT +* .. +* .. External Functions .. + LOGICAL DISNAN + INTEGER IDAMAX + DOUBLE PRECISION DLAMCH, DZNRM2 + EXTERNAL DISNAN, DLAMCH, IDAMAX, DZNRM2 +* .. +* .. Executable Statements .. +* +* Initialize INFO +* + INFO = 0 +* +* MINMNFACT in the smallest dimension of the submatrix +* A(IOFFSET+1:M,1:N) to be factorized. +* + MINMNFACT = MIN( M-IOFFSET, N ) + MINMNUPDT = MIN( M-IOFFSET, N+NRHS ) + NB = MIN( NB, MINMNFACT ) + TOL3Z = SQRT( DLAMCH( 'Epsilon' ) ) + HUGEVAL = DLAMCH( 'Overflow' ) +* +* Compute factorization in a while loop over NB columns, +* K is the column index in the block A(1:M,1:N). +* + K = 0 + LSTICC = 0 + DONE = .FALSE. +* + DO WHILE ( K.LT.NB .AND. LSTICC.EQ.0 ) + K = K + 1 + I = IOFFSET + K +* + IF( I.EQ.1 ) THEN +* +* We are at the first column of the original whole matrix A_orig, +* therefore we use the computed KP1 and MAXC2NRM from the +* main routine. +* + KP = KP1 +* + ELSE +* +* Determine the pivot column in K-th step, i.e. the index +* of the column with the maximum 2-norm in the +* submatrix A(I:M,K:N). +* + KP = ( K-1 ) + IDAMAX( N-K+1, VN1( K ), 1 ) +* +* Determine the maximum column 2-norm and the relative maximum +* column 2-norm of the submatrix A(I:M,K:N) in step K. +* + MAXC2NRMK = VN1( KP ) +* +* ============================================================ +* +* Check if the submatrix A(I:M,K:N) contains NaN, set +* INFO parameter to the column number, where the first NaN +* is found and return from the routine. +* We need to check the condition only if the +* column index (same as row index) of the original whole +* matrix is larger than 1, since the condition for whole +* original matrix is checked in the main routine. +* + IF( DISNAN( MAXC2NRMK ) ) THEN +* + DONE = .TRUE. +* +* Set KB, the number of factorized partial columns +* that are non-zero in each step in the block, +* i.e. the rank of the factor R. +* Set IF, the number of processed rows in the block, which +* is the same as the number of processed rows in +* the original whole matrix A_orig. +* + KB = K - 1 + IF = I - 1 + INFO = KB + KP +* +* Set RELMAXC2NRMK to NaN. +* + RELMAXC2NRMK = MAXC2NRMK +* +* There is no need to apply the block reflector to the +* residual of the matrix A stored in A(KB+1:M,KB+1:N), +* since the submatrix contains NaN and we stop +* the computation. +* But, we need to apply the block reflector to the residual +* right hand sides stored in A(KB+1:M,N+1:N+NRHS), if the +* residual right hand sides exist. This occurs +* when ( NRHS != 0 AND KB <= (M-IOFFSET) ): +* +* A(I+1:M,N+1:N+NRHS) := A(I+1:M,N+1:N+NRHS) - +* A(I+1:M,1:KB) * F(N+1:N+NRHS,1:KB)**H. + + IF( NRHS.GT.0 .AND. KB.LT.(M-IOFFSET) ) THEN + CALL ZGEMM( 'No transpose', 'Conjugate transpose', + $ M-IF, NRHS, KB, -CONE, A( IF+1, 1 ), LDA, + $ F( N+1, 1 ), LDF, CONE, A( IF+1, N+1 ), LDA ) + END IF +* +* There is no need to recompute the 2-norm of the +* difficult columns, since we stop the factorization. +* +* Array TAU(KF+1:MINMNFACT) is not set and contains +* undefined elements. +* +* Return from the routine. +* + RETURN + END IF +* +* Quick return, if the submatrix A(I:M,K:N) is +* a zero matrix. We need to check it only if the column index +* (same as row index) is larger than 1, since the condition +* for the whole original matrix A_orig is checked in the main +* routine. +* + IF( MAXC2NRMK.EQ.ZERO ) THEN +* + DONE = .TRUE. +* +* Set KB, the number of factorized partial columns +* that are non-zero in each step in the block, +* i.e. the rank of the factor R. +* Set IF, the number of processed rows in the block, which +* is the same as the number of processed rows in +* the original whole matrix A_orig. +* + KB = K - 1 + IF = I - 1 + RELMAXC2NRMK = ZERO +* +* There is no need to apply the block reflector to the +* residual of the matrix A stored in A(KB+1:M,KB+1:N), +* since the submatrix is zero and we stop the computation. +* But, we need to apply the block reflector to the residual +* right hand sides stored in A(KB+1:M,N+1:N+NRHS), if the +* residual right hand sides exist. This occurs +* when ( NRHS != 0 AND KB <= (M-IOFFSET) ): +* +* A(I+1:M,N+1:N+NRHS) := A(I+1:M,N+1:N+NRHS) - +* A(I+1:M,1:KB) * F(N+1:N+NRHS,1:KB)**H. +* + IF( NRHS.GT.0 .AND. KB.LT.(M-IOFFSET) ) THEN + CALL ZGEMM( 'No transpose', 'Conjugate transpose', + $ M-IF, NRHS, KB, -CONE, A( IF+1, 1 ), LDA, + $ F( N+1, 1 ), LDF, CONE, A( IF+1, N+1 ), LDA ) + END IF +* +* There is no need to recompute the 2-norm of the +* difficult columns, since we stop the factorization. +* +* Set TAUs corresponding to the columns that were not +* factorized to ZERO, i.e. set TAU(KB+1:MINMNFACT) = CZERO, +* which is equivalent to seting TAU(K:MINMNFACT) = CZERO. +* + DO J = K, MINMNFACT + TAU( J ) = CZERO + END DO +* +* Return from the routine. +* + RETURN +* + END IF +* +* ============================================================ +* +* Check if the submatrix A(I:M,K:N) contains Inf, +* set INFO parameter to the column number, where +* the first Inf is found plus N, and continue +* the computation. +* We need to check the condition only if the +* column index (same as row index) of the original whole +* matrix is larger than 1, since the condition for whole +* original matrix is checked in the main routine. +* + IF( INFO.EQ.0 .AND. MAXC2NRMK.GT.HUGEVAL ) THEN + INFO = N + K - 1 + KP + END IF +* +* ============================================================ +* +* Test for the second and third tolerance stopping criteria. +* NOTE: There is no need to test for ABSTOL.GE.ZERO, since +* MAXC2NRMK is non-negative. Similarly, there is no need +* to test for RELTOL.GE.ZERO, since RELMAXC2NRMK is +* non-negative. +* We need to check the condition only if the +* column index (same as row index) of the original whole +* matrix is larger than 1, since the condition for whole +* original matrix is checked in the main routine. +* + RELMAXC2NRMK = MAXC2NRMK / MAXC2NRM +* + IF( MAXC2NRMK.LE.ABSTOL .OR. RELMAXC2NRMK.LE.RELTOL ) THEN +* + DONE = .TRUE. +* +* Set KB, the number of factorized partial columns +* that are non-zero in each step in the block, +* i.e. the rank of the factor R. +* Set IF, the number of processed rows in the block, which +* is the same as the number of processed rows in +* the original whole matrix A_orig; +* + KB = K - 1 + IF = I - 1 +* +* Apply the block reflector to the residual of the +* matrix A and the residual of the right hand sides B, if +* the residual matrix and and/or the residual of the right +* hand sides exist, i.e. if the submatrix +* A(I+1:M,KB+1:N+NRHS) exists. This occurs when +* KB < MINMNUPDT = min( M-IOFFSET, N+NRHS ): +* +* A(IF+1:M,K+1:N+NRHS) := A(IF+1:M,KB+1:N+NRHS) - +* A(IF+1:M,1:KB) * F(KB+1:N+NRHS,1:KB)**H. +* + IF( KB.LT.MINMNUPDT ) THEN + CALL ZGEMM( 'No transpose', 'Conjugate transpose', + $ M-IF, N+NRHS-KB, KB,-CONE, A( IF+1, 1 ), LDA, + $ F( KB+1, 1 ), LDF, CONE, A( IF+1, KB+1 ), LDA ) + END IF +* +* There is no need to recompute the 2-norm of the +* difficult columns, since we stop the factorization. +* +* Set TAUs corresponding to the columns that were not +* factorized to ZERO, i.e. set TAU(KB+1:MINMNFACT) = CZERO, +* which is equivalent to seting TAU(K:MINMNFACT) = CZERO. +* + DO J = K, MINMNFACT + TAU( J ) = CZERO + END DO +* +* Return from the routine. +* + RETURN +* + END IF +* +* ============================================================ +* +* End ELSE of IF(I.EQ.1) +* + END IF +* +* =============================================================== +* +* If the pivot column is not the first column of the +* subblock A(1:M,K:N): +* 1) swap the K-th column and the KP-th pivot column +* in A(1:M,1:N); +* 2) swap the K-th row and the KP-th row in F(1:N,1:K-1) +* 3) copy the K-th element into the KP-th element of the partial +* and exact 2-norm vectors VN1 and VN2. (Swap is not needed +* for VN1 and VN2 since we use the element with the index +* larger than K in the next loop step.) +* 4) Save the pivot interchange with the indices relative to the +* the original matrix A_orig, not the block A(1:M,1:N). +* + IF( KP.NE.K ) THEN + CALL ZSWAP( M, A( 1, KP ), 1, A( 1, K ), 1 ) + CALL ZSWAP( K-1, F( KP, 1 ), LDF, F( K, 1 ), LDF ) + VN1( KP ) = VN1( K ) + VN2( KP ) = VN2( K ) + ITEMP = JPIV( KP ) + JPIV( KP ) = JPIV( K ) + JPIV( K ) = ITEMP + END IF +* +* Apply previous Householder reflectors to column K: +* A(I:M,K) := A(I:M,K) - A(I:M,1:K-1)*F(K,1:K-1)**H. +* + IF( K.GT.1 ) THEN + DO J = 1, K - 1 + F( K, J ) = DCONJG( F( K, J ) ) + END DO + CALL ZGEMV( 'No transpose', M-I+1, K-1, -CONE, A( I, 1 ), + $ LDA, F( K, 1 ), LDF, CONE, A( I, K ), 1 ) + DO J = 1, K - 1 + F( K, J ) = DCONJG( F( K, J ) ) + END DO + END IF +* +* Generate elementary reflector H(k) using the column A(I:M,K). +* + IF( I.LT.M ) THEN + CALL ZLARFG( M-I+1, A( I, K ), A( I+1, K ), 1, TAU( K ) ) + ELSE + TAU( K ) = CZERO + END IF +* +* Check if TAU(K) contains NaN, set INFO parameter +* to the column number where NaN is found and return from +* the routine. +* NOTE: There is no need to check TAU(K) for Inf, +* since ZLARFG cannot produce TAU(KK) or Householder vector +* below the diagonal containing Inf. Only BETA on the diagonal, +* returned by ZLARFG can contain Inf, which requires +* TAU(K) to contain NaN. Therefore, this case of generating Inf +* by ZLARFG is covered by checking TAU(K) for NaN. +* + IF( DISNAN( DBLE( TAU(K) ) ) ) THEN + TAUNAN = DBLE( TAU(K) ) + ELSE IF( DISNAN( DIMAG( TAU(K) ) ) ) THEN + TAUNAN = DIMAG( TAU(K) ) + ELSE + TAUNAN = ZERO + END IF +* + IF( DISNAN( TAUNAN ) ) THEN +* + DONE = .TRUE. +* +* Set KB, the number of factorized partial columns +* that are non-zero in each step in the block, +* i.e. the rank of the factor R. +* Set IF, the number of processed rows in the block, which +* is the same as the number of processed rows in +* the original whole matrix A_orig. +* + KB = K - 1 + IF = I - 1 + INFO = K +* +* Set MAXC2NRMK and RELMAXC2NRMK to NaN. +* + MAXC2NRMK = TAUNAN + RELMAXC2NRMK = TAUNAN +* +* There is no need to apply the block reflector to the +* residual of the matrix A stored in A(KB+1:M,KB+1:N), +* since the submatrix contains NaN and we stop +* the computation. +* But, we need to apply the block reflector to the residual +* right hand sides stored in A(KB+1:M,N+1:N+NRHS), if the +* residual right hand sides exist. This occurs +* when ( NRHS != 0 AND KB <= (M-IOFFSET) ): +* +* A(I+1:M,N+1:N+NRHS) := A(I+1:M,N+1:N+NRHS) - +* A(I+1:M,1:KB) * F(N+1:N+NRHS,1:KB)**H. +* + IF( NRHS.GT.0 .AND. KB.LT.(M-IOFFSET) ) THEN + CALL ZGEMM( 'No transpose', 'Conjugate transpose', + $ M-IF, NRHS, KB, -CONE, A( IF+1, 1 ), LDA, + $ F( N+1, 1 ), LDF, CONE, A( IF+1, N+1 ), LDA ) + END IF +* +* There is no need to recompute the 2-norm of the +* difficult columns, since we stop the factorization. +* +* Array TAU(KF+1:MINMNFACT) is not set and contains +* undefined elements. +* +* Return from the routine. +* + RETURN + END IF +* +* =============================================================== +* + AIK = A( I, K ) + A( I, K ) = CONE +* +* =============================================================== +* +* Compute the current K-th column of F: +* 1) F(K+1:N,K) := tau(K) * A(I:M,K+1:N)**H * A(I:M,K). +* + IF( K.LT.N+NRHS ) THEN + CALL ZGEMV( 'Conjugate transpose', M-I+1, N+NRHS-K, + $ TAU( K ), A( I, K+1 ), LDA, A( I, K ), 1, + $ CZERO, F( K+1, K ), 1 ) + END IF +* +* 2) Zero out elements above and on the diagonal of the +* column K in matrix F, i.e elements F(1:K,K). +* + DO J = 1, K + F( J, K ) = CZERO + END DO +* +* 3) Incremental updating of the K-th column of F: +* F(1:N,K) := F(1:N,K) - tau(K) * F(1:N,1:K-1) * A(I:M,1:K-1)**H +* * A(I:M,K). +* + IF( K.GT.1 ) THEN + CALL ZGEMV( 'Conjugate Transpose', M-I+1, K-1, -TAU( K ), + $ A( I, 1 ), LDA, A( I, K ), 1, CZERO, + $ AUXV( 1 ), 1 ) +* + CALL ZGEMV( 'No transpose', N+NRHS, K-1, CONE, + $ F( 1, 1 ), LDF, AUXV( 1 ), 1, CONE, + $ F( 1, K ), 1 ) + END IF +* +* =============================================================== +* +* Update the current I-th row of A: +* A(I,K+1:N+NRHS) := A(I,K+1:N+NRHS) +* - A(I,1:K)*F(K+1:N+NRHS,1:K)**H. +* + IF( K.LT.N+NRHS ) THEN + CALL ZGEMM( 'No transpose', 'Conjugate transpose', + $ 1, N+NRHS-K, K, -CONE, A( I, 1 ), LDA, + $ F( K+1, 1 ), LDF, CONE, A( I, K+1 ), LDA ) + END IF +* + A( I, K ) = AIK +* +* Update the partial column 2-norms for the residual matrix, +* only if the residual matrix A(I+1:M,K+1:N) exists, i.e. +* when K < MINMNFACT = min( M-IOFFSET, N ). +* + IF( K.LT.MINMNFACT ) THEN +* + DO J = K + 1, N + IF( VN1( J ).NE.ZERO ) THEN +* +* NOTE: The following lines follow from the analysis in +* Lapack Working Note 176. +* + TEMP = ABS( A( I, J ) ) / VN1( J ) + TEMP = MAX( ZERO, ( ONE+TEMP )*( ONE-TEMP ) ) + TEMP2 = TEMP*( VN1( J ) / VN2( J ) )**2 + IF( TEMP2.LE.TOL3Z ) THEN +* +* At J-index, we have a difficult column for the +* update of the 2-norm. Save the index of the previous +* difficult column in IWORK(J-1). +* NOTE: ILSTCC > 1, threfore we can use IWORK only +* with N-1 elements, where the elements are +* shifted by 1 to the left. +* + IWORK( J-1 ) = LSTICC +* +* Set the index of the last difficult column LSTICC. +* + LSTICC = J +* + ELSE + VN1( J ) = VN1( J )*SQRT( TEMP ) + END IF + END IF + END DO +* + END IF +* +* End of while loop. +* + END DO +* +* Now, afler the loop: +* Set KB, the number of factorized columns in the block; +* Set IF, the number of processed rows in the block, which +* is the same as the number of processed rows in +* the original whole matrix A_orig, IF = IOFFSET + KB. +* + KB = K + IF = I +* +* Apply the block reflector to the residual of the matrix A +* and the residual of the right hand sides B, if the residual +* matrix and and/or the residual of the right hand sides +* exist, i.e. if the submatrix A(I+1:M,KB+1:N+NRHS) exists. +* This occurs when KB < MINMNUPDT = min( M-IOFFSET, N+NRHS ): +* +* A(IF+1:M,K+1:N+NRHS) := A(IF+1:M,KB+1:N+NRHS) - +* A(IF+1:M,1:KB) * F(KB+1:N+NRHS,1:KB)**H. +* + IF( KB.LT.MINMNUPDT ) THEN + CALL ZGEMM( 'No transpose', 'Conjugate transpose', + $ M-IF, N+NRHS-KB, KB, -CONE, A( IF+1, 1 ), LDA, + $ F( KB+1, 1 ), LDF, CONE, A( IF+1, KB+1 ), LDA ) + END IF +* +* Recompute the 2-norm of the difficult columns. +* Loop over the index of the difficult columns from the largest +* to the smallest index. +* + DO WHILE( LSTICC.GT.0 ) +* +* LSTICC is the index of the last difficult column is greater +* than 1. +* ITEMP is the index of the previous difficult column. +* + ITEMP = IWORK( LSTICC-1 ) +* +* Compute the 2-norm explicilty for the last difficult column and +* save it in the partial and exact 2-norm vectors VN1 and VN2. +* +* NOTE: The computation of VN1( LSTICC ) relies on the fact that +* DZNRM2 does not fail on vectors with norm below the value of +* SQRT(DLAMCH('S')) +* + VN1( LSTICC ) = DZNRM2( M-IF, A( IF+1, LSTICC ), 1 ) + VN2( LSTICC ) = VN1( LSTICC ) +* +* Downdate the index of the last difficult column to +* the index of the previous difficult column. +* + LSTICC = ITEMP +* + END DO +* + RETURN +* +* End of ZLAQP3RK +* + END diff --git a/TESTING/EIG/alareq.f b/TESTING/EIG/alareq.f index 2585a686a0..2cbe6db382 100644 --- a/TESTING/EIG/alareq.f +++ b/TESTING/EIG/alareq.f @@ -28,12 +28,12 @@ *> to evaluate the input line which requested NMATS matrix types for *> PATH. The flow of control is as follows: *> -*> If NMATS = NTYPES then +*> IF NMATS = NTYPES THEN *> DOTYPE(1:NTYPES) = .TRUE. -*> else +*> ELSE *> Read the next input line for NMATS matrix types *> Set DOTYPE(I) = .TRUE. for each valid type I -*> endif +*> END IF *> \endverbatim * * Arguments: diff --git a/TESTING/EIG/alarqg.f b/TESTING/EIG/alarqg.f index 6e2e6e7ecf..b9fb88c651 100644 --- a/TESTING/EIG/alarqg.f +++ b/TESTING/EIG/alarqg.f @@ -28,12 +28,12 @@ *> to evaluate the input line which requested NMATS matrix types for *> PATH. The flow of control is as follows: *> -*> If NMATS = NTYPES then +*> IF NMATS = NTYPES THEN *> DOTYPE(1:NTYPES) = .TRUE. -*> else +*> ELSE *> Read the next input line for NMATS matrix types *> Set DOTYPE(I) = .TRUE. for each valid type I -*> endif +*> END IF *> \endverbatim * * Arguments: diff --git a/TESTING/LIN/CMakeLists.txt b/TESTING/LIN/CMakeLists.txt old mode 100644 new mode 100755 index 2c3e2a5fd7..5e691c3bd0 --- a/TESTING/LIN/CMakeLists.txt +++ b/TESTING/LIN/CMakeLists.txt @@ -9,7 +9,7 @@ set(DZLNTST dlaord.f) set(SLINTST schkaa.F schkeq.f schkgb.f schkge.f schkgt.f schklq.f schkpb.f schkpo.f schkps.f schkpp.f - schkpt.f schkq3.f schkql.f schkqr.f schkrq.f + schkpt.f schkq3.f schkqp3rk.f schkql.f schkqr.f schkrq.f schksp.f schksy.f schksy_rook.f schksy_rk.f schksy_aa.f schksy_aa_2stage.f schktb.f schktp.f schktr.f @@ -56,7 +56,7 @@ set(CLINTST cchkaa.F cchkhe.f cchkhe_rook.f cchkhe_rk.f cchkhe_aa.f cchkhe_aa_2stage.f cchkhp.f cchklq.f cchkpb.f - cchkpo.f cchkps.f cchkpp.f cchkpt.f cchkq3.f cchkql.f + cchkpo.f cchkps.f cchkpp.f cchkpt.f cchkq3.f cchkqp3rk.f cchkql.f cchkqr.f cchkrq.f cchksp.f cchksy.f cchksy_rook.f cchksy_rk.f cchksy_aa.f cchksy_aa_2stage.f cchktb.f @@ -110,7 +110,7 @@ endif() set(DLINTST dchkaa.F dchkeq.f dchkgb.f dchkge.f dchkgt.f dchklq.f dchkpb.f dchkpo.f dchkps.f dchkpp.f - dchkpt.f dchkq3.f dchkql.f dchkqr.f dchkrq.f + dchkpt.f dchkq3.f dchkqp3rk.f dchkql.f dchkqr.f dchkrq.f dchksp.f dchksy.f dchksy_rook.f dchksy_rk.f dchksy_aa.f dchksy_aa_2stage.f dchktb.f dchktp.f dchktr.f @@ -158,7 +158,7 @@ set(ZLINTST zchkaa.F zchkhe.f zchkhe_rook.f zchkhe_rk.f zchkhe_aa.f zchkhe_aa_2stage.f zchkhp.f zchklq.f zchkpb.f - zchkpo.f zchkps.f zchkpp.f zchkpt.f zchkq3.f zchkql.f + zchkpo.f zchkps.f zchkpp.f zchkpt.f zchkq3.f zchkqp3rk.f zchkql.f zchkqr.f zchkrq.f zchksp.f zchksy.f zchksy_rook.f zchksy_rk.f zchksy_aa.f zchksy_aa_2stage.f zchktb.f diff --git a/TESTING/LIN/Makefile b/TESTING/LIN/Makefile old mode 100644 new mode 100755 index 2474d04db1..e6bcafbb3d --- a/TESTING/LIN/Makefile +++ b/TESTING/LIN/Makefile @@ -45,7 +45,7 @@ DZLNTST = dlaord.o SLINTST = schkaa.o \ schkeq.o schkgb.o schkge.o schkgt.o \ schklq.o schkpb.o schkpo.o schkps.o schkpp.o \ - schkpt.o schkq3.o schkql.o schkqr.o schkrq.o \ + schkpt.o schkq3.o schkqp3rk.o schkql.o schkqr.o schkrq.o \ schksp.o schksy.o schksy_rook.o schksy_rk.o \ schksy_aa.o schksy_aa_2stage.o schktb.o schktp.o schktr.o \ schktz.o \ @@ -89,7 +89,7 @@ CLINTST = cchkaa.o \ cchkeq.o cchkgb.o cchkge.o cchkgt.o \ cchkhe.o cchkhe_rook.o cchkhe_rk.o \ cchkhe_aa.o cchkhe_aa_2stage.o cchkhp.o cchklq.o cchkpb.o \ - cchkpo.o cchkps.o cchkpp.o cchkpt.o cchkq3.o cchkql.o \ + cchkpo.o cchkps.o cchkpp.o cchkpt.o cchkq3.o cchkqp3rk.o cchkql.o \ cchkqr.o cchkrq.o cchksp.o cchksy.o cchksy_rook.o cchksy_rk.o \ cchksy_aa.o cchksy_aa_2stage.o cchktb.o \ cchktp.o cchktr.o cchktz.o \ @@ -137,7 +137,7 @@ endif DLINTST = dchkaa.o \ dchkeq.o dchkgb.o dchkge.o dchkgt.o \ dchklq.o dchkpb.o dchkpo.o dchkps.o dchkpp.o \ - dchkpt.o dchkq3.o dchkql.o dchkqr.o dchkrq.o \ + dchkpt.o dchkq3.o dchkqp3rk.o dchkql.o dchkqr.o dchkrq.o \ dchksp.o dchksy.o dchksy_rook.o dchksy_rk.o \ dchksy_aa.o dchksy_aa_2stage.o dchktb.o dchktp.o dchktr.o \ dchktz.o \ @@ -182,7 +182,7 @@ ZLINTST = zchkaa.o \ zchkeq.o zchkgb.o zchkge.o zchkgt.o \ zchkhe.o zchkhe_rook.o zchkhe_rk.o zchkhe_aa.o zchkhe_aa_2stage.o \ zchkhp.o zchklq.o zchkpb.o \ - zchkpo.o zchkps.o zchkpp.o zchkpt.o zchkq3.o zchkql.o \ + zchkpo.o zchkps.o zchkpp.o zchkpt.o zchkq3.o zchkqp3rk.o zchkql.o \ zchkqr.o zchkrq.o zchksp.o zchksy.o zchksy_rook.o zchksy_rk.o \ zchksy_aa.o zchksy_aa_2stage.o zchktb.o \ zchktp.o zchktr.o zchktz.o \ diff --git a/TESTING/LIN/alaerh.f b/TESTING/LIN/alaerh.f old mode 100644 new mode 100755 index 1845888a66..6c8a47f1e2 --- a/TESTING/LIN/alaerh.f +++ b/TESTING/LIN/alaerh.f @@ -797,6 +797,18 @@ SUBROUTINE ALAERH( PATH, SUBNAM, INFO, INFOE, OPTS, M, N, KL, KU, WRITE( NOUT, FMT = 9978 ) $ SUBNAM(1:LEN_TRIM( SUBNAM )), INFO, M, N, IMAT END IF +* + ELSE IF( LSAMEN( 2, P2, 'QK' ) ) THEN +* +* xQK: truncated QR factorization with pivoting +* + IF( LSAMEN( 7, SUBNAM( 2: 8 ), 'GEQP3RK' ) ) THEN + WRITE( NOUT, FMT = 9930 ) + $ SUBNAM(1:LEN_TRIM( SUBNAM )), INFO, M, N, KL, N5, IMAT + ELSE IF( LSAMEN( 5, SUBNAM( 2: 6 ), 'LATMS' ) ) THEN + WRITE( NOUT, FMT = 9978 ) + $ SUBNAM(1:LEN_TRIM( SUBNAM )), INFO, M, N, IMAT + END IF * ELSE IF( LSAMEN( 2, P2, 'LQ' ) ) THEN * @@ -1147,6 +1159,11 @@ SUBROUTINE ALAERH( PATH, SUBNAM, INFO, INFOE, OPTS, M, N, KL, KU, * What we do next * 9949 FORMAT( ' ==> Doing only the condition estimate for this case' ) +* +* SUBNAM, INFO, M, N, NB, IMAT +* + 9930 FORMAT( ' *** Error code from ', A, '=', I5, / ' ==> M =', I5, + $ ', N =', I5, ', NX =', I5, ', NB =', I4, ', type ', I2 ) * RETURN * diff --git a/TESTING/LIN/alahd.f b/TESTING/LIN/alahd.f old mode 100644 new mode 100755 index dd75394b3a..8f966c5841 --- a/TESTING/LIN/alahd.f +++ b/TESTING/LIN/alahd.f @@ -584,13 +584,27 @@ SUBROUTINE ALAHD( IOUNIT, PATH ) * * QR decomposition with column pivoting * - WRITE( IOUNIT, FMT = 9986 )PATH + WRITE( IOUNIT, FMT = 8006 )PATH WRITE( IOUNIT, FMT = 9969 ) WRITE( IOUNIT, FMT = '( '' Test ratios:'' )' ) WRITE( IOUNIT, FMT = 9940 )1 WRITE( IOUNIT, FMT = 9939 )2 WRITE( IOUNIT, FMT = 9938 )3 WRITE( IOUNIT, FMT = '( '' Messages:'' )' ) +* + ELSE IF( LSAMEN( 2, P2, 'QK' ) ) THEN +* +* truncated QR decomposition with column pivoting +* + WRITE( IOUNIT, FMT = 8006 )PATH + WRITE( IOUNIT, FMT = 9871 ) + WRITE( IOUNIT, FMT = '( '' Test ratios:'' )' ) + WRITE( IOUNIT, FMT = 8060 )1 + WRITE( IOUNIT, FMT = 8061 )2 + WRITE( IOUNIT, FMT = 8062 )3 + WRITE( IOUNIT, FMT = 8063 )4 + WRITE( IOUNIT, FMT = 8064 )5 + WRITE( IOUNIT, FMT = '( '' Messages:'' )' ) * ELSE IF( LSAMEN( 2, P2, 'TZ' ) ) THEN * @@ -779,6 +793,8 @@ SUBROUTINE ALAHD( IOUNIT, PATH ) $ 'tall-skinny or short-wide matrices' ) 8005 FORMAT( / 1X, A3, ': Householder reconstruction from TSQR', $ ' factorization output ', /,' for tall-skinny matrices.' ) + 8006 FORMAT( / 1X, A3, ': truncated QR factorization', + $ ' with column pivoting' ) * * GE matrix types * @@ -922,6 +938,36 @@ SUBROUTINE ALAHD( IOUNIT, PATH ) $ / 4X, '3. Geometric distribution', 10X, $ '6. Every second column fixed' ) * +* QK matrix types +* + 9871 FORMAT( 4X, ' 1. Zero matrix', / + $ 4X, ' 2. Random, Diagonal, CNDNUM = 2', / + $ 4X, ' 3. Random, Upper triangular, CNDNUM = 2', / + $ 4X, ' 4. Random, Lower triangular, CNDNUM = 2', / + $ 4X, ' 5. Random, First column is zero, CNDNUM = 2', / + $ 4X, ' 6. Random, Last MINMN column is zero, CNDNUM = 2', / + $ 4X, ' 7. Random, Last N column is zero, CNDNUM = 2', / + $ 4X, ' 8. Random, Middle column in MINMN is zero,', + $ ' CNDNUM = 2', / + $ 4X, ' 9. Random, First half of MINMN columns are zero,', + $ ' CNDNUM = 2', / + $ 4X, '10. Random, Last columns are zero starting from', + $ ' MINMN/2+1, CNDNUM = 2', / + $ 4X, '11. Random, Half MINMN columns in the middle are', + $ ' zero starting from MINMN/2-(MINMN/2)/2+1,' + $ ' CNDNUM = 2', / + $ 4X, '12. Random, Odd columns are ZERO, CNDNUM = 2', / + $ 4X, '13. Random, Even columns are ZERO, CNDNUM = 2', / + $ 4X, '14. Random, CNDNUM = 2', / + $ 4X, '15. Random, CNDNUM = sqrt(0.1/EPS)', / + $ 4X, '16. Random, CNDNUM = 0.1/EPS', / + $ 4X, '17. Random, CNDNUM = 0.1/EPS,', + $ ' one small singular value S(N)=1/CNDNUM', / + $ 4X, '18. Random, CNDNUM = 2, scaled near underflow,', + $ ' NORM = SMALL = SAFMIN', / + $ 4X, '19. Random, CNDNUM = 2, scaled near overflow,', + $ ' NORM = LARGE = 1.0/( 0.25 * ( SAFMIN / EPS ) )' ) +* * TZ matrix types * 9968 FORMAT( ' Matrix types (2-3 have condition 1/EPS):', / 4X, @@ -1030,9 +1076,8 @@ SUBROUTINE ALAHD( IOUNIT, PATH ) $ ' * norm(C) * EPS )' ) 9940 FORMAT( 3X, I2, ': norm(svd(A) - svd(R)) / ', $ '( M * norm(svd(R)) * EPS )' ) - 9939 FORMAT( 3X, I2, ': norm( A*P - Q*R ) / ( M * norm(A) * EPS )' - $ ) - 9938 FORMAT( 3X, I2, ': norm( I - Q''*Q ) / ( M * EPS )' ) + 9939 FORMAT( 3X, I2, ': norm( A*P - Q*R ) / ( M * norm(A) * EPS )') + 9938 FORMAT( 3X, I2, ': norm( I - Q''*Q ) / ( M * EPS )' ) 9937 FORMAT( 3X, I2, ': norm( A - R*Q ) / ( M * norm(A) * EPS )' $ ) 9935 FORMAT( 3X, I2, ': norm( B - A * X ) / ', @@ -1105,6 +1150,15 @@ SUBROUTINE ALAHD( IOUNIT, PATH ) 8054 FORMAT(3X,I2,': norm( C*Q - C*Q ) / ( M * norm(C) * EPS )' ) 8055 FORMAT(3X,I2,': norm( C*Q'' - C*Q'' ) / ( M * norm(C) * EPS )') + 8060 FORMAT( 3X, I2, ': 2-norm(svd(A) - svd(R)) / ', + $ '( max(M,N) * 2-norm(svd(R)) * EPS )' ) + 8061 FORMAT( 3X, I2, ': 1-norm( A*P - Q*R ) / ( max(M,N) * 1-norm(A)', + $ ' * EPS )') + 8062 FORMAT( 3X, I2, ': 1-norm( I - Q''*Q ) / ( M * EPS )' ) + 8063 FORMAT( 3X, I2, ': Returns 1.0D+100, if abs(R(K+1,K+1))', + $ ' > abs(R(K,K)), where K=1:KFACT-1' ) + 8064 FORMAT( 3X, I2, ': 1-norm(Q**T * B - Q**T * B ) / ( M * EPS )') + * RETURN * diff --git a/TESTING/LIN/alareq.f b/TESTING/LIN/alareq.f index db18775ebc..3f057fa486 100644 --- a/TESTING/LIN/alareq.f +++ b/TESTING/LIN/alareq.f @@ -28,12 +28,12 @@ *> to evaluate the input line which requested NMATS matrix types for *> PATH. The flow of control is as follows: *> -*> If NMATS = NTYPES then +*> IF NMATS = NTYPES THEN *> DOTYPE(1:NTYPES) = .TRUE. -*> else +*> ELSE *> Read the next input line for NMATS matrix types *> Set DOTYPE(I) = .TRUE. for each valid type I -*> endif +*> END IF *> \endverbatim * * Arguments: diff --git a/TESTING/LIN/cchkaa.F b/TESTING/LIN/cchkaa.F index ec1534ed4e..474454a519 100644 --- a/TESTING/LIN/cchkaa.F +++ b/TESTING/LIN/cchkaa.F @@ -69,6 +69,7 @@ *> CLQ 8 List types on next line if 0 < NTYPES < 8 *> CQL 8 List types on next line if 0 < NTYPES < 8 *> CQP 6 List types on next line if 0 < NTYPES < 6 +*> ZQK 19 List types on next line if 0 < NTYPES < 19 *> CTZ 3 List types on next line if 0 < NTYPES < 3 *> CLS 6 List types on next line if 0 < NTYPES < 6 *> CEQ @@ -153,12 +154,11 @@ PROGRAM CCHKAA $ NBVAL( MAXIN ), NBVAL2( MAXIN ), $ NSVAL( MAXIN ), NVAL( MAXIN ), NXVAL( MAXIN ), $ RANKVAL( MAXIN ), PIV( NMAX ) - REAL S( 2*NMAX ) - COMPLEX E( NMAX ) * .. * .. Allocatable Arrays .. INTEGER AllocateStatus - REAL, DIMENSION(:), ALLOCATABLE :: RWORK + REAL, DIMENSION(:), ALLOCATABLE :: RWORK, S + COMPLEX, DIMENSION(:), ALLOCATABLE :: E COMPLEX, DIMENSION(:,:), ALLOCATABLE :: A, B, WORK * .. * .. External Functions .. @@ -170,14 +170,14 @@ PROGRAM CCHKAA EXTERNAL ALAREQ, CCHKEQ, CCHKGB, CCHKGE, CCHKGT, CCHKHE, $ CCHKHE_ROOK, CCHKHE_RK, CCHKHE_AA, CCHKHP, $ CCHKLQ, CCHKUNHR_COL, CCHKPB, CCHKPO, CCHKPS, - $ CCHKPP, CCHKPT, CCHKQ3, CCHKQL, CCHKQR, CCHKRQ, - $ CCHKSP, CCHKSY, CCHKSY_ROOK, CCHKSY_RK, - $ CCHKSY_AA, CCHKTB, CCHKTP, CCHKTR, CCHKTZ, - $ CDRVGB, CDRVGE, CDRVGT, CDRVHE, CDRVHE_ROOK, - $ CDRVHE_RK, CDRVHE_AA, CDRVHP, CDRVLS, CDRVPB, - $ CDRVPO, CDRVPP, CDRVPT, CDRVSP, CDRVSY, - $ CDRVSY_ROOK, CDRVSY_RK, CDRVSY_AA, ILAVER, - $ CCHKQRT, CCHKQRTP + $ CCHKPP, CCHKPT, CCHKQ3, CCHKQP3RK, CCHKQL, + $ CCHKQR, CCHKRQ, CCHKSP, CCHKSY, CCHKSY_ROOK, + $ CCHKSY_RK, CCHKSY_AA, CCHKTB, CCHKTP, CCHKTR, + $ CCHKTZ, CDRVGB, CDRVGE, CDRVGT, CDRVHE, + $ CDRVHE_ROOK, CDRVHE_RK, CDRVHE_AA, CDRVHP, + $ CDRVLS, CDRVPB, CDRVPO, CDRVPP, CDRVPT, CDRVSP, + $ CDRVSY, CDRVSY_ROOK, CDRVSY_RK, CDRVSY_AA, + $ ILAVER, CCHKQRT, CCHKQRTP * .. * .. Scalars in Common .. LOGICAL LERR, OK @@ -203,6 +203,10 @@ PROGRAM CCHKAA IF (AllocateStatus /= 0) STOP "*** Not enough memory ***" ALLOCATE ( WORK( NMAX, NMAX+MAXRHS+10 ), STAT = AllocateStatus ) IF (AllocateStatus /= 0) STOP "*** Not enough memory ***" + ALLOCATE ( E( NMAX ), STAT = AllocateStatus ) + IF (AllocateStatus /= 0) STOP "*** Not enough memory ***" + ALLOCATE ( S( 2*NMAX ), STAT = AllocateStatus) + IF (AllocateStatus /= 0) STOP "*** Not enough memory ***" ALLOCATE ( RWORK( 150*NMAX+2*MAXRHS ), STAT = AllocateStatus ) IF (AllocateStatus /= 0) STOP "*** Not enough memory ***" * .. @@ -1109,6 +1113,23 @@ PROGRAM CCHKAA ELSE WRITE( NOUT, FMT = 9989 )PATH END IF +* + ELSE IF( LSAMEN( 2, C2, 'QK' ) ) THEN +* +* QK: truncated QR factorization with pivoting +* + NTYPES = 19 + CALL ALAREQ( PATH, NMATS, DOTYPE, NTYPES, NIN, NOUT ) +* + IF( TSTCHK ) THEN + CALL CCHKQP3RK( DOTYPE, NM, MVAL, NN, NVAL, NNS, NSVAL, + $ NNB, NBVAL, NXVAL, THRESH, A( 1, 1 ), + $ A( 1, 2 ), B( 1, 1 ), B( 1, 2 ), + $ S( 1 ), B( 1, 4 ), + $ WORK, RWORK, IWORK, NOUT ) + ELSE + WRITE( NOUT, FMT = 9989 )PATH + END IF * ELSE IF( LSAMEN( 2, C2, 'LS' ) ) THEN * diff --git a/TESTING/LIN/cchkqp3rk.f b/TESTING/LIN/cchkqp3rk.f new file mode 100644 index 0000000000..79d6add72e --- /dev/null +++ b/TESTING/LIN/cchkqp3rk.f @@ -0,0 +1,836 @@ +*> \brief \b CCHKQP3RK +* +* =========== DOCUMENTATION =========== +* +* Online html documentation available at +* http://www.netlib.org/lapack/explore-html/ +* +* Definition: +* =========== +* +* SUBROUTINE CCHKQP3RK( DOTYPE, NM, MVAL, NN, NVAL, NNS, NSVAL, +* $ NNB, NBVAL, NXVAL, THRESH, A, COPYA, +* $ B, COPYB, S, TAU, +* $ WORK, RWORK, IWORK, NOUT ) +* IMPLICIT NONE +* +* .. Scalar Arguments .. +* INTEGER NM, NN, NNB, NOUT +* REAL THRESH +* .. +* .. Array Arguments .. +* LOGICAL DOTYPE( * ) +* INTEGER IWORK( * ), MVAL( * ), NBVAL( * ), NVAL( * ), +* $ NXVAL( * ) +* REAL S( * ), RWORK( * ) +* COMPLEX A( * ), COPYA( * ), TAU( * ), WORK( * ) +* .. +* +* +*> \par Purpose: +* ============= +*> +*> \verbatim +*> +*> CCHKQP3RK tests CGEQP3RK. +*> \endverbatim +* +* Arguments: +* ========== +* +*> \param[in] DOTYPE +*> \verbatim +*> DOTYPE is LOGICAL array, dimension (NTYPES) +*> The matrix types to be used for testing. Matrices of type j +*> (for 1 <= j <= NTYPES) are used for testing if DOTYPE(j) = +*> .TRUE.; if DOTYPE(j) = .FALSE., then type j is not used. +*> \endverbatim +*> +*> \param[in] NM +*> \verbatim +*> NM is INTEGER +*> The number of values of M contained in the vector MVAL. +*> \endverbatim +*> +*> \param[in] MVAL +*> \verbatim +*> MVAL is INTEGER array, dimension (NM) +*> The values of the matrix row dimension M. +*> \endverbatim +*> +*> \param[in] NN +*> \verbatim +*> NN is INTEGER +*> The number of values of N contained in the vector NVAL. +*> \endverbatim +*> +*> \param[in] NVAL +*> \verbatim +*> NVAL is INTEGER array, dimension (NN) +*> The values of the matrix column dimension N. +*> \endverbatim +*> +*> \param[in] NNS +*> \verbatim +*> NNS is INTEGER +*> The number of values of NRHS contained in the vector NSVAL. +*> \endverbatim +*> +*> \param[in] NSVAL +*> \verbatim +*> NSVAL is INTEGER array, dimension (NNS) +*> The values of the number of right hand sides NRHS. +*> \endverbatim +*> \param[in] NNB +*> \verbatim +*> NNB is INTEGER +*> The number of values of NB and NX contained in the +*> vectors NBVAL and NXVAL. The blocking parameters are used +*> in pairs (NB,NX). +*> \endverbatim +*> +*> \param[in] NBVAL +*> \verbatim +*> NBVAL is INTEGER array, dimension (NNB) +*> The values of the blocksize NB. +*> \endverbatim +*> +*> \param[in] NXVAL +*> \verbatim +*> NXVAL is INTEGER array, dimension (NNB) +*> The values of the crossover point NX. +*> \endverbatim +*> +*> \param[in] THRESH +*> \verbatim +*> THRESH is REAL +*> The threshold value for the test ratios. A result is +*> included in the output file if RESULT >= THRESH. To have +*> every test ratio printed, use THRESH = 0. +*> \endverbatim +*> +*> \param[out] A +*> \verbatim +*> A is COMPLEX array, dimension (MMAX*NMAX) +*> where MMAX is the maximum value of M in MVAL and NMAX is the +*> maximum value of N in NVAL. +*> \endverbatim +*> +*> \param[out] COPYA +*> \verbatim +*> COPYA is COMPLEX array, dimension (MMAX*NMAX) +*> \endverbatim +*> +*> \param[out] B +*> \verbatim +*> B is COMPLEX array, dimension (MMAX*NSMAX) +*> where MMAX is the maximum value of M in MVAL and NSMAX is the +*> maximum value of NRHS in NSVAL. +*> \endverbatim +*> +*> \param[out] COPYB +*> \verbatim +*> COPYB is COMPLEX array, dimension (MMAX*NSMAX) +*> \endverbatim +*> +*> \param[out] S +*> \verbatim +*> S is REAL array, dimension +*> (min(MMAX,NMAX)) +*> \endverbatim +*> +*> \param[out] TAU +*> \verbatim +*> TAU is COMPLEX array, dimension (MMAX) +*> \endverbatim +*> +*> \param[out] WORK +*> \verbatim +*> WORK is COMPLEX array, dimension +*> (max(M*max(M,N) + 4*min(M,N) + max(M,N))) +*> \endverbatim +*> +*> \param[out] RWORK +*> \verbatim +*> RWORK is REAL array, dimension (4*NMAX) +*> \endverbatim +*> +*> \param[out] IWORK +*> \verbatim +*> IWORK is INTEGER array, dimension (2*NMAX) +*> \endverbatim +*> +*> \param[in] NOUT +*> \verbatim +*> NOUT is INTEGER +*> The unit number for output. +*> \endverbatim +* +* Authors: +* ======== +* +*> \author Univ. of Tennessee +*> \author Univ. of California Berkeley +*> \author Univ. of Colorado Denver +*> \author NAG Ltd. +* +*> \ingroup complex_lin +* +* ===================================================================== + SUBROUTINE CCHKQP3RK( DOTYPE, NM, MVAL, NN, NVAL, NNS, NSVAL, + $ NNB, NBVAL, NXVAL, THRESH, A, COPYA, + $ B, COPYB, S, TAU, + $ WORK, RWORK, IWORK, NOUT ) + IMPLICIT NONE +* +* -- LAPACK test routine -- +* -- LAPACK is a software package provided by Univ. of Tennessee, -- +* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- +* +* .. Scalar Arguments .. + INTEGER NM, NN, NNB, NNS, NOUT + REAL THRESH +* .. +* .. Array Arguments .. + LOGICAL DOTYPE( * ) + INTEGER IWORK( * ), NBVAL( * ), MVAL( * ), NVAL( * ), + $ NSVAL( * ), NXVAL( * ) + REAL S( * ), RWORK( * ) + COMPLEX A( * ), COPYA( * ), B( * ), COPYB( * ), + $ TAU( * ), WORK( * ) +* .. +* +* ===================================================================== +* +* .. Parameters .. + INTEGER NTYPES + PARAMETER ( NTYPES = 19 ) + INTEGER NTESTS + PARAMETER ( NTESTS = 5 ) + REAL ONE, ZERO, BIGNUM + COMPLEX CONE, CZERO + PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0, + $ CZERO = ( 0.0E+0, 0.0E+0 ), + $ CONE = ( 1.0E+0, 0.0E+0 ), + $ BIGNUM = 1.0E+38 ) +* .. +* .. Local Scalars .. + CHARACTER DIST, TYPE + CHARACTER*3 PATH + INTEGER I, IHIGH, ILOW, IM, IMAT, IN, INC_ZERO, + $ INB, IND_OFFSET_GEN, + $ IND_IN, IND_OUT, INS, INFO, + $ ISTEP, J, J_INC, J_FIRST_NZ, JB_ZERO, + $ KFACT, KL, KMAX, KU, LDA, LW, LWORK, + $ LWORK_MQR, M, MINMN, MINMNB_GEN, MODE, N, + $ NB, NB_ZERO, NERRS, NFAIL, NB_GEN, NRHS, + $ NRUN, NX, T + REAL ANORM, CNDNUM, EPS, ABSTOL, RELTOL, + $ DTEMP, MAXC2NRMK, RELMAXC2NRMK +* .. +* .. Local Arrays .. + INTEGER ISEED( 4 ), ISEEDY( 4 ) + REAL RESULT( NTESTS ), RDUMMY( 1 ) +* .. +* .. External Functions .. + REAL SLAMCH, CQPT01, CQRT11, CQRT12, CLANGE + EXTERNAL SLAMCH, CQPT01, CQRT11, CQRT12, CLANGE +* .. +* .. External Subroutines .. + EXTERNAL ALAERH, ALAHD, ALASUM, SLAORD, ICOPY, CAXPY, + $ XLAENV, CGEQP3RK, CLACPY, CLASET, CLATB4, + $ CLATMS, CUNMQR, CSWAP +* .. +* .. Intrinsic Functions .. + INTRINSIC ABS, MAX, MIN, MOD, REAL +* .. +* .. Scalars in Common .. + LOGICAL LERR, OK + CHARACTER*32 SRNAMT + INTEGER INFOT, IOUNIT, CUNMQR_LWORK +* .. +* .. Common blocks .. + COMMON / INFOC / INFOT, IOUNIT, OK, LERR + COMMON / SRNAMC / SRNAMT +* .. +* .. Data statements .. + DATA ISEEDY / 1988, 1989, 1990, 1991 / +* .. +* .. Executable Statements .. +* +* Initialize constants and the random number seed. +* + PATH( 1: 1 ) = 'Complex precision' + PATH( 2: 3 ) = 'QK' + NRUN = 0 + NFAIL = 0 + NERRS = 0 + DO I = 1, 4 + ISEED( I ) = ISEEDY( I ) + END DO + EPS = SLAMCH( 'Epsilon' ) + INFOT = 0 +* + DO IM = 1, NM +* +* Do for each value of M in MVAL. +* + M = MVAL( IM ) + LDA = MAX( 1, M ) +* + DO IN = 1, NN +* +* Do for each value of N in NVAL. +* + N = NVAL( IN ) + MINMN = MIN( M, N ) + LWORK = MAX( 1, M*MAX( M, N )+4*MINMN+MAX( M, N ), + $ M*N + 2*MINMN + 4*N ) +* + DO INS = 1, NNS + NRHS = NSVAL( INS ) +* +* Set up parameters with CLATB4 and generate +* M-by-NRHS B matrix with CLATMS. +* IMAT = 14: +* Random matrix, CNDNUM = 2, NORM = ONE, +* MODE = 3 (geometric distribution of singular values). +* + CALL CLATB4( PATH, 14, M, NRHS, TYPE, KL, KU, ANORM, + $ MODE, CNDNUM, DIST ) +* + SRNAMT = 'CLATMS' + CALL CLATMS( M, NRHS, DIST, ISEED, TYPE, S, MODE, + $ CNDNUM, ANORM, KL, KU, 'No packing', + $ COPYB, LDA, WORK, INFO ) +* +* Check error code from CLATMS. +* + IF( INFO.NE.0 ) THEN + CALL ALAERH( PATH, 'CLATMS', INFO, 0, ' ', M, + $ NRHS, -1, -1, -1, 6, NFAIL, NERRS, + $ NOUT ) + CYCLE + END IF +* + DO IMAT = 1, NTYPES +* +* Do the tests only if DOTYPE( IMAT ) is true. +* + IF( .NOT.DOTYPE( IMAT ) ) + $ CYCLE +* +* The type of distribution used to generate the random +* eigen-/singular values: +* ( 'S' for symmetric distribution ) => UNIFORM( -1, 1 ) +* +* Do for each type of NON-SYMMETRIC matrix: CNDNUM NORM MODE +* 1. Zero matrix +* 2. Random, Diagonal, CNDNUM = 2 CNDNUM = 2 ONE 3 ( geometric distribution of singular values ) +* 3. Random, Upper triangular, CNDNUM = 2 CNDNUM = 2 ONE 3 ( geometric distribution of singular values ) +* 4. Random, Lower triangular, CNDNUM = 2 CNDNUM = 2 ONE 3 ( geometric distribution of singular values ) +* 5. Random, First column is zero, CNDNUM = 2 CNDNUM = 2 ONE 3 ( geometric distribution of singular values ) +* 6. Random, Last MINMN column is zero, CNDNUM = 2 CNDNUM = 2 ONE 3 ( geometric distribution of singular values ) +* 7. Random, Last N column is zero, CNDNUM = 2 CNDNUM = 2 ONE 3 ( geometric distribution of singular values ) +* 8. Random, Middle column in MINMN is zero, CNDNUM = 2 CNDNUM = 2 ONE 3 ( geometric distribution of singular values ) +* 9. Random, First half of MINMN columns are zero, CNDNUM = 2 CNDNUM = 2 ONE 3 ( geometric distribution of singular values ) +* 10. Random, Last columns are zero starting from MINMN/2+1, CNDNUM = 2 CNDNUM = 2 ONE 3 ( geometric distribution of singular values ) +* 11. Random, Half MINMN columns in the middle are zero starting +* from MINMN/2-(MINMN/2)/2+1, CNDNUM = 2 CNDNUM = 2 ONE 3 ( geometric distribution of singular values ) +* 12. Random, Odd columns are ZERO, CNDNUM = 2 CNDNUM = 2 ONE 3 ( geometric distribution of singular values ) +* 13. Random, Even columns are ZERO, CNDNUM = 2 CNDNUM = 2 ONE 3 ( geometric distribution of singular values ) +* 14. Random, CNDNUM = 2 CNDNUM = 2 ONE 3 ( geometric distribution of singular values ) +* 15. Random, CNDNUM = sqrt(0.1/EPS) CNDNUM = BADC1 = sqrt(0.1/EPS) ONE 3 ( geometric distribution of singular values ) +* 16. Random, CNDNUM = 0.1/EPS CNDNUM = BADC2 = 0.1/EPS ONE 3 ( geometric distribution of singular values ) +* 17. Random, CNDNUM = 0.1/EPS, CNDNUM = BADC2 = 0.1/EPS ONE 2 ( one small singular value, S(N)=1/CNDNUM ) +* one small singular value S(N)=1/CNDNUM +* 18. Random, CNDNUM = 2, scaled near underflow CNDNUM = 2 SMALL = SAFMIN +* 19. Random, CNDNUM = 2, scaled near overflow CNDNUM = 2 LARGE = 1.0/( 0.25 * ( SAFMIN / EPS ) ) 3 ( geometric distribution of singular values ) +* + IF( IMAT.EQ.1 ) THEN +* +* Matrix 1: Zero matrix +* + CALL CLASET( 'Full', M, N, CZERO, CZERO, COPYA, LDA ) + DO I = 1, MINMN + S( I ) = ZERO + END DO +* + ELSE IF( (IMAT.GE.2 .AND. IMAT.LE.4 ) + $ .OR. (IMAT.GE.14 .AND. IMAT.LE.19 ) ) THEN +* +* Matrices 2-5. +* +* Set up parameters with DLATB4 and generate a test +* matrix with CLATMS. +* + CALL CLATB4( PATH, IMAT, M, N, TYPE, KL, KU, ANORM, + $ MODE, CNDNUM, DIST ) +* + SRNAMT = 'CLATMS' + CALL CLATMS( M, N, DIST, ISEED, TYPE, S, MODE, + $ CNDNUM, ANORM, KL, KU, 'No packing', + $ COPYA, LDA, WORK, INFO ) +* +* Check error code from CLATMS. +* + IF( INFO.NE.0 ) THEN + CALL ALAERH( PATH, 'CLATMS', INFO, 0, ' ', M, N, + $ -1, -1, -1, IMAT, NFAIL, NERRS, + $ NOUT ) + CYCLE + END IF +* + CALL SLAORD( 'Decreasing', MINMN, S, 1 ) +* + ELSE IF( MINMN.GE.2 + $ .AND. IMAT.GE.5 .AND. IMAT.LE.13 ) THEN +* +* Rectangular matrices 5-13 that contain zero columns, +* only for matrices MINMN >=2. +* +* JB_ZERO is the column index of ZERO block. +* NB_ZERO is the column block size of ZERO block. +* NB_GEN is the column blcok size of the +* generated block. +* J_INC in the non_zero column index increment +* for matrix 12 and 13. +* J_FIRS_NZ is the index of the first non-zero +* column. +* + IF( IMAT.EQ.5 ) THEN +* +* First column is zero. +* + JB_ZERO = 1 + NB_ZERO = 1 + NB_GEN = N - NB_ZERO +* + ELSE IF( IMAT.EQ.6 ) THEN +* +* Last column MINMN is zero. +* + JB_ZERO = MINMN + NB_ZERO = 1 + NB_GEN = N - NB_ZERO +* + ELSE IF( IMAT.EQ.7 ) THEN +* +* Last column N is zero. +* + JB_ZERO = N + NB_ZERO = 1 + NB_GEN = N - NB_ZERO +* + ELSE IF( IMAT.EQ.8 ) THEN +* +* Middle column in MINMN is zero. +* + JB_ZERO = MINMN / 2 + 1 + NB_ZERO = 1 + NB_GEN = N - NB_ZERO +* + ELSE IF( IMAT.EQ.9 ) THEN +* +* First half of MINMN columns is zero. +* + JB_ZERO = 1 + NB_ZERO = MINMN / 2 + NB_GEN = N - NB_ZERO +* + ELSE IF( IMAT.EQ.10 ) THEN +* +* Last columns are zero columns, +* starting from (MINMN / 2 + 1) column. +* + JB_ZERO = MINMN / 2 + 1 + NB_ZERO = N - JB_ZERO + 1 + NB_GEN = N - NB_ZERO +* + ELSE IF( IMAT.EQ.11 ) THEN +* +* Half of the columns in the middle of MINMN +* columns is zero, starting from +* MINMN/2 - (MINMN/2)/2 + 1 column. +* + JB_ZERO = MINMN / 2 - (MINMN / 2) / 2 + 1 + NB_ZERO = MINMN / 2 + NB_GEN = N - NB_ZERO +* + ELSE IF( IMAT.EQ.12 ) THEN +* +* Odd-numbered columns are zero, +* + NB_GEN = N / 2 + NB_ZERO = N - NB_GEN + J_INC = 2 + J_FIRST_NZ = 2 +* + ELSE IF( IMAT.EQ.13 ) THEN +* +* Even-numbered columns are zero. +* + NB_ZERO = N / 2 + NB_GEN = N - NB_ZERO + J_INC = 2 + J_FIRST_NZ = 1 +* + END IF +* +* +* 1) Set the first NB_ZERO columns in COPYA(1:M,1:N) +* to zero. +* + CALL CLASET( 'Full', M, NB_ZERO, CZERO, CZERO, + $ COPYA, LDA ) +* +* 2) Generate an M-by-(N-NB_ZERO) matrix with the +* chosen singular value distribution +* in COPYA(1:M,NB_ZERO+1:N). +* + CALL CLATB4( PATH, IMAT, M, NB_GEN, TYPE, KL, KU, + $ ANORM, MODE, CNDNUM, DIST ) +* + SRNAMT = 'CLATMS' +* + IND_OFFSET_GEN = NB_ZERO * LDA +* + CALL CLATMS( M, NB_GEN, DIST, ISEED, TYPE, S, MODE, + $ CNDNUM, ANORM, KL, KU, 'No packing', + $ COPYA( IND_OFFSET_GEN + 1 ), LDA, + $ WORK, INFO ) +* +* Check error code from CLATMS. +* + IF( INFO.NE.0 ) THEN + CALL ALAERH( PATH, 'CLATMS', INFO, 0, ' ', M, + $ NB_GEN, -1, -1, -1, IMAT, NFAIL, + $ NERRS, NOUT ) + CYCLE + END IF +* +* 3) Swap the gererated colums from the right side +* NB_GEN-size block in COPYA into correct column +* positions. +* + IF( IMAT.EQ.6 + $ .OR. IMAT.EQ.7 + $ .OR. IMAT.EQ.8 + $ .OR. IMAT.EQ.10 + $ .OR. IMAT.EQ.11 ) THEN +* +* Move by swapping the generated columns +* from the right NB_GEN-size block from +* (NB_ZERO+1:NB_ZERO+JB_ZERO) +* into columns (1:JB_ZERO-1). +* + DO J = 1, JB_ZERO-1, 1 + CALL CSWAP( M, + $ COPYA( ( NB_ZERO+J-1)*LDA+1), 1, + $ COPYA( (J-1)*LDA + 1 ), 1 ) + END DO +* + ELSE IF( IMAT.EQ.12 .OR. IMAT.EQ.13 ) THEN +* +* ( IMAT = 12, Odd-numbered ZERO columns. ) +* Swap the generated columns from the right +* NB_GEN-size block into the even zero colums in the +* left NB_ZERO-size block. +* +* ( IMAT = 13, Even-numbered ZERO columns. ) +* Swap the generated columns from the right +* NB_GEN-size block into the odd zero colums in the +* left NB_ZERO-size block. +* + DO J = 1, NB_GEN, 1 + IND_OUT = ( NB_ZERO+J-1 )*LDA + 1 + IND_IN = ( J_INC*(J-1)+(J_FIRST_NZ-1) )*LDA + $ + 1 + CALL CSWAP( M, + $ COPYA( IND_OUT ), 1, + $ COPYA( IND_IN), 1 ) + END DO +* + END IF +* +* 5) Order the singular values generated by +* DLAMTS in decreasing order and add trailing zeros +* that correspond to zero columns. +* The total number of singular values is MINMN. +* + MINMNB_GEN = MIN( M, NB_GEN ) +* + CALL SLAORD( 'Decreasing', MINMNB_GEN, S, 1 ) + + DO I = MINMNB_GEN+1, MINMN + S( I ) = ZERO + END DO +* + ELSE +* +* IF(MINMN.LT.2) skip this size for this matrix type. +* + CYCLE + END IF +* +* Initialize a copy array for a pivot array for DGEQP3RK. +* + DO I = 1, N + IWORK( I ) = 0 + END DO +* + DO INB = 1, NNB +* +* Do for each pair of values (NB,NX) in NBVAL and NXVAL. +* + NB = NBVAL( INB ) + CALL XLAENV( 1, NB ) + NX = NXVAL( INB ) + CALL XLAENV( 3, NX ) +* +* We do MIN(M,N)+1 because we need a test for KMAX > N, +* when KMAX is larger than MIN(M,N), KMAX should be +* KMAX = MIN(M,N) +* + DO KMAX = 0, MIN(M,N)+1 +* +* Get a working copy of COPYA into A( 1:M,1:N ). +* Get a working copy of COPYB into A( 1:M, (N+1):NRHS ). +* Get a working copy of COPYB into into B( 1:M, 1:NRHS ). +* Get a working copy of IWORK(1:N) awith zeroes into +* which is going to be used as pivot array IWORK( N+1:2N ). +* NOTE: IWORK(2N+1:3N) is going to be used as a WORK array +* for the routine. +* + CALL CLACPY( 'All', M, N, COPYA, LDA, A, LDA ) + CALL CLACPY( 'All', M, NRHS, COPYB, LDA, + $ A( LDA*N + 1 ), LDA ) + CALL CLACPY( 'All', M, NRHS, COPYB, LDA, + $ B, LDA ) + CALL ICOPY( N, IWORK( 1 ), 1, IWORK( N+1 ), 1 ) +* + ABSTOL = -1.0 + RELTOl = -1.0 +* +* Compute the QR factorization with pivoting of A +* + LW = MAX( 1, MAX( 2*N + NB*( N+NRHS+1 ), + $ 3*N + NRHS - 1 ) ) +* +* Compute CGEQP3RK factorization of A. +* + SRNAMT = 'CGEQP3RK' + CALL CGEQP3RK( M, N, NRHS, KMAX, ABSTOL, RELTOL, + $ A, LDA, KFACT, MAXC2NRMK, + $ RELMAXC2NRMK, IWORK( N+1 ), TAU, + $ WORK, LW, RWORK, IWORK( 2*N+1 ), + $ INFO ) +* +* Check error code from CGEQP3RK. +* + IF( INFO.LT.0 ) + $ CALL ALAERH( PATH, 'CGEQP3RK', INFO, 0, ' ', + $ M, N, NX, -1, NB, IMAT, + $ NFAIL, NERRS, NOUT ) +* + IF( KFACT.EQ.MINMN ) THEN +* +* Compute test 1: +* +* This test in only for the full rank factorization of +* the matrix A. +* +* Array S(1:min(M,N)) contains svd(A) the sigular values +* of the original matrix A in decreasing absolute value +* order. The test computes svd(R), the vector sigular +* values of the upper trapezoid of A(1:M,1:N) that +* contains the factor R, in decreasing order. The test +* returns the ratio: +* +* 2-norm(svd(R) - svd(A)) / ( max(M,N) * 2-norm(svd(A)) * EPS ) +* + RESULT( 1 ) = CQRT12( M, N, A, LDA, S, WORK, + $ LWORK , RWORK ) +* + DO T = 1, 1 + IF( RESULT( T ).GE.THRESH ) THEN + IF( NFAIL.EQ.0 .AND. NERRS.EQ.0 ) + $ CALL ALAHD( NOUT, PATH ) + WRITE( NOUT, FMT = 9999 ) 'CGEQP3RK', M, N, + $ NRHS, KMAX, ABSTOL, RELTOL, NB, NX, + $ IMAT, T, RESULT( T ) + NFAIL = NFAIL + 1 + END IF + END DO + NRUN = NRUN + 1 +* +* End test 1 +* + END IF + +* Compute test 2: +* +* The test returns the ratio: +* +* 1-norm( A*P - Q*R ) / ( max(M,N) * 1-norm(A) * EPS ) +* + RESULT( 2 ) = CQPT01( M, N, KFACT, COPYA, A, LDA, TAU, + $ IWORK( N+1 ), WORK, LWORK ) +* +* Compute test 3: +* +* The test returns the ratio: +* +* 1-norm( Q**T * Q - I ) / ( M * EPS ) +* + RESULT( 3 ) = CQRT11( M, KFACT, A, LDA, TAU, WORK, + $ LWORK ) +* +* Print information about the tests that did not pass +* the threshold. +* + DO T = 2, 3 + IF( RESULT( T ).GE.THRESH ) THEN + IF( NFAIL.EQ.0 .AND. NERRS.EQ.0 ) + $ CALL ALAHD( NOUT, PATH ) + WRITE( NOUT, FMT = 9999 ) 'CGEQP3RK', M, N, + $ NRHS, KMAX, ABSTOL, RELTOL, + $ NB, NX, IMAT, T, RESULT( T ) + NFAIL = NFAIL + 1 + END IF + END DO + NRUN = NRUN + 2 +* +* Compute test 4: +* +* This test is only for the factorizations with the +* rank greater than 2. +* The elements on the diagonal of R should be non- +* increasing. +* +* The test returns the ratio: +* +* Returns 1.0D+100 if abs(R(K+1,K+1)) > abs(R(K,K)), +* K=1:KFACT-1 +* + IF( MIN(KFACT, MINMN).GE.2 ) THEN +* + DO J = 1, KFACT-1, 1 +* + DTEMP = (( ABS( A( (J-1)*M+J ) ) - + $ ABS( A( (J)*M+J+1 ) ) ) / + $ ABS( A(1) ) ) +* + IF( DTEMP.LT.ZERO ) THEN + RESULT( 4 ) = BIGNUM + END IF +* + END DO +* +* Print information about the tests that did not +* pass the threshold. +* + DO T = 4, 4 + IF( RESULT( T ).GE.THRESH ) THEN + IF( NFAIL.EQ.0 .AND. NERRS.EQ.0 ) + $ CALL ALAHD( NOUT, PATH ) + WRITE( NOUT, FMT = 9999 ) 'CGEQP3RK', + $ M, N, NRHS, KMAX, ABSTOL, RELTOL, + $ NB, NX, IMAT, T, + $ RESULT( T ) + NFAIL = NFAIL + 1 + END IF + END DO + NRUN = NRUN + 1 +* +* End test 4. +* + END IF +* +* Compute test 5: +* +* This test in only for matrix A with min(M,N) > 0. +* +* The test returns the ratio: +* +* 1-norm(Q**T * B - Q**T * B ) / +* ( M * EPS ) +* +* (1) Compute B:=Q**T * B in the matrix B. +* + IF( MINMN.GT.0 ) THEN +* + LWORK_MQR = MAX(1, NRHS) + CALL CUNMQR( 'Left', 'Conjugate transpose', + $ M, NRHS, KFACT, A, LDA, TAU, B, LDA, + $ WORK, LWORK_MQR, INFO ) +* + DO I = 1, NRHS +* +* Compare N+J-th column of A and J-column of B. +* + CALL CAXPY( M, -CONE, A( ( N+I-1 )*LDA+1 ), 1, + $ B( ( I-1 )*LDA+1 ), 1 ) + END DO +* + RESULT( 5 ) = + $ ABS( + $ CLANGE( 'One-norm', M, NRHS, B, LDA, RDUMMY ) / + $ ( REAL( M )*SLAMCH( 'Epsilon' ) ) + $ ) +* +* Print information about the tests that did not pass +* the threshold. +* + DO T = 5, 5 + IF( RESULT( T ).GE.THRESH ) THEN + IF( NFAIL.EQ.0 .AND. NERRS.EQ.0 ) + $ CALL ALAHD( NOUT, PATH ) + WRITE( NOUT, FMT = 9999 ) 'CGEQP3RK', M, N, + $ NRHS, KMAX, ABSTOL, RELTOL, + $ NB, NX, IMAT, T, RESULT( T ) + NFAIL = NFAIL + 1 + END IF + END DO + NRUN = NRUN + 1 +* +* End compute test 5. +* + END IF +* +* END DO KMAX = 1, MIN(M,N)+1 +* + END DO +* +* END DO for INB = 1, NNB +* + END DO +* +* END DO for IMAT = 1, NTYPES +* + END DO +* +* END DO for INS = 1, NNS +* + END DO +* +* END DO for IN = 1, NN +* + END DO +* +* END DO for IM = 1, NM +* + END DO +* +* Print a summary of the results. +* + CALL ALASUM( PATH, NOUT, NFAIL, NRUN, NERRS ) +* + 9999 FORMAT( 1X, A, ' M =', I5, ', N =', I5, ', NRHS =', I5, + $ ', KMAX =', I5, ', ABSTOL =', G12.5, + $ ', RELTOL =', G12.5, ', NB =', I4, ', NX =', I4, + $ ', type ', I2, ', test ', I2, ', ratio =', G12.5 ) +* +* End of CCHKQP3RK +* + END diff --git a/TESTING/LIN/clatb4.f b/TESTING/LIN/clatb4.f index e04ba3dfe1..233a8631a8 100644 --- a/TESTING/LIN/clatb4.f +++ b/TESTING/LIN/clatb4.f @@ -225,6 +225,110 @@ SUBROUTINE CLATB4( PATH, IMAT, M, N, TYPE, KL, KU, ANORM, MODE, ELSE ANORM = ONE END IF +* + ELSE IF( LSAMEN( 2, C2, 'QK' ) ) THEN +* +* xQK: truncated QR with pivoting. +* Set parameters to generate a general +* M x N matrix. +* +* Set TYPE, the type of matrix to be generated. 'N' is nonsymmetric. +* + TYPE = 'N' +* +* Set DIST, the type of distribution for the random +* number generator. 'S' is +* + DIST = 'S' +* +* Set the lower and upper bandwidths. +* + IF( IMAT.EQ.2 ) THEN +* +* 2. Random, Diagonal, CNDNUM = 2 +* + KL = 0 + KU = 0 + CNDNUM = TWO + ANORM = ONE + MODE = 3 + ELSE IF( IMAT.EQ.3 ) THEN +* +* 3. Random, Upper triangular, CNDNUM = 2 +* + KL = 0 + KU = MAX( N-1, 0 ) + CNDNUM = TWO + ANORM = ONE + MODE = 3 + ELSE IF( IMAT.EQ.4 ) THEN +* +* 4. Random, Lower triangular, CNDNUM = 2 +* + KL = MAX( M-1, 0 ) + KU = 0 + CNDNUM = TWO + ANORM = ONE + MODE = 3 + ELSE +* +* 5.-19. Rectangular matrix +* + KL = MAX( M-1, 0 ) + KU = MAX( N-1, 0 ) +* + IF( IMAT.GE.5 .AND. IMAT.LE.14 ) THEN +* +* 5.-14. Random, CNDNUM = 2. +* + CNDNUM = TWO + ANORM = ONE + MODE = 3 +* + ELSE IF( IMAT.EQ.15 ) THEN +* +* 15. Random, CNDNUM = sqrt(0.1/EPS) +* + CNDNUM = BADC1 + ANORM = ONE + MODE = 3 +* + ELSE IF( IMAT.EQ.16 ) THEN +* +* 16. Random, CNDNUM = 0.1/EPS +* + CNDNUM = BADC2 + ANORM = ONE + MODE = 3 +* + ELSE IF( IMAT.EQ.17 ) THEN +* +* 17. Random, CNDNUM = 0.1/EPS, +* one small singular value S(N)=1/CNDNUM +* + CNDNUM = BADC2 + ANORM = ONE + MODE = 2 +* + ELSE IF( IMAT.EQ.18 ) THEN +* +* 18. Random, scaled near underflow +* + CNDNUM = TWO + ANORM = SMALL + MODE = 3 +* + ELSE IF( IMAT.EQ.19 ) THEN +* +* 19. Random, scaled near overflow +* + CNDNUM = TWO + ANORM = LARGE + MODE = 3 +* + END IF +* + END IF * ELSE IF( LSAMEN( 2, C2, 'GE' ) ) THEN * diff --git a/TESTING/LIN/cqpt01.f b/TESTING/LIN/cqpt01.f index 79fc2dc66c..149c5bb7c7 100644 --- a/TESTING/LIN/cqpt01.f +++ b/TESTING/LIN/cqpt01.f @@ -33,7 +33,8 @@ *> Householder vectors, and the rest of AF contains a partially updated *> matrix. *> -*> This function returns ||A*P - Q*R||/(||norm(A)||*eps*M) +*> This function returns ||A*P - Q*R|| / ( ||norm(A)||*eps*max(M,N) ) +*> where || . || is matrix one norm. *> \endverbatim * * Arguments: @@ -172,28 +173,28 @@ REAL FUNCTION CQPT01( M, N, K, A, AF, LDA, TAU, JPVT, * NORMA = CLANGE( 'One-norm', M, N, A, LDA, RWORK ) * - DO 30 J = 1, K - DO 10 I = 1, MIN( J, M ) + DO J = 1, K + DO I = 1, MIN( J, M ) WORK( ( J-1 )*M+I ) = AF( I, J ) - 10 CONTINUE - DO 20 I = J + 1, M + END DO + DO I = J + 1, M WORK( ( J-1 )*M+I ) = ZERO - 20 CONTINUE - 30 CONTINUE - DO 40 J = K + 1, N + END DO + END DO + DO J = K + 1, N CALL CCOPY( M, AF( 1, J ), 1, WORK( ( J-1 )*M+1 ), 1 ) - 40 CONTINUE + END DO * CALL CUNMQR( 'Left', 'No transpose', M, N, K, AF, LDA, TAU, WORK, $ M, WORK( M*N+1 ), LWORK-M*N, INFO ) * - DO 50 J = 1, N + DO J = 1, N * * Compare i-th column of QR and jpvt(i)-th column of A * CALL CAXPY( M, CMPLX( -ONE ), A( 1, JPVT( J ) ), 1, $ WORK( ( J-1 )*M+1 ), 1 ) - 50 CONTINUE + END DO * CQPT01 = CLANGE( 'One-norm', M, N, WORK, M, RWORK ) / $ ( REAL( MAX( M, N ) )*SLAMCH( 'Epsilon' ) ) diff --git a/TESTING/LIN/cqrt11.f b/TESTING/LIN/cqrt11.f index 494d5e9cd7..a520849737 100644 --- a/TESTING/LIN/cqrt11.f +++ b/TESTING/LIN/cqrt11.f @@ -157,9 +157,9 @@ REAL FUNCTION CQRT11( M, K, A, LDA, TAU, WORK, LWORK ) CALL CUNM2R( 'Left', 'Conjugate transpose', M, M, K, A, LDA, TAU, $ WORK, M, WORK( M*M+1 ), INFO ) * - DO 10 J = 1, M + DO J = 1, M WORK( ( J-1 )*M+J ) = WORK( ( J-1 )*M+J ) - ONE - 10 CONTINUE + END DO * CQRT11 = CLANGE( 'One-norm', M, M, WORK, M, RDUMMY ) / $ ( REAL( M )*SLAMCH( 'Epsilon' ) ) diff --git a/TESTING/LIN/cqrt12.f b/TESTING/LIN/cqrt12.f index 80ff6dbdf9..0df2d833b9 100644 --- a/TESTING/LIN/cqrt12.f +++ b/TESTING/LIN/cqrt12.f @@ -28,7 +28,7 @@ *> CQRT12 computes the singular values `svlues' of the upper trapezoid *> of A(1:M,1:N) and returns the ratio *> -*> || s - svlues||/(||svlues||*eps*max(M,N)) +*> || svlues -s ||/( ||s||*eps*max(M,N) ) *> \endverbatim * * Arguments: @@ -153,11 +153,11 @@ REAL FUNCTION CQRT12( M, N, A, LDA, S, WORK, LWORK, * Copy upper triangle of A into work * CALL CLASET( 'Full', M, N, CMPLX( ZERO ), CMPLX( ZERO ), WORK, M ) - DO 20 J = 1, N - DO 10 I = 1, MIN( J, M ) + DO J = 1, N + DO I = 1, MIN( J, M ) WORK( ( J-1 )*M+I ) = A( I, J ) - 10 CONTINUE - 20 CONTINUE + END DO + END DO * * Get machine parameters * @@ -206,9 +206,9 @@ REAL FUNCTION CQRT12( M, N, A, LDA, S, WORK, LWORK, * ELSE * - DO 30 I = 1, MN + DO I = 1, MN RWORK( I ) = ZERO - 30 CONTINUE + END DO END IF * * Compare s and singular values of work diff --git a/TESTING/LIN/dchkaa.F b/TESTING/LIN/dchkaa.F old mode 100644 new mode 100755 index ef9d7808ce..74077eb94e --- a/TESTING/LIN/dchkaa.F +++ b/TESTING/LIN/dchkaa.F @@ -63,6 +63,7 @@ *> DLQ 8 List types on next line if 0 < NTYPES < 8 *> DQL 8 List types on next line if 0 < NTYPES < 8 *> DQP 6 List types on next line if 0 < NTYPES < 6 +*> DQK 19 List types on next line if 0 < NTYPES < 19 *> DTZ 3 List types on next line if 0 < NTYPES < 3 *> DLS 6 List types on next line if 0 < NTYPES < 6 *> DEQ @@ -149,12 +150,12 @@ PROGRAM DCHKAA $ NBVAL( MAXIN ), NBVAL2( MAXIN ), $ NSVAL( MAXIN ), NVAL( MAXIN ), NXVAL( MAXIN ), $ RANKVAL( MAXIN ), PIV( NMAX ) - DOUBLE PRECISION E( NMAX ), S( 2*NMAX ) * .. * .. Allocatable Arrays .. INTEGER AllocateStatus - DOUBLE PRECISION, DIMENSION(:), ALLOCATABLE :: RWORK - DOUBLE PRECISION, DIMENSION(:,:), ALLOCATABLE :: A, B, WORK + DOUBLE PRECISION, DIMENSION(:), ALLOCATABLE :: RWORK, S + DOUBLE PRECISION, DIMENSION(:), ALLOCATABLE :: E + DOUBLE PRECISION, DIMENSION(:,:), ALLOCATABLE :: A, B, WORK * .. * .. External Functions .. LOGICAL LSAME, LSAMEN @@ -164,13 +165,13 @@ PROGRAM DCHKAA * .. External Subroutines .. EXTERNAL ALAREQ, DCHKEQ, DCHKGB, DCHKGE, DCHKGT, DCHKLQ, $ DCHKORHR_COL, DCHKPB, DCHKPO, DCHKPS, DCHKPP, - $ DCHKPT, DCHKQ3, DCHKQL, DCHKQR, DCHKRQ, DCHKSP, - $ DCHKSY, DCHKSY_ROOK, DCHKSY_RK, DCHKSY_AA, - $ DCHKTB, DCHKTP, DCHKTR, DCHKTZ, DDRVGB, DDRVGE, - $ DDRVGT, DDRVLS, DDRVPB, DDRVPO, DDRVPP, DDRVPT, - $ DDRVSP, DDRVSY, DDRVSY_ROOK, DDRVSY_RK, - $ DDRVSY_AA, ILAVER, DCHKLQTP, DCHKQRT, DCHKQRTP, - $ DCHKLQT,DCHKTSQR + $ DCHKPT, DCHKQ3, DCHKQP3RK, DCHKQL, DCHKQR, + $ DCHKRQ, DCHKSP, DCHKSY, DCHKSY_ROOK, DCHKSY_RK, + $ DCHKSY_AA, DCHKTB, DCHKTP, DCHKTR, DCHKTZ, + $ DDRVGB, DDRVGE, DDRVGT, DDRVLS, DDRVPB, DDRVPO, + $ DDRVPP, DDRVPT, DDRVSP, DDRVSY, DDRVSY_ROOK, + $ DDRVSY_RK, DDRVSY_AA, ILAVER, DCHKLQTP, DCHKQRT, + $ DCHKQRTP, DCHKLQT,DCHKTSQR * .. * .. Scalars in Common .. LOGICAL LERR, OK @@ -197,6 +198,10 @@ PROGRAM DCHKAA IF (AllocateStatus /= 0) STOP "*** Not enough memory ***" ALLOCATE ( WORK( NMAX, 3*NMAX+MAXRHS+30 ), STAT = AllocateStatus ) IF (AllocateStatus /= 0) STOP "*** Not enough memory ***" + ALLOCATE ( E( NMAX ), STAT = AllocateStatus ) + IF (AllocateStatus /= 0) STOP "*** Not enough memory ***" + ALLOCATE ( S( 2*NMAX ), STAT = AllocateStatus ) + IF (AllocateStatus /= 0) STOP "*** Not enough memory ***" ALLOCATE ( RWORK( 5*NMAX+2*MAXRHS ), STAT = AllocateStatus ) IF (AllocateStatus /= 0) STOP "*** Not enough memory ***" * @@ -919,9 +924,26 @@ PROGRAM DCHKAA CALL ALAREQ( PATH, NMATS, DOTYPE, NTYPES, NIN, NOUT ) * IF( TSTCHK ) THEN - CALL DCHKQ3( DOTYPE, NM, MVAL, NN, NVAL, NNB, NBVAL, NXVAL, - $ THRESH, A( 1, 1 ), A( 1, 2 ), B( 1, 1 ), - $ B( 1, 3 ), WORK, IWORK, NOUT ) + CALL DCHKQ3( DOTYPE, NM, MVAL, NN, NVAL, NNB, NBVAL, + $ NXVAL, THRESH, A( 1, 1 ), A( 1, 2 ), + $ B( 1, 1 ), B( 1, 3 ), WORK, IWORK, NOUT ) + ELSE + WRITE( NOUT, FMT = 9989 )PATH + END IF +* + ELSE IF( LSAMEN( 2, C2, 'QK' ) ) THEN +* +* QK: truncated QR factorization with pivoting +* + NTYPES = 19 + CALL ALAREQ( PATH, NMATS, DOTYPE, NTYPES, NIN, NOUT ) +* + IF( TSTCHK ) THEN + CALL DCHKQP3RK( DOTYPE, NM, MVAL, NN, NVAL, NNS, NSVAL, + $ NNB, NBVAL, NXVAL, THRESH, A( 1, 1 ), + $ A( 1, 2 ), B( 1, 1 ), B( 1, 2 ), + $ B( 1, 3 ), B( 1, 4 ), + $ WORK, IWORK, NOUT ) ELSE WRITE( NOUT, FMT = 9989 )PATH END IF diff --git a/TESTING/LIN/dchkq3.f b/TESTING/LIN/dchkq3.f index 1fdf07252b..494008fa85 100644 --- a/TESTING/LIN/dchkq3.f +++ b/TESTING/LIN/dchkq3.f @@ -30,7 +30,7 @@ *> *> \verbatim *> -*> DCHKQ3 tests DGEQP3. +*> DCHKQ3 tests DGEQP3. *> \endverbatim * * Arguments: diff --git a/TESTING/LIN/dchkqp3rk.f b/TESTING/LIN/dchkqp3rk.f new file mode 100755 index 0000000000..434d2067e2 --- /dev/null +++ b/TESTING/LIN/dchkqp3rk.f @@ -0,0 +1,832 @@ +*> \brief \b DCHKQP3RK +* +* =========== DOCUMENTATION =========== +* +* Online html documentation available at +* http://www.netlib.org/lapack/explore-html/ +* +* Definition: +* =========== +* +* SUBROUTINE DCHKQP3RK( DOTYPE, NM, MVAL, NN, NVAL, NNS, NSVAL, +* $ NNB, NBVAL, NXVAL, THRESH, A, COPYA, +* $ B, COPYB, S, TAU, +* $ WORK, IWORK, NOUT ) +* IMPLICIT NONE +* +* -- LAPACK test routine -- +* -- LAPACK is a software package provided by Univ. of Tennessee, -- +* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- +* +* .. Scalar Arguments .. +* INTEGER NM, NN, NNS, NNB, NOUT +* DOUBLE PRECISION THRESH +* .. +* .. Array Arguments .. +* LOGICAL DOTYPE( * ) +* INTEGER IWORK( * ), MVAL( * ), NBVAL( * ), NSVAL( * ), +* $ NVAL( * ), NXVAL( * ) +* DOUBLE PRECISION A( * ), COPYA( * ), B( * ), COPYB( * ), +* $ S( * ), TAU( * ), WORK( * ) +* .. +* +* +*> \par Purpose: +* ============= +*> +*> \verbatim +*> +*> DCHKQP3RK tests DGEQP3RK. +*> \endverbatim +* +* Arguments: +* ========== +* +*> \param[in] DOTYPE +*> \verbatim +*> DOTYPE is LOGICAL array, dimension (NTYPES) +*> The matrix types to be used for testing. Matrices of type j +*> (for 1 <= j <= NTYPES) are used for testing if DOTYPE(j) = +*> .TRUE.; if DOTYPE(j) = .FALSE., then type j is not used. +*> \endverbatim +*> +*> \param[in] NM +*> \verbatim +*> NM is INTEGER +*> The number of values of M contained in the vector MVAL. +*> \endverbatim +*> +*> \param[in] MVAL +*> \verbatim +*> MVAL is INTEGER array, dimension (NM) +*> The values of the matrix row dimension M. +*> \endverbatim +*> +*> \param[in] NN +*> \verbatim +*> NN is INTEGER +*> The number of values of N contained in the vector NVAL. +*> \endverbatim +*> +*> \param[in] NVAL +*> \verbatim +*> NVAL is INTEGER array, dimension (NN) +*> The values of the matrix column dimension N. +*> \endverbatim +*> +*> \param[in] NNS +*> \verbatim +*> NNS is INTEGER +*> The number of values of NRHS contained in the vector NSVAL. +*> \endverbatim +*> +*> \param[in] NSVAL +*> \verbatim +*> NSVAL is INTEGER array, dimension (NNS) +*> The values of the number of right hand sides NRHS. +*> \endverbatim +*> +*> \param[in] NNB +*> \verbatim +*> NNB is INTEGER +*> The number of values of NB and NX contained in the +*> vectors NBVAL and NXVAL. The blocking parameters are used +*> in pairs (NB,NX). +*> \endverbatim +*> +*> \param[in] NBVAL +*> \verbatim +*> NBVAL is INTEGER array, dimension (NNB) +*> The values of the blocksize NB. +*> \endverbatim +*> +*> \param[in] NXVAL +*> \verbatim +*> NXVAL is INTEGER array, dimension (NNB) +*> The values of the crossover point NX. +*> \endverbatim +*> +*> \param[in] THRESH +*> \verbatim +*> THRESH is DOUBLE PRECISION +*> The threshold value for the test ratios. A result is +*> included in the output file if RESULT >= THRESH. To have +*> every test ratio printed, use THRESH = 0. +*> \endverbatim +*> +*> \param[out] A +*> \verbatim +*> A is DOUBLE PRECISION array, dimension (MMAX*NMAX) +*> where MMAX is the maximum value of M in MVAL and NMAX is the +*> maximum value of N in NVAL. +*> \endverbatim +*> +*> \param[out] COPYA +*> \verbatim +*> COPYA is DOUBLE PRECISION array, dimension (MMAX*NMAX) +*> \endverbatim +*> +*> \param[out] B +*> \verbatim +*> B is DOUBLE PRECISION array, dimension (MMAX*NSMAX) +*> where MMAX is the maximum value of M in MVAL and NSMAX is the +*> maximum value of NRHS in NSVAL. +*> \endverbatim +*> +*> \param[out] COPYB +*> \verbatim +*> COPYB is DOUBLE PRECISION array, dimension (MMAX*NSMAX) +*> \endverbatim +*> +*> \param[out] S +*> \verbatim +*> S is DOUBLE PRECISION array, dimension +*> (min(MMAX,NMAX)) +*> \endverbatim +*> +*> \param[out] TAU +*> \verbatim +*> TAU is DOUBLE PRECISION array, dimension (MMAX) +*> \endverbatim +*> +*> \param[out] WORK +*> \verbatim +*> WORK is DOUBLE PRECISION array, dimension +*> (MMAX*NMAX + 4*NMAX + MMAX) +*> \endverbatim +*> +*> \param[out] IWORK +*> \verbatim +*> IWORK is INTEGER array, dimension (2*NMAX) +*> \endverbatim +*> +*> \param[in] NOUT +*> \verbatim +*> NOUT is INTEGER +*> The unit number for output. +*> \endverbatim +* +* Authors: +* ======== +* +*> \author Univ. of Tennessee +*> \author Univ. of California Berkeley +*> \author Univ. of Colorado Denver +*> \author NAG Ltd. +* +*> \ingroup double_lin +* +* ===================================================================== + SUBROUTINE DCHKQP3RK( DOTYPE, NM, MVAL, NN, NVAL, NNS, NSVAL, + $ NNB, NBVAL, NXVAL, THRESH, A, COPYA, + $ B, COPYB, S, TAU, + $ WORK, IWORK, NOUT ) + IMPLICIT NONE +* +* -- LAPACK test routine -- +* -- LAPACK is a software package provided by Univ. of Tennessee, -- +* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- +* +* .. Scalar Arguments .. + INTEGER NM, NN, NNB, NNS, NOUT + DOUBLE PRECISION THRESH +* .. +* .. Array Arguments .. + LOGICAL DOTYPE( * ) + INTEGER IWORK( * ), NBVAL( * ), MVAL( * ), NVAL( * ), + $ NSVAL( * ), NXVAL( * ) + DOUBLE PRECISION A( * ), COPYA( * ), B( * ), COPYB( * ), + $ S( * ), TAU( * ), WORK( * ) +* .. +* +* ===================================================================== +* +* .. Parameters .. + INTEGER NTYPES + PARAMETER ( NTYPES = 19 ) + INTEGER NTESTS + PARAMETER ( NTESTS = 5 ) + DOUBLE PRECISION ONE, ZERO, BIGNUM + PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0, + $ BIGNUM = 1.0D+38 ) +* .. +* .. Local Scalars .. + CHARACTER DIST, TYPE + CHARACTER*3 PATH + INTEGER I, IHIGH, ILOW, IM, IMAT, IN, INC_ZERO, + $ INB, IND_OFFSET_GEN, + $ IND_IN, IND_OUT, INS, INFO, + $ ISTEP, J, J_INC, J_FIRST_NZ, JB_ZERO, + $ KFACT, KL, KMAX, KU, LDA, LW, LWORK, + $ LWORK_MQR, M, MINMN, MINMNB_GEN, MODE, N, + $ NB, NB_ZERO, NERRS, NFAIL, NB_GEN, NRHS, + $ NRUN, NX, T + DOUBLE PRECISION ANORM, CNDNUM, EPS, ABSTOL, RELTOL, + $ DTEMP, MAXC2NRMK, RELMAXC2NRMK +* .. +* .. Local Arrays .. + INTEGER ISEED( 4 ), ISEEDY( 4 ) + DOUBLE PRECISION RESULT( NTESTS ), RDUMMY( 1 ) +* .. +* .. External Functions .. + DOUBLE PRECISION DLAMCH, DQPT01, DQRT11, DQRT12, DLANGE, + $ DLAPY2 + EXTERNAL DLAMCH, DQPT01, DQRT11, DQRT12, DLANGE +* .. +* .. External Subroutines .. + EXTERNAL ALAERH, ALAHD, ALASUM, DAXPY, DGEQP3RK, + $ DLACPY, DLAORD, DLASET, DLATB4, DLATMS, + $ DORMQR, DSWAP, ICOPY, XLAENV +* .. +* .. Intrinsic Functions .. + INTRINSIC ABS, DBLE, MAX, MIN, MOD +* .. +* .. Scalars in Common .. + LOGICAL LERR, OK + CHARACTER*32 SRNAMT + INTEGER INFOT, IOUNIT +* .. +* .. Common blocks .. + COMMON / INFOC / INFOT, IOUNIT, OK, LERR + COMMON / SRNAMC / SRNAMT +* .. +* .. Data statements .. + DATA ISEEDY / 1988, 1989, 1990, 1991 / +* .. +* .. Executable Statements .. +* +* Initialize constants and the random number seed. +* + PATH( 1: 1 ) = 'Double precision' + PATH( 2: 3 ) = 'QK' + NRUN = 0 + NFAIL = 0 + NERRS = 0 + DO I = 1, 4 + ISEED( I ) = ISEEDY( I ) + END DO + EPS = DLAMCH( 'Epsilon' ) + INFOT = 0 +* + DO IM = 1, NM +* +* Do for each value of M in MVAL. +* + M = MVAL( IM ) + LDA = MAX( 1, M ) +* + DO IN = 1, NN +* +* Do for each value of N in NVAL. +* + N = NVAL( IN ) + MINMN = MIN( M, N ) + LWORK = MAX( 1, M*MAX( M, N )+4*MINMN+MAX( M, N ), + $ M*N + 2*MINMN + 4*N ) +* + DO INS = 1, NNS + NRHS = NSVAL( INS ) +* +* Set up parameters with DLATB4 and generate +* M-by-NRHS B matrix with DLATMS. +* IMAT = 14: +* Random matrix, CNDNUM = 2, NORM = ONE, +* MODE = 3 (geometric distribution of singular values). +* + CALL DLATB4( PATH, 14, M, NRHS, TYPE, KL, KU, ANORM, + $ MODE, CNDNUM, DIST ) +* + SRNAMT = 'DLATMS' + CALL DLATMS( M, NRHS, DIST, ISEED, TYPE, S, MODE, + $ CNDNUM, ANORM, KL, KU, 'No packing', + $ COPYB, LDA, WORK, INFO ) + + +* +* Check error code from DLATMS. +* + IF( INFO.NE.0 ) THEN + CALL ALAERH( PATH, 'DLATMS', INFO, 0, ' ', M, + $ NRHS, -1, -1, -1, 6, NFAIL, NERRS, + $ NOUT ) + CYCLE + END IF +* + DO IMAT = 1, NTYPES +* +* Do the tests only if DOTYPE( IMAT ) is true. +* + IF( .NOT.DOTYPE( IMAT ) ) + $ CYCLE +* +* The type of distribution used to generate the random +* eigen-/singular values: +* ( 'S' for symmetric distribution ) => UNIFORM( -1, 1 ) +* +* Do for each type of NON-SYMMETRIC matrix: CNDNUM NORM MODE +* 1. Zero matrix +* 2. Random, Diagonal, CNDNUM = 2 CNDNUM = 2 ONE 3 ( geometric distribution of singular values ) +* 3. Random, Upper triangular, CNDNUM = 2 CNDNUM = 2 ONE 3 ( geometric distribution of singular values ) +* 4. Random, Lower triangular, CNDNUM = 2 CNDNUM = 2 ONE 3 ( geometric distribution of singular values ) +* 5. Random, First column is zero, CNDNUM = 2 CNDNUM = 2 ONE 3 ( geometric distribution of singular values ) +* 6. Random, Last MINMN column is zero, CNDNUM = 2 CNDNUM = 2 ONE 3 ( geometric distribution of singular values ) +* 7. Random, Last N column is zero, CNDNUM = 2 CNDNUM = 2 ONE 3 ( geometric distribution of singular values ) +* 8. Random, Middle column in MINMN is zero, CNDNUM = 2 CNDNUM = 2 ONE 3 ( geometric distribution of singular values ) +* 9. Random, First half of MINMN columns are zero, CNDNUM = 2 CNDNUM = 2 ONE 3 ( geometric distribution of singular values ) +* 10. Random, Last columns are zero starting from MINMN/2+1, CNDNUM = 2 CNDNUM = 2 ONE 3 ( geometric distribution of singular values ) +* 11. Random, Half MINMN columns in the middle are zero starting +* from MINMN/2-(MINMN/2)/2+1, CNDNUM = 2 CNDNUM = 2 ONE 3 ( geometric distribution of singular values ) +* 12. Random, Odd columns are ZERO, CNDNUM = 2 CNDNUM = 2 ONE 3 ( geometric distribution of singular values ) +* 13. Random, Even columns are ZERO, CNDNUM = 2 CNDNUM = 2 ONE 3 ( geometric distribution of singular values ) +* 14. Random, CNDNUM = 2 CNDNUM = 2 ONE 3 ( geometric distribution of singular values ) +* 15. Random, CNDNUM = sqrt(0.1/EPS) CNDNUM = BADC1 = sqrt(0.1/EPS) ONE 3 ( geometric distribution of singular values ) +* 16. Random, CNDNUM = 0.1/EPS CNDNUM = BADC2 = 0.1/EPS ONE 3 ( geometric distribution of singular values ) +* 17. Random, CNDNUM = 0.1/EPS, CNDNUM = BADC2 = 0.1/EPS ONE 2 ( one small singular value, S(N)=1/CNDNUM ) +* one small singular value S(N)=1/CNDNUM +* 18. Random, CNDNUM = 2, scaled near underflow CNDNUM = 2 SMALL = SAFMIN +* 19. Random, CNDNUM = 2, scaled near overflow CNDNUM = 2 LARGE = 1.0/( 0.25 * ( SAFMIN / EPS ) ) 3 ( geometric distribution of singular values ) +* + IF( IMAT.EQ.1 ) THEN +* +* Matrix 1: Zero matrix +* + CALL DLASET( 'Full', M, N, ZERO, ZERO, COPYA, LDA ) + DO I = 1, MINMN + S( I ) = ZERO + END DO +* + ELSE IF( (IMAT.GE.2 .AND. IMAT.LE.4 ) + $ .OR. (IMAT.GE.14 .AND. IMAT.LE.19 ) ) THEN +* +* Matrices 2-5. +* +* Set up parameters with DLATB4 and generate a test +* matrix with DLATMS. +* + CALL DLATB4( PATH, IMAT, M, N, TYPE, KL, KU, ANORM, + $ MODE, CNDNUM, DIST ) +* + SRNAMT = 'DLATMS' + CALL DLATMS( M, N, DIST, ISEED, TYPE, S, MODE, + $ CNDNUM, ANORM, KL, KU, 'No packing', + $ COPYA, LDA, WORK, INFO ) +* +* Check error code from DLATMS. +* + IF( INFO.NE.0 ) THEN + CALL ALAERH( PATH, 'DLATMS', INFO, 0, ' ', M, N, + $ -1, -1, -1, IMAT, NFAIL, NERRS, + $ NOUT ) + CYCLE + END IF +* + CALL DLAORD( 'Decreasing', MINMN, S, 1 ) +* + ELSE IF( MINMN.GE.2 + $ .AND. IMAT.GE.5 .AND. IMAT.LE.13 ) THEN +* +* Rectangular matrices 5-13 that contain zero columns, +* only for matrices MINMN >=2. +* +* JB_ZERO is the column index of ZERO block. +* NB_ZERO is the column block size of ZERO block. +* NB_GEN is the column blcok size of the +* generated block. +* J_INC in the non_zero column index increment +* for matrix 12 and 13. +* J_FIRS_NZ is the index of the first non-zero +* column. +* + IF( IMAT.EQ.5 ) THEN +* +* First column is zero. +* + JB_ZERO = 1 + NB_ZERO = 1 + NB_GEN = N - NB_ZERO +* + ELSE IF( IMAT.EQ.6 ) THEN +* +* Last column MINMN is zero. +* + JB_ZERO = MINMN + NB_ZERO = 1 + NB_GEN = N - NB_ZERO +* + ELSE IF( IMAT.EQ.7 ) THEN +* +* Last column N is zero. +* + JB_ZERO = N + NB_ZERO = 1 + NB_GEN = N - NB_ZERO +* + ELSE IF( IMAT.EQ.8 ) THEN +* +* Middle column in MINMN is zero. +* + JB_ZERO = MINMN / 2 + 1 + NB_ZERO = 1 + NB_GEN = N - NB_ZERO +* + ELSE IF( IMAT.EQ.9 ) THEN +* +* First half of MINMN columns is zero. +* + JB_ZERO = 1 + NB_ZERO = MINMN / 2 + NB_GEN = N - NB_ZERO +* + ELSE IF( IMAT.EQ.10 ) THEN +* +* Last columns are zero columns, +* starting from (MINMN / 2 + 1) column. +* + JB_ZERO = MINMN / 2 + 1 + NB_ZERO = N - JB_ZERO + 1 + NB_GEN = N - NB_ZERO +* + ELSE IF( IMAT.EQ.11 ) THEN +* +* Half of the columns in the middle of MINMN +* columns is zero, starting from +* MINMN/2 - (MINMN/2)/2 + 1 column. +* + JB_ZERO = MINMN / 2 - (MINMN / 2) / 2 + 1 + NB_ZERO = MINMN / 2 + NB_GEN = N - NB_ZERO +* + ELSE IF( IMAT.EQ.12 ) THEN +* +* Odd-numbered columns are zero, +* + NB_GEN = N / 2 + NB_ZERO = N - NB_GEN + J_INC = 2 + J_FIRST_NZ = 2 +* + ELSE IF( IMAT.EQ.13 ) THEN +* +* Even-numbered columns are zero. +* + NB_ZERO = N / 2 + NB_GEN = N - NB_ZERO + J_INC = 2 + J_FIRST_NZ = 1 +* + END IF +* +* +* 1) Set the first NB_ZERO columns in COPYA(1:M,1:N) +* to zero. +* + CALL DLASET( 'Full', M, NB_ZERO, ZERO, ZERO, + $ COPYA, LDA ) +* +* 2) Generate an M-by-(N-NB_ZERO) matrix with the +* chosen singular value distribution +* in COPYA(1:M,NB_ZERO+1:N). +* + CALL DLATB4( PATH, IMAT, M, NB_GEN, TYPE, KL, KU, + $ ANORM, MODE, CNDNUM, DIST ) +* + SRNAMT = 'DLATMS' +* + IND_OFFSET_GEN = NB_ZERO * LDA +* + CALL DLATMS( M, NB_GEN, DIST, ISEED, TYPE, S, MODE, + $ CNDNUM, ANORM, KL, KU, 'No packing', + $ COPYA( IND_OFFSET_GEN + 1 ), LDA, + $ WORK, INFO ) +* +* Check error code from DLATMS. +* + IF( INFO.NE.0 ) THEN + CALL ALAERH( PATH, 'DLATMS', INFO, 0, ' ', M, + $ NB_GEN, -1, -1, -1, IMAT, NFAIL, + $ NERRS, NOUT ) + CYCLE + END IF +* +* 3) Swap the gererated colums from the right side +* NB_GEN-size block in COPYA into correct column +* positions. +* + IF( IMAT.EQ.6 + $ .OR. IMAT.EQ.7 + $ .OR. IMAT.EQ.8 + $ .OR. IMAT.EQ.10 + $ .OR. IMAT.EQ.11 ) THEN +* +* Move by swapping the generated columns +* from the right NB_GEN-size block from +* (NB_ZERO+1:NB_ZERO+JB_ZERO) +* into columns (1:JB_ZERO-1). +* + DO J = 1, JB_ZERO-1, 1 + CALL DSWAP( M, + $ COPYA( ( NB_ZERO+J-1)*LDA+1), 1, + $ COPYA( (J-1)*LDA + 1 ), 1 ) + END DO +* + ELSE IF( IMAT.EQ.12 .OR. IMAT.EQ.13 ) THEN +* +* ( IMAT = 12, Odd-numbered ZERO columns. ) +* Swap the generated columns from the right +* NB_GEN-size block into the even zero colums in the +* left NB_ZERO-size block. +* +* ( IMAT = 13, Even-numbered ZERO columns. ) +* Swap the generated columns from the right +* NB_GEN-size block into the odd zero colums in the +* left NB_ZERO-size block. +* + DO J = 1, NB_GEN, 1 + IND_OUT = ( NB_ZERO+J-1 )*LDA + 1 + IND_IN = ( J_INC*(J-1)+(J_FIRST_NZ-1) )*LDA + $ + 1 + CALL DSWAP( M, + $ COPYA( IND_OUT ), 1, + $ COPYA( IND_IN), 1 ) + END DO +* + END IF +* +* 5) Order the singular values generated by +* DLAMTS in decreasing order and add trailing zeros +* that correspond to zero columns. +* The total number of singular values is MINMN. +* + MINMNB_GEN = MIN( M, NB_GEN ) +* + DO I = MINMNB_GEN+1, MINMN + S( I ) = ZERO + END DO +* + ELSE +* +* IF(MINMN.LT.2) skip this size for this matrix type. +* + CYCLE + END IF +* +* Initialize a copy array for a pivot array for DGEQP3RK. +* + DO I = 1, N + IWORK( I ) = 0 + END DO +* + DO INB = 1, NNB +* +* Do for each pair of values (NB,NX) in NBVAL and NXVAL. +* + NB = NBVAL( INB ) + CALL XLAENV( 1, NB ) + NX = NXVAL( INB ) + CALL XLAENV( 3, NX ) +* +* We do MIN(M,N)+1 because we need a test for KMAX > N, +* when KMAX is larger than MIN(M,N), KMAX should be +* KMAX = MIN(M,N) +* + DO KMAX = 0, MIN(M,N)+1 +* +* Get a working copy of COPYA into A( 1:M,1:N ). +* Get a working copy of COPYB into A( 1:M, (N+1):NRHS ). +* Get a working copy of COPYB into into B( 1:M, 1:NRHS ). +* Get a working copy of IWORK(1:N) awith zeroes into +* which is going to be used as pivot array IWORK( N+1:2N ). +* NOTE: IWORK(2N+1:3N) is going to be used as a WORK array +* for the routine. +* + CALL DLACPY( 'All', M, N, COPYA, LDA, A, LDA ) + CALL DLACPY( 'All', M, NRHS, COPYB, LDA, + $ A( LDA*N + 1 ), LDA ) + CALL DLACPY( 'All', M, NRHS, COPYB, LDA, + $ B, LDA ) + CALL ICOPY( N, IWORK( 1 ), 1, IWORK( N+1 ), 1 ) +* + ABSTOL = -1.0 + RELTOL = -1.0 +* +* Compute the QR factorization with pivoting of A +* + LW = MAX( 1, MAX( 2*N + NB*( N+NRHS+1 ), + $ 3*N + NRHS - 1 ) ) +* +* Compute DGEQP3RK factorization of A. +* + SRNAMT = 'DGEQP3RK' + CALL DGEQP3RK( M, N, NRHS, KMAX, ABSTOL, RELTOL, + $ A, LDA, KFACT, MAXC2NRMK, + $ RELMAXC2NRMK, IWORK( N+1 ), TAU, + $ WORK, LW, IWORK( 2*N+1 ), INFO ) +* +* Check error code from DGEQP3RK. +* + IF( INFO.LT.0 ) + $ CALL ALAERH( PATH, 'DGEQP3RK', INFO, 0, ' ', + $ M, N, NX, -1, NB, IMAT, + $ NFAIL, NERRS, NOUT ) +* +* Compute test 1: +* +* This test in only for the full rank factorization of +* the matrix A. +* +* Array S(1:min(M,N)) contains svd(A) the sigular values +* of the original matrix A in decreasing absolute value +* order. The test computes svd(R), the vector sigular +* values of the upper trapezoid of A(1:M,1:N) that +* contains the factor R, in decreasing order. The test +* returns the ratio: +* +* 2-norm(svd(R) - svd(A)) / ( max(M,N) * 2-norm(svd(A)) * EPS ) +* + IF( KFACT.EQ.MINMN ) THEN +* + RESULT( 1 ) = DQRT12( M, N, A, LDA, S, WORK, + $ LWORK ) +* + DO T = 1, 1 + IF( RESULT( T ).GE.THRESH ) THEN + IF( NFAIL.EQ.0 .AND. NERRS.EQ.0 ) + $ CALL ALAHD( NOUT, PATH ) + WRITE( NOUT, FMT = 9999 ) 'DGEQP3RK', M, N, + $ NRHS, KMAX, ABSTOL, RELTOL, NB, NX, + $ IMAT, T, RESULT( T ) + NFAIL = NFAIL + 1 + END IF + END DO + NRUN = NRUN + 1 +* +* End test 1 +* + END IF +* +* Compute test 2: +* +* The test returns the ratio: +* +* 1-norm( A*P - Q*R ) / ( max(M,N) * 1-norm(A) * EPS ) +* + RESULT( 2 ) = DQPT01( M, N, KFACT, COPYA, A, LDA, TAU, + $ IWORK( N+1 ), WORK, LWORK ) +* +* Compute test 3: +* +* The test returns the ratio: +* +* 1-norm( Q**T * Q - I ) / ( M * EPS ) +* + RESULT( 3 ) = DQRT11( M, KFACT, A, LDA, TAU, WORK, + $ LWORK ) +* +* Print information about the tests that did not pass +* the threshold. +* + DO T = 2, 3 + IF( RESULT( T ).GE.THRESH ) THEN + IF( NFAIL.EQ.0 .AND. NERRS.EQ.0 ) + $ CALL ALAHD( NOUT, PATH ) + WRITE( NOUT, FMT = 9999 ) 'DGEQP3RK', M, N, + $ NRHS, KMAX, ABSTOL, RELTOL, + $ NB, NX, IMAT, T, RESULT( T ) + NFAIL = NFAIL + 1 + END IF + END DO + NRUN = NRUN + 2 +* +* Compute test 4: +* +* This test is only for the factorizations with the +* rank greater than 2. +* The elements on the diagonal of R should be non- +* increasing. +* +* The test returns the ratio: +* +* Returns 1.0D+100 if abs(R(K+1,K+1)) > abs(R(K,K)), +* K=1:KFACT-1 +* + IF( MIN(KFACT, MINMN).GE.2 ) THEN +* + DO J = 1, KFACT-1, 1 + + DTEMP = (( ABS( A( (J-1)*M+J ) ) - + $ ABS( A( (J)*M+J+1 ) ) ) / + $ ABS( A(1) ) ) +* + IF( DTEMP.LT.ZERO ) THEN + RESULT( 4 ) = BIGNUM + END IF +* + END DO +* +* Print information about the tests that did not +* pass the threshold. +* + DO T = 4, 4 + IF( RESULT( T ).GE.THRESH ) THEN + IF( NFAIL.EQ.0 .AND. NERRS.EQ.0 ) + $ CALL ALAHD( NOUT, PATH ) + WRITE( NOUT, FMT = 9999 ) 'DGEQP3RK', + $ M, N, NRHS, KMAX, ABSTOL, RELTOL, + $ NB, NX, IMAT, T, + $ RESULT( T ) + NFAIL = NFAIL + 1 + END IF + END DO + NRUN = NRUN + 1 +* +* End test 4. +* + END IF +* +* Compute test 5: +* +* This test in only for matrix A with min(M,N) > 0. +* +* The test returns the ratio: +* +* 1-norm(Q**T * B - Q**T * B ) / +* ( M * EPS ) +* +* (1) Compute B:=Q**T * B in the matrix B. +* + IF( MINMN.GT.0 ) THEN +* + LWORK_MQR = MAX(1, NRHS) + CALL DORMQR( 'Left', 'Transpose', + $ M, NRHS, KFACT, A, LDA, TAU, B, LDA, + $ WORK, LWORK_MQR, INFO ) +* + DO I = 1, NRHS +* +* Compare N+J-th column of A and J-column of B. +* + CALL DAXPY( M, -ONE, A( ( N+I-1 )*LDA+1 ), 1, + $ B( ( I-1 )*LDA+1 ), 1 ) + END DO +* + RESULT( 5 ) = + $ ABS( + $ DLANGE( 'One-norm', M, NRHS, B, LDA, RDUMMY ) / + $ ( DBLE( M )*DLAMCH( 'Epsilon' ) ) + $ ) +* +* Print information about the tests that did not pass +* the threshold. +* + DO T = 5, 5 + IF( RESULT( T ).GE.THRESH ) THEN + IF( NFAIL.EQ.0 .AND. NERRS.EQ.0 ) + $ CALL ALAHD( NOUT, PATH ) + WRITE( NOUT, FMT = 9999 ) 'DGEQP3RK', M, N, + $ NRHS, KMAX, ABSTOL, RELTOL, + $ NB, NX, IMAT, T, RESULT( T ) + NFAIL = NFAIL + 1 + END IF + END DO + NRUN = NRUN + 1 +* +* End compute test 5. +* + END IF +* +* END DO KMAX = 1, MIN(M,N)+1 +* + END DO +* +* END DO for INB = 1, NNB +* + END DO +* +* END DO for IMAT = 1, NTYPES +* + END DO +* +* END DO for INS = 1, NNS +* + END DO +* +* END DO for IN = 1, NN +* + END DO +* +* END DO for IM = 1, NM +* + END DO +* +* Print a summary of the results. +* + CALL ALASUM( PATH, NOUT, NFAIL, NRUN, NERRS ) +* + 9999 FORMAT( 1X, A, ' M =', I5, ', N =', I5, ', NRHS =', I5, + $ ', KMAX =', I5, ', ABSTOL =', G12.5, + $ ', RELTOL =', G12.5, ', NB =', I4, ', NX =', I4, + $ ', type ', I2, ', test ', I2, ', ratio =', G12.5 ) +* +* End of DCHKQP3RK +* + END diff --git a/TESTING/LIN/dlatb4.f b/TESTING/LIN/dlatb4.f old mode 100644 new mode 100755 index 28689877c7..f3bccd45b2 --- a/TESTING/LIN/dlatb4.f +++ b/TESTING/LIN/dlatb4.f @@ -133,7 +133,7 @@ SUBROUTINE DLATB4( PATH, IMAT, M, N, TYPE, KL, KU, ANORM, MODE, * * .. Parameters .. DOUBLE PRECISION SHRINK, TENTH - PARAMETER ( SHRINK = 0.25D0, TENTH = 0.1D+0 ) + PARAMETER ( SHRINK = 0.25D+0, TENTH = 0.1D+0 ) DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) DOUBLE PRECISION TWO @@ -224,6 +224,110 @@ SUBROUTINE DLATB4( PATH, IMAT, M, N, TYPE, KL, KU, ANORM, MODE, ELSE ANORM = ONE END IF +* + ELSE IF( LSAMEN( 2, C2, 'QK' ) ) THEN +* +* xQK: truncated QR with pivoting. +* Set parameters to generate a general +* M x N matrix. +* +* Set TYPE, the type of matrix to be generated. 'N' is nonsymmetric. +* + TYPE = 'N' +* +* Set DIST, the type of distribution for the random +* number generator. 'S' is +* + DIST = 'S' +* +* Set the lower and upper bandwidths. +* + IF( IMAT.EQ.2 ) THEN +* +* 2. Random, Diagonal, CNDNUM = 2 +* + KL = 0 + KU = 0 + CNDNUM = TWO + ANORM = ONE + MODE = 3 + ELSE IF( IMAT.EQ.3 ) THEN +* +* 3. Random, Upper triangular, CNDNUM = 2 +* + KL = 0 + KU = MAX( N-1, 0 ) + CNDNUM = TWO + ANORM = ONE + MODE = 3 + ELSE IF( IMAT.EQ.4 ) THEN +* +* 4. Random, Lower triangular, CNDNUM = 2 +* + KL = MAX( M-1, 0 ) + KU = 0 + CNDNUM = TWO + ANORM = ONE + MODE = 3 + ELSE +* +* 5.-19. Rectangular matrix +* + KL = MAX( M-1, 0 ) + KU = MAX( N-1, 0 ) +* + IF( IMAT.GE.5 .AND. IMAT.LE.14 ) THEN +* +* 5.-14. Random, CNDNUM = 2. +* + CNDNUM = TWO + ANORM = ONE + MODE = 3 +* + ELSE IF( IMAT.EQ.15 ) THEN +* +* 15. Random, CNDNUM = sqrt(0.1/EPS) +* + CNDNUM = BADC1 + ANORM = ONE + MODE = 3 +* + ELSE IF( IMAT.EQ.16 ) THEN +* +* 16. Random, CNDNUM = 0.1/EPS +* + CNDNUM = BADC2 + ANORM = ONE + MODE = 3 +* + ELSE IF( IMAT.EQ.17 ) THEN +* +* 17. Random, CNDNUM = 0.1/EPS, +* one small singular value S(N)=1/CNDNUM +* + CNDNUM = BADC2 + ANORM = ONE + MODE = 2 +* + ELSE IF( IMAT.EQ.18 ) THEN +* +* 18. Random, scaled near underflow +* + CNDNUM = TWO + ANORM = SMALL + MODE = 3 +* + ELSE IF( IMAT.EQ.19 ) THEN +* +* 19. Random, scaled near overflow +* + CNDNUM = TWO + ANORM = LARGE + MODE = 3 +* + END IF +* + END IF * ELSE IF( LSAMEN( 2, C2, 'GE' ) ) THEN * diff --git a/TESTING/LIN/dqpt01.f b/TESTING/LIN/dqpt01.f old mode 100644 new mode 100755 index 8efbdc774d..af3f5dd364 --- a/TESTING/LIN/dqpt01.f +++ b/TESTING/LIN/dqpt01.f @@ -28,12 +28,13 @@ *> *> DQPT01 tests the QR-factorization with pivoting of a matrix A. The *> array AF contains the (possibly partial) QR-factorization of A, where -*> the upper triangle of AF(1:k,1:k) is a partial triangular factor, -*> the entries below the diagonal in the first k columns are the +*> the upper triangle of AF(1:K,1:K) is a partial triangular factor, +*> the entries below the diagonal in the first K columns are the *> Householder vectors, and the rest of AF contains a partially updated *> matrix. *> -*> This function returns ||A*P - Q*R||/(||norm(A)||*eps*M) +*> This function returns ||A*P - Q*R|| / ( ||norm(A)||*eps*max(M,N) ), +*> where || . || is matrix one norm. *> \endverbatim * * Arguments: @@ -172,28 +173,41 @@ DOUBLE PRECISION FUNCTION DQPT01( M, N, K, A, AF, LDA, TAU, JPVT, * NORMA = DLANGE( 'One-norm', M, N, A, LDA, RWORK ) * - DO 30 J = 1, K - DO 10 I = 1, MIN( J, M ) + DO J = 1, K +* +* Copy the upper triangular part of the factor R stored +* in AF(1:K,1:K) into the work array WORK. +* + DO I = 1, MIN( J, M ) WORK( ( J-1 )*M+I ) = AF( I, J ) - 10 CONTINUE - DO 20 I = J + 1, M + END DO +* +* Zero out the elements below the diagonal in the work array. +* + DO I = J + 1, M WORK( ( J-1 )*M+I ) = ZERO - 20 CONTINUE - 30 CONTINUE - DO 40 J = K + 1, N + END DO + END DO +* +* Copy columns (K+1,N) from AF into the work array WORK. +* AF(1:K,K+1:N) contains the rectangular block of the upper trapezoidal +* factor R, AF(K+1:M,K+1:N) contains the partially updated residual +* matrix of R. +* + DO J = K + 1, N CALL DCOPY( M, AF( 1, J ), 1, WORK( ( J-1 )*M+1 ), 1 ) - 40 CONTINUE + END DO * CALL DORMQR( 'Left', 'No transpose', M, N, K, AF, LDA, TAU, WORK, $ M, WORK( M*N+1 ), LWORK-M*N, INFO ) * - DO 50 J = 1, N + DO J = 1, N * -* Compare i-th column of QR and jpvt(i)-th column of A +* Compare J-th column of QR and JPVT(J)-th column of A. * CALL DAXPY( M, -ONE, A( 1, JPVT( J ) ), 1, WORK( ( J-1 )*M+1 ), $ 1 ) - 50 CONTINUE + END DO * DQPT01 = DLANGE( 'One-norm', M, N, WORK, M, RWORK ) / $ ( DBLE( MAX( M, N ) )*DLAMCH( 'Epsilon' ) ) diff --git a/TESTING/LIN/dqrt11.f b/TESTING/LIN/dqrt11.f old mode 100644 new mode 100755 index 33c7fab378..38bbeb8228 --- a/TESTING/LIN/dqrt11.f +++ b/TESTING/LIN/dqrt11.f @@ -157,9 +157,9 @@ DOUBLE PRECISION FUNCTION DQRT11( M, K, A, LDA, TAU, WORK, LWORK ) CALL DORM2R( 'Left', 'Transpose', M, M, K, A, LDA, TAU, WORK, M, $ WORK( M*M+1 ), INFO ) * - DO 10 J = 1, M + DO J = 1, M WORK( ( J-1 )*M+J ) = WORK( ( J-1 )*M+J ) - ONE - 10 CONTINUE + END DO * DQRT11 = DLANGE( 'One-norm', M, M, WORK, M, RDUMMY ) / $ ( DBLE( M )*DLAMCH( 'Epsilon' ) ) diff --git a/TESTING/LIN/dqrt12.f b/TESTING/LIN/dqrt12.f old mode 100644 new mode 100755 index a3bfbebb3d..b8a124c591 --- a/TESTING/LIN/dqrt12.f +++ b/TESTING/LIN/dqrt12.f @@ -26,7 +26,7 @@ *> DQRT12 computes the singular values `svlues' of the upper trapezoid *> of A(1:M,1:N) and returns the ratio *> -*> || s - svlues||/(||svlues||*eps*max(M,N)) +*> || svlues - s ||/(||s||*eps*max(M,N)) *> \endverbatim * * Arguments: @@ -144,11 +144,11 @@ DOUBLE PRECISION FUNCTION DQRT12( M, N, A, LDA, S, WORK, LWORK ) * Copy upper triangle of A into work * CALL DLASET( 'Full', M, N, ZERO, ZERO, WORK, M ) - DO 20 J = 1, N - DO 10 I = 1, MIN( J, M ) + DO J = 1, N + DO I = 1, MIN( J, M ) WORK( ( J-1 )*M+I ) = A( I, J ) - 10 CONTINUE - 20 CONTINUE + END DO + END DO * * Get machine parameters * @@ -197,16 +197,18 @@ DOUBLE PRECISION FUNCTION DQRT12( M, N, A, LDA, S, WORK, LWORK ) * ELSE * - DO 30 I = 1, MN + DO I = 1, MN WORK( M*N+I ) = ZERO - 30 CONTINUE + END DO END IF * * Compare s and singular values of work * CALL DAXPY( MN, -ONE, S, 1, WORK( M*N+1 ), 1 ) +* DQRT12 = DASUM( MN, WORK( M*N+1 ), 1 ) / - $ ( DLAMCH( 'Epsilon' )*DBLE( MAX( M, N ) ) ) + $ ( DLAMCH('Epsilon') * DBLE( MAX( M, N ) ) ) +* IF( NRMSVL.NE.ZERO ) $ DQRT12 = DQRT12 / NRMSVL * diff --git a/TESTING/LIN/schkaa.F b/TESTING/LIN/schkaa.F index a5b826d06e..2b9f2ea452 100644 --- a/TESTING/LIN/schkaa.F +++ b/TESTING/LIN/schkaa.F @@ -63,6 +63,7 @@ *> SLQ 8 List types on next line if 0 < NTYPES < 8 *> SQL 8 List types on next line if 0 < NTYPES < 8 *> SQP 6 List types on next line if 0 < NTYPES < 6 +*> DQK 19 List types on next line if 0 < NTYPES < 19 *> STZ 3 List types on next line if 0 < NTYPES < 3 *> SLS 6 List types on next line if 0 < NTYPES < 6 *> SEQ @@ -147,11 +148,11 @@ PROGRAM SCHKAA $ NBVAL( MAXIN ), NBVAL2( MAXIN ), $ NSVAL( MAXIN ), NVAL( MAXIN ), NXVAL( MAXIN ), $ RANKVAL( MAXIN ), PIV( NMAX ) - REAL E( NMAX ), S( 2*NMAX ) * .. * .. Allocatable Arrays .. INTEGER AllocateStatus - REAL, DIMENSION(:), ALLOCATABLE :: RWORK + REAL, DIMENSION(:), ALLOCATABLE :: RWORK, S + REAL, DIMENSION(:), ALLOCATABLE :: E REAL, DIMENSION(:,:), ALLOCATABLE :: A, B, WORK * .. * .. External Functions .. @@ -162,13 +163,13 @@ PROGRAM SCHKAA * .. External Subroutines .. EXTERNAL ALAREQ, SCHKEQ, SCHKGB, SCHKGE, SCHKGT, SCHKLQ, $ SCHKORHR_COL, SCHKPB, SCHKPO, SCHKPS, SCHKPP, - $ SCHKPT, SCHKQ3, SCHKQL, SCHKQR, SCHKRQ, SCHKSP, - $ SCHKSY, SCHKSY_ROOK, SCHKSY_RK, SCHKSY_AA, - $ SCHKTB, SCHKTP, SCHKTR, SCHKTZ, SDRVGB, SDRVGE, - $ SDRVGT, SDRVLS, SDRVPB, SDRVPO, SDRVPP, SDRVPT, - $ SDRVSP, SDRVSY, SDRVSY_ROOK, SDRVSY_RK, - $ SDRVSY_AA, ILAVER, SCHKLQTP, SCHKQRT, SCHKQRTP, - $ SCHKLQT, SCHKTSQR + $ SCHKPT, SCHKQ3, SCHKQP3RK, SCHKQL, SCHKQR, + $ SCHKRQ, SCHKSP, SCHKSY, SCHKSY_ROOK, SCHKSY_RK, + $ SCHKSY_AA, SCHKTB, SCHKTP, SCHKTR, SCHKTZ, + $ SDRVGB, SDRVGE, SDRVGT, SDRVLS, SDRVPB, SDRVPO, + $ SDRVPP, SDRVPT, SDRVSP, SDRVSY, SDRVSY_ROOK, + $ SDRVSY_RK, SDRVSY_AA, ILAVER, SCHKLQTP, SCHKQRT, + $ SCHKQRTP, SCHKLQT, SCHKTSQR * .. * .. Scalars in Common .. LOGICAL LERR, OK @@ -188,13 +189,17 @@ PROGRAM SCHKAA * .. * .. Allocate memory dynamically .. * - ALLOCATE (A( ( KDMAX+1 )*NMAX, 7 ), STAT = AllocateStatus ) + ALLOCATE ( A( ( KDMAX+1 )*NMAX, 7 ), STAT = AllocateStatus ) IF (AllocateStatus /= 0) STOP "*** Not enough memory ***" - ALLOCATE (B( NMAX*MAXRHS, 4 ), STAT = AllocateStatus ) + ALLOCATE ( B( NMAX*MAXRHS, 4 ), STAT = AllocateStatus ) IF (AllocateStatus /= 0) STOP "*** Not enough memory ***" - ALLOCATE (WORK( NMAX, NMAX+MAXRHS+30 ) , STAT = AllocateStatus ) + ALLOCATE ( WORK( NMAX, 3*NMAX+MAXRHS+30 ), STAT = AllocateStatus ) IF (AllocateStatus /= 0) STOP "*** Not enough memory ***" - ALLOCATE (RWORK( 5*NMAX+2*MAXRHS ), STAT = AllocateStatus ) + ALLOCATE ( E( NMAX ), STAT = AllocateStatus ) + IF (AllocateStatus /= 0) STOP "*** Not enough memory ***" + ALLOCATE ( S( 2*NMAX ), STAT = AllocateStatus ) + IF (AllocateStatus /= 0) STOP "*** Not enough memory ***" + ALLOCATE ( RWORK( 5*NMAX+2*MAXRHS ), STAT = AllocateStatus ) IF (AllocateStatus /= 0) STOP "*** Not enough memory ***" * .. * .. Executable Statements .. @@ -920,6 +925,23 @@ PROGRAM SCHKAA ELSE WRITE( NOUT, FMT = 9989 )PATH END IF +* + ELSE IF( LSAMEN( 2, C2, 'QK' ) ) THEN +* +* QK: truncated QR factorization with pivoting +* + NTYPES = 19 + CALL ALAREQ( PATH, NMATS, DOTYPE, NTYPES, NIN, NOUT ) +* + IF( TSTCHK ) THEN + CALL SCHKQP3RK( DOTYPE, NM, MVAL, NN, NVAL, NNS, NSVAL, + $ NNB, NBVAL, NXVAL, THRESH, A( 1, 1 ), + $ A( 1, 2 ), B( 1, 1 ), B( 1, 2 ), + $ B( 1, 3 ), B( 1, 4 ), + $ WORK, IWORK, NOUT ) + ELSE + WRITE( NOUT, FMT = 9989 )PATH + END IF * ELSE IF( LSAMEN( 2, C2, 'TZ' ) ) THEN * diff --git a/TESTING/LIN/schkqp3rk.f b/TESTING/LIN/schkqp3rk.f new file mode 100755 index 0000000000..36cf9370ea --- /dev/null +++ b/TESTING/LIN/schkqp3rk.f @@ -0,0 +1,831 @@ +*> \brief \b SCHKQP3RK +* +* =========== DOCUMENTATION =========== +* +* Online html documentation available at +* http://www.netlib.org/lapack/explore-html/ +* +* Definition: +* =========== +* +* SUBROUTINE SCHKQP3RK( DOTYPE, NM, MVAL, NN, NVAL, NNS, NSVAL, +* $ NNB, NBVAL, NXVAL, THRESH, A, COPYA, +* $ B, COPYB, S, TAU, +* $ WORK, IWORK, NOUT ) +* IMPLICIT NONE +* +* -- LAPACK test routine -- +* -- LAPACK is a software package provided by Univ. of Tennessee, -- +* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- +* +* .. Scalar Arguments .. +* INTEGER NM, NN, NNS, NNB, NOUT +* REAL THRESH +* .. +* .. Array Arguments .. +* LOGICAL DOTYPE( * ) +* INTEGER IWORK( * ), MVAL( * ), NBVAL( * ), NSVAL( * ), +* $ NVAL( * ), NXVAL( * ) +* REAL A( * ), COPYA( * ), B( * ), COPYB( * ), +* $ S( * ), TAU( * ), WORK( * ) +* .. +* +* +*> \par Purpose: +* ============= +*> +*> \verbatim +*> +*> SCHKQP3RK tests SGEQP3RK. +*> \endverbatim +* +* Arguments: +* ========== +* +*> \param[in] DOTYPE +*> \verbatim +*> DOTYPE is LOGICAL array, dimension (NTYPES) +*> The matrix types to be used for testing. Matrices of type j +*> (for 1 <= j <= NTYPES) are used for testing if DOTYPE(j) = +*> .TRUE.; if DOTYPE(j) = .FALSE., then type j is not used. +*> \endverbatim +*> +*> \param[in] NM +*> \verbatim +*> NM is INTEGER +*> The number of values of M contained in the vector MVAL. +*> \endverbatim +*> +*> \param[in] MVAL +*> \verbatim +*> MVAL is INTEGER array, dimension (NM) +*> The values of the matrix row dimension M. +*> \endverbatim +*> +*> \param[in] NN +*> \verbatim +*> NN is INTEGER +*> The number of values of N contained in the vector NVAL. +*> \endverbatim +*> +*> \param[in] NVAL +*> \verbatim +*> NVAL is INTEGER array, dimension (NN) +*> The values of the matrix column dimension N. +*> \endverbatim +*> +*> \param[in] NNS +*> \verbatim +*> NNS is INTEGER +*> The number of values of NRHS contained in the vector NSVAL. +*> \endverbatim +*> +*> \param[in] NSVAL +*> \verbatim +*> NSVAL is INTEGER array, dimension (NNS) +*> The values of the number of right hand sides NRHS. +*> \endverbatim +*> +*> \param[in] NNB +*> \verbatim +*> NNB is INTEGER +*> The number of values of NB and NX contained in the +*> vectors NBVAL and NXVAL. The blocking parameters are used +*> in pairs (NB,NX). +*> \endverbatim +*> +*> \param[in] NBVAL +*> \verbatim +*> NBVAL is INTEGER array, dimension (NNB) +*> The values of the blocksize NB. +*> \endverbatim +*> +*> \param[in] NXVAL +*> \verbatim +*> NXVAL is INTEGER array, dimension (NNB) +*> The values of the crossover point NX. +*> \endverbatim +*> +*> \param[in] THRESH +*> \verbatim +*> THRESH is REAL +*> The threshold value for the test ratios. A result is +*> included in the output file if RESULT >= THRESH. To have +*> every test ratio printed, use THRESH = 0. +*> \endverbatim +*> +*> \param[out] A +*> \verbatim +*> A is REAL array, dimension (MMAX*NMAX) +*> where MMAX is the maximum value of M in MVAL and NMAX is the +*> maximum value of N in NVAL. +*> \endverbatim +*> +*> \param[out] COPYA +*> \verbatim +*> COPYA is REAL array, dimension (MMAX*NMAX) +*> \endverbatim +*> +*> \param[out] B +*> \verbatim +*> B is REAL array, dimension (MMAX*NSMAX) +*> where MMAX is the maximum value of M in MVAL and NSMAX is the +*> maximum value of NRHS in NSVAL. +*> \endverbatim +*> +*> \param[out] COPYB +*> \verbatim +*> COPYB is REAL array, dimension (MMAX*NSMAX) +*> \endverbatim +*> +*> \param[out] S +*> \verbatim +*> S is REAL array, dimension +*> (min(MMAX,NMAX)) +*> \endverbatim +*> +*> \param[out] TAU +*> \verbatim +*> TAU is REAL array, dimension (MMAX) +*> \endverbatim +*> +*> \param[out] WORK +*> \verbatim +*> WORK is REAL array, dimension +*> (MMAX*NMAX + 4*NMAX + MMAX) +*> \endverbatim +*> +*> \param[out] IWORK +*> \verbatim +*> IWORK is INTEGER array, dimension (2*NMAX) +*> \endverbatim +*> +*> \param[in] NOUT +*> \verbatim +*> NOUT is INTEGER +*> The unit number for output. +*> \endverbatim +* +* Authors: +* ======== +* +*> \author Univ. of Tennessee +*> \author Univ. of California Berkeley +*> \author Univ. of Colorado Denver +*> \author NAG Ltd. +* +*> \ingroup single_lin +* +* ===================================================================== + SUBROUTINE SCHKQP3RK( DOTYPE, NM, MVAL, NN, NVAL, NNS, NSVAL, + $ NNB, NBVAL, NXVAL, THRESH, A, COPYA, + $ B, COPYB, S, TAU, + $ WORK, IWORK, NOUT ) + IMPLICIT NONE +* +* -- LAPACK test routine -- +* -- LAPACK is a software package provided by Univ. of Tennessee, -- +* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- +* +* .. Scalar Arguments .. + INTEGER NM, NN, NNB, NNS, NOUT + REAL THRESH +* .. +* .. Array Arguments .. + LOGICAL DOTYPE( * ) + INTEGER IWORK( * ), NBVAL( * ), MVAL( * ), NVAL( * ), + $ NSVAL( * ), NXVAL( * ) + REAL A( * ), COPYA( * ), B( * ), COPYB( * ), + $ S( * ), TAU( * ), WORK( * ) +* .. +* +* ===================================================================== +* +* .. Parameters .. + INTEGER NTYPES + PARAMETER ( NTYPES = 19 ) + INTEGER NTESTS + PARAMETER ( NTESTS = 5 ) + REAL ONE, ZERO, BIGNUM + PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0, + $ BIGNUM = 1.0E+38 ) +* .. +* .. Local Scalars .. + CHARACTER DIST, TYPE + CHARACTER*3 PATH + INTEGER I, IHIGH, ILOW, IM, IMAT, IN, INC_ZERO, + $ INB, IND_OFFSET_GEN, + $ IND_IN, IND_OUT, INS, INFO, + $ ISTEP, J, J_INC, J_FIRST_NZ, JB_ZERO, + $ KFACT, KL, KMAX, KU, LDA, LW, LWORK, + $ LWORK_MQR, M, MINMN, MINMNB_GEN, MODE, N, + $ NB, NB_ZERO, NERRS, NFAIL, NB_GEN, NRHS, + $ NRUN, NX, T + REAL ANORM, CNDNUM, EPS, ABSTOL, RELTOL, + $ DTEMP, MAXC2NRMK, RELMAXC2NRMK +* .. +* .. Local Arrays .. + INTEGER ISEED( 4 ), ISEEDY( 4 ) + REAL RESULT( NTESTS ), RDUMMY( 1 ) +* .. +* .. External Functions .. + REAL SLAMCH, SQPT01, SQRT11, SQRT12, SLANGE + EXTERNAL SLAMCH, SQPT01, SQRT11, SQRT12, SLANGE +* .. +* .. External Subroutines .. + EXTERNAL ALAERH, ALAHD, ALASUM, SAXPY, SGEQP3RK, + $ SLACPY, SLAORD, SLASET, SLATB4, SLATMS, + $ SORMQR, SSWAP, ICOPY, XLAENV +* .. +* .. Intrinsic Functions .. + INTRINSIC ABS, MAX, MIN, MOD, REAL +* .. +* .. Scalars in Common .. + LOGICAL LERR, OK + CHARACTER*32 SRNAMT + INTEGER INFOT, IOUNIT +* .. +* .. Common blocks .. + COMMON / INFOC / INFOT, IOUNIT, OK, LERR + COMMON / SRNAMC / SRNAMT +* .. +* .. Data statements .. + DATA ISEEDY / 1988, 1989, 1990, 1991 / +* .. +* .. Executable Statements .. +* +* Initialize constants and the random number seed. +* + PATH( 1: 1 ) = 'Single precision' + PATH( 2: 3 ) = 'QK' + NRUN = 0 + NFAIL = 0 + NERRS = 0 + DO I = 1, 4 + ISEED( I ) = ISEEDY( I ) + END DO + EPS = SLAMCH( 'Epsilon' ) + INFOT = 0 +* + DO IM = 1, NM +* +* Do for each value of M in MVAL. +* + M = MVAL( IM ) + LDA = MAX( 1, M ) +* + DO IN = 1, NN +* +* Do for each value of N in NVAL. +* + N = NVAL( IN ) + MINMN = MIN( M, N ) + LWORK = MAX( 1, M*MAX( M, N )+4*MINMN+MAX( M, N ), + $ M*N + 2*MINMN + 4*N ) +* + DO INS = 1, NNS + NRHS = NSVAL( INS ) +* +* Set up parameters with SLATB4 and generate +* M-by-NRHS B matrix with SLATMS. +* IMAT = 14: +* Random matrix, CNDNUM = 2, NORM = ONE, +* MODE = 3 (geometric distribution of singular values). +* + CALL SLATB4( PATH, 14, M, NRHS, TYPE, KL, KU, ANORM, + $ MODE, CNDNUM, DIST ) +* + SRNAMT = 'SLATMS' + CALL SLATMS( M, NRHS, DIST, ISEED, TYPE, S, MODE, + $ CNDNUM, ANORM, KL, KU, 'No packing', + $ COPYB, LDA, WORK, INFO ) + + +* +* Check error code from SLATMS. +* + IF( INFO.NE.0 ) THEN + CALL ALAERH( PATH, 'SLATMS', INFO, 0, ' ', M, + $ NRHS, -1, -1, -1, 6, NFAIL, NERRS, + $ NOUT ) + CYCLE + END IF +* + DO IMAT = 1, NTYPES +* +* Do the tests only if DOTYPE( IMAT ) is true. +* + IF( .NOT.DOTYPE( IMAT ) ) + $ CYCLE +* +* The type of distribution used to generate the random +* eigen-/singular values: +* ( 'S' for symmetric distribution ) => UNIFORM( -1, 1 ) +* +* Do for each type of NON-SYMMETRIC matrix: CNDNUM NORM MODE +* 1. Zero matrix +* 2. Random, Diagonal, CNDNUM = 2 CNDNUM = 2 ONE 3 ( geometric distribution of singular values ) +* 3. Random, Upper triangular, CNDNUM = 2 CNDNUM = 2 ONE 3 ( geometric distribution of singular values ) +* 4. Random, Lower triangular, CNDNUM = 2 CNDNUM = 2 ONE 3 ( geometric distribution of singular values ) +* 5. Random, First column is zero, CNDNUM = 2 CNDNUM = 2 ONE 3 ( geometric distribution of singular values ) +* 6. Random, Last MINMN column is zero, CNDNUM = 2 CNDNUM = 2 ONE 3 ( geometric distribution of singular values ) +* 7. Random, Last N column is zero, CNDNUM = 2 CNDNUM = 2 ONE 3 ( geometric distribution of singular values ) +* 8. Random, Middle column in MINMN is zero, CNDNUM = 2 CNDNUM = 2 ONE 3 ( geometric distribution of singular values ) +* 9. Random, First half of MINMN columns are zero, CNDNUM = 2 CNDNUM = 2 ONE 3 ( geometric distribution of singular values ) +* 10. Random, Last columns are zero starting from MINMN/2+1, CNDNUM = 2 CNDNUM = 2 ONE 3 ( geometric distribution of singular values ) +* 11. Random, Half MINMN columns in the middle are zero starting +* from MINMN/2-(MINMN/2)/2+1, CNDNUM = 2 CNDNUM = 2 ONE 3 ( geometric distribution of singular values ) +* 12. Random, Odd columns are ZERO, CNDNUM = 2 CNDNUM = 2 ONE 3 ( geometric distribution of singular values ) +* 13. Random, Even columns are ZERO, CNDNUM = 2 CNDNUM = 2 ONE 3 ( geometric distribution of singular values ) +* 14. Random, CNDNUM = 2 CNDNUM = 2 ONE 3 ( geometric distribution of singular values ) +* 15. Random, CNDNUM = sqrt(0.1/EPS) CNDNUM = BADC1 = sqrt(0.1/EPS) ONE 3 ( geometric distribution of singular values ) +* 16. Random, CNDNUM = 0.1/EPS CNDNUM = BADC2 = 0.1/EPS ONE 3 ( geometric distribution of singular values ) +* 17. Random, CNDNUM = 0.1/EPS, CNDNUM = BADC2 = 0.1/EPS ONE 2 ( one small singular value, S(N)=1/CNDNUM ) +* one small singular value S(N)=1/CNDNUM +* 18. Random, CNDNUM = 2, scaled near underflow CNDNUM = 2 SMALL = SAFMIN +* 19. Random, CNDNUM = 2, scaled near overflow CNDNUM = 2 LARGE = 1.0/( 0.25 * ( SAFMIN / EPS ) ) 3 ( geometric distribution of singular values ) +* + IF( IMAT.EQ.1 ) THEN +* +* Matrix 1: Zero matrix +* + CALL SLASET( 'Full', M, N, ZERO, ZERO, COPYA, LDA ) + DO I = 1, MINMN + S( I ) = ZERO + END DO +* + ELSE IF( (IMAT.GE.2 .AND. IMAT.LE.4 ) + $ .OR. (IMAT.GE.14 .AND. IMAT.LE.19 ) ) THEN +* +* Matrices 2-5. +* +* Set up parameters with SLATB4 and generate a test +* matrix with SLATMS. +* + CALL SLATB4( PATH, IMAT, M, N, TYPE, KL, KU, ANORM, + $ MODE, CNDNUM, DIST ) +* + SRNAMT = 'SLATMS' + CALL SLATMS( M, N, DIST, ISEED, TYPE, S, MODE, + $ CNDNUM, ANORM, KL, KU, 'No packing', + $ COPYA, LDA, WORK, INFO ) +* +* Check error code from SLATMS. +* + IF( INFO.NE.0 ) THEN + CALL ALAERH( PATH, 'SLATMS', INFO, 0, ' ', M, N, + $ -1, -1, -1, IMAT, NFAIL, NERRS, + $ NOUT ) + CYCLE + END IF +* + CALL SLAORD( 'Decreasing', MINMN, S, 1 ) +* + ELSE IF( MINMN.GE.2 + $ .AND. IMAT.GE.5 .AND. IMAT.LE.13 ) THEN +* +* Rectangular matrices 5-13 that contain zero columns, +* only for matrices MINMN >=2. +* +* JB_ZERO is the column index of ZERO block. +* NB_ZERO is the column block size of ZERO block. +* NB_GEN is the column blcok size of the +* generated block. +* J_INC in the non_zero column index increment +* for matrix 12 and 13. +* J_FIRS_NZ is the index of the first non-zero +* column. +* + IF( IMAT.EQ.5 ) THEN +* +* First column is zero. +* + JB_ZERO = 1 + NB_ZERO = 1 + NB_GEN = N - NB_ZERO +* + ELSE IF( IMAT.EQ.6 ) THEN +* +* Last column MINMN is zero. +* + JB_ZERO = MINMN + NB_ZERO = 1 + NB_GEN = N - NB_ZERO +* + ELSE IF( IMAT.EQ.7 ) THEN +* +* Last column N is zero. +* + JB_ZERO = N + NB_ZERO = 1 + NB_GEN = N - NB_ZERO +* + ELSE IF( IMAT.EQ.8 ) THEN +* +* Middle column in MINMN is zero. +* + JB_ZERO = MINMN / 2 + 1 + NB_ZERO = 1 + NB_GEN = N - NB_ZERO +* + ELSE IF( IMAT.EQ.9 ) THEN +* +* First half of MINMN columns is zero. +* + JB_ZERO = 1 + NB_ZERO = MINMN / 2 + NB_GEN = N - NB_ZERO +* + ELSE IF( IMAT.EQ.10 ) THEN +* +* Last columns are zero columns, +* starting from (MINMN / 2 + 1) column. +* + JB_ZERO = MINMN / 2 + 1 + NB_ZERO = N - JB_ZERO + 1 + NB_GEN = N - NB_ZERO +* + ELSE IF( IMAT.EQ.11 ) THEN +* +* Half of the columns in the middle of MINMN +* columns is zero, starting from +* MINMN/2 - (MINMN/2)/2 + 1 column. +* + JB_ZERO = MINMN / 2 - (MINMN / 2) / 2 + 1 + NB_ZERO = MINMN / 2 + NB_GEN = N - NB_ZERO +* + ELSE IF( IMAT.EQ.12 ) THEN +* +* Odd-numbered columns are zero, +* + NB_GEN = N / 2 + NB_ZERO = N - NB_GEN + J_INC = 2 + J_FIRST_NZ = 2 +* + ELSE IF( IMAT.EQ.13 ) THEN +* +* Even-numbered columns are zero. +* + NB_ZERO = N / 2 + NB_GEN = N - NB_ZERO + J_INC = 2 + J_FIRST_NZ = 1 +* + END IF +* +* +* 1) Set the first NB_ZERO columns in COPYA(1:M,1:N) +* to zero. +* + CALL SLASET( 'Full', M, NB_ZERO, ZERO, ZERO, + $ COPYA, LDA ) +* +* 2) Generate an M-by-(N-NB_ZERO) matrix with the +* chosen singular value distribution +* in COPYA(1:M,NB_ZERO+1:N). +* + CALL SLATB4( PATH, IMAT, M, NB_GEN, TYPE, KL, KU, + $ ANORM, MODE, CNDNUM, DIST ) +* + SRNAMT = 'SLATMS' +* + IND_OFFSET_GEN = NB_ZERO * LDA +* + CALL SLATMS( M, NB_GEN, DIST, ISEED, TYPE, S, MODE, + $ CNDNUM, ANORM, KL, KU, 'No packing', + $ COPYA( IND_OFFSET_GEN + 1 ), LDA, + $ WORK, INFO ) +* +* Check error code from SLATMS. +* + IF( INFO.NE.0 ) THEN + CALL ALAERH( PATH, 'SLATMS', INFO, 0, ' ', M, + $ NB_GEN, -1, -1, -1, IMAT, NFAIL, + $ NERRS, NOUT ) + CYCLE + END IF +* +* 3) Swap the gererated colums from the right side +* NB_GEN-size block in COPYA into correct column +* positions. +* + IF( IMAT.EQ.6 + $ .OR. IMAT.EQ.7 + $ .OR. IMAT.EQ.8 + $ .OR. IMAT.EQ.10 + $ .OR. IMAT.EQ.11 ) THEN +* +* Move by swapping the generated columns +* from the right NB_GEN-size block from +* (NB_ZERO+1:NB_ZERO+JB_ZERO) +* into columns (1:JB_ZERO-1). +* + DO J = 1, JB_ZERO-1, 1 + CALL SSWAP( M, + $ COPYA( ( NB_ZERO+J-1)*LDA+1), 1, + $ COPYA( (J-1)*LDA + 1 ), 1 ) + END DO +* + ELSE IF( IMAT.EQ.12 .OR. IMAT.EQ.13 ) THEN +* +* ( IMAT = 12, Odd-numbered ZERO columns. ) +* Swap the generated columns from the right +* NB_GEN-size block into the even zero colums in the +* left NB_ZERO-size block. +* +* ( IMAT = 13, Even-numbered ZERO columns. ) +* Swap the generated columns from the right +* NB_GEN-size block into the odd zero colums in the +* left NB_ZERO-size block. +* + DO J = 1, NB_GEN, 1 + IND_OUT = ( NB_ZERO+J-1 )*LDA + 1 + IND_IN = ( J_INC*(J-1)+(J_FIRST_NZ-1) )*LDA + $ + 1 + CALL SSWAP( M, + $ COPYA( IND_OUT ), 1, + $ COPYA( IND_IN), 1 ) + END DO +* + END IF +* +* 5) Order the singular values generated by +* DLAMTS in decreasing order and add trailing zeros +* that correspond to zero columns. +* The total number of singular values is MINMN. +* + MINMNB_GEN = MIN( M, NB_GEN ) +* + DO I = MINMNB_GEN+1, MINMN + S( I ) = ZERO + END DO +* + ELSE +* +* IF(MINMN.LT.2) skip this size for this matrix type. +* + CYCLE + END IF +* +* Initialize a copy array for a pivot array for SGEQP3RK. +* + DO I = 1, N + IWORK( I ) = 0 + END DO +* + DO INB = 1, NNB +* +* Do for each pair of values (NB,NX) in NBVAL and NXVAL. +* + NB = NBVAL( INB ) + CALL XLAENV( 1, NB ) + NX = NXVAL( INB ) + CALL XLAENV( 3, NX ) +* +* We do MIN(M,N)+1 because we need a test for KMAX > N, +* when KMAX is larger than MIN(M,N), KMAX should be +* KMAX = MIN(M,N) +* + DO KMAX = 0, MIN(M,N)+1 +* +* Get a working copy of COPYA into A( 1:M,1:N ). +* Get a working copy of COPYB into A( 1:M, (N+1):NRHS ). +* Get a working copy of COPYB into into B( 1:M, 1:NRHS ). +* Get a working copy of IWORK(1:N) awith zeroes into +* which is going to be used as pivot array IWORK( N+1:2N ). +* NOTE: IWORK(2N+1:3N) is going to be used as a WORK array +* for the routine. +* + CALL SLACPY( 'All', M, N, COPYA, LDA, A, LDA ) + CALL SLACPY( 'All', M, NRHS, COPYB, LDA, + $ A( LDA*N + 1 ), LDA ) + CALL SLACPY( 'All', M, NRHS, COPYB, LDA, + $ B, LDA ) + CALL ICOPY( N, IWORK( 1 ), 1, IWORK( N+1 ), 1 ) +* + ABSTOL = -1.0 + RELTOL = -1.0 +* +* Compute the QR factorization with pivoting of A +* + LW = MAX( 1, MAX( 2*N + NB*( N+NRHS+1 ), + $ 3*N + NRHS - 1 ) ) +* +* Compute SGEQP3RK factorization of A. +* + SRNAMT = 'SGEQP3RK' + CALL SGEQP3RK( M, N, NRHS, KMAX, ABSTOL, RELTOL, + $ A, LDA, KFACT, MAXC2NRMK, + $ RELMAXC2NRMK, IWORK( N+1 ), TAU, + $ WORK, LW, IWORK( 2*N+1 ), INFO ) +* +* Check error code from SGEQP3RK. +* + IF( INFO.LT.0 ) + $ CALL ALAERH( PATH, 'SGEQP3RK', INFO, 0, ' ', + $ M, N, NX, -1, NB, IMAT, + $ NFAIL, NERRS, NOUT ) +* +* Compute test 1: +* +* This test in only for the full rank factorization of +* the matrix A. +* +* Array S(1:min(M,N)) contains svd(A) the sigular values +* of the original matrix A in decreasing absolute value +* order. The test computes svd(R), the vector sigular +* values of the upper trapezoid of A(1:M,1:N) that +* contains the factor R, in decreasing order. The test +* returns the ratio: +* +* 2-norm(svd(R) - svd(A)) / ( max(M,N) * 2-norm(svd(A)) * EPS ) +* + IF( KFACT.EQ.MINMN ) THEN +* + RESULT( 1 ) = SQRT12( M, N, A, LDA, S, WORK, + $ LWORK ) +* + DO T = 1, 1 + IF( RESULT( T ).GE.THRESH ) THEN + IF( NFAIL.EQ.0 .AND. NERRS.EQ.0 ) + $ CALL ALAHD( NOUT, PATH ) + WRITE( NOUT, FMT = 9999 ) 'SGEQP3RK', M, N, + $ NRHS, KMAX, ABSTOL, RELTOL, NB, NX, + $ IMAT, T, RESULT( T ) + NFAIL = NFAIL + 1 + END IF + END DO + NRUN = NRUN + 1 +* +* End test 1 +* + END IF +* +* Compute test 2: +* +* The test returns the ratio: +* +* 1-norm( A*P - Q*R ) / ( max(M,N) * 1-norm(A) * EPS ) +* + RESULT( 2 ) = SQPT01( M, N, KFACT, COPYA, A, LDA, TAU, + $ IWORK( N+1 ), WORK, LWORK ) +* +* Compute test 3: +* +* The test returns the ratio: +* +* 1-norm( Q**T * Q - I ) / ( M * EPS ) +* + RESULT( 3 ) = SQRT11( M, KFACT, A, LDA, TAU, WORK, + $ LWORK ) +* +* Print information about the tests that did not pass +* the threshold. +* + DO T = 2, 3 + IF( RESULT( T ).GE.THRESH ) THEN + IF( NFAIL.EQ.0 .AND. NERRS.EQ.0 ) + $ CALL ALAHD( NOUT, PATH ) + WRITE( NOUT, FMT = 9999 ) 'SGEQP3RK', M, N, + $ NRHS, KMAX, ABSTOL, RELTOL, + $ NB, NX, IMAT, T, RESULT( T ) + NFAIL = NFAIL + 1 + END IF + END DO + NRUN = NRUN + 2 +* +* Compute test 4: +* +* This test is only for the factorizations with the +* rank greater than 2. +* The elements on the diagonal of R should be non- +* increasing. +* +* The test returns the ratio: +* +* Returns 1.0D+100 if abs(R(K+1,K+1)) > abs(R(K,K)), +* K=1:KFACT-1 +* + IF( MIN(KFACT, MINMN).GE.2 ) THEN +* + DO J = 1, KFACT-1, 1 + + DTEMP = (( ABS( A( (J-1)*M+J ) ) - + $ ABS( A( (J)*M+J+1 ) ) ) / + $ ABS( A(1) ) ) +* + IF( DTEMP.LT.ZERO ) THEN + RESULT( 4 ) = BIGNUM + END IF +* + END DO +* +* Print information about the tests that did not +* pass the threshold. +* + DO T = 4, 4 + IF( RESULT( T ).GE.THRESH ) THEN + IF( NFAIL.EQ.0 .AND. NERRS.EQ.0 ) + $ CALL ALAHD( NOUT, PATH ) + WRITE( NOUT, FMT = 9999 ) 'SGEQP3RK', + $ M, N, NRHS, KMAX, ABSTOL, RELTOL, + $ NB, NX, IMAT, T, + $ RESULT( T ) + NFAIL = NFAIL + 1 + END IF + END DO + NRUN = NRUN + 1 +* +* End test 4. +* + END IF +* +* Compute test 5: +* +* This test in only for matrix A with min(M,N) > 0. +* +* The test returns the ratio: +* +* 1-norm(Q**T * B - Q**T * B ) / +* ( M * EPS ) +* +* (1) Compute B:=Q**T * B in the matrix B. +* + IF( MINMN.GT.0 ) THEN +* + LWORK_MQR = MAX(1, NRHS) + CALL SORMQR( 'Left', 'Transpose', + $ M, NRHS, KFACT, A, LDA, TAU, B, LDA, + $ WORK, LWORK_MQR, INFO ) +* + DO I = 1, NRHS +* +* Compare N+J-th column of A and J-column of B. +* + CALL SAXPY( M, -ONE, A( ( N+I-1 )*LDA+1 ), 1, + $ B( ( I-1 )*LDA+1 ), 1 ) + END DO +* + RESULT( 5 ) = + $ ABS( + $ SLANGE( 'One-norm', M, NRHS, B, LDA, RDUMMY ) / + $ ( REAL( M )*SLAMCH( 'Epsilon' ) ) + $ ) +* +* Print information about the tests that did not pass +* the threshold. +* + DO T = 5, 5 + IF( RESULT( T ).GE.THRESH ) THEN + IF( NFAIL.EQ.0 .AND. NERRS.EQ.0 ) + $ CALL ALAHD( NOUT, PATH ) + WRITE( NOUT, FMT = 9999 ) 'SGEQP3RK', M, N, + $ NRHS, KMAX, ABSTOL, RELTOL, + $ NB, NX, IMAT, T, RESULT( T ) + NFAIL = NFAIL + 1 + END IF + END DO + NRUN = NRUN + 1 +* +* End compute test 5. +* + END IF +* +* END DO KMAX = 1, MIN(M,N)+1 +* + END DO +* +* END DO for INB = 1, NNB +* + END DO +* +* END DO for IMAT = 1, NTYPES +* + END DO +* +* END DO for INS = 1, NNS +* + END DO +* +* END DO for IN = 1, NN +* + END DO +* +* END DO for IM = 1, NM +* + END DO +* +* Print a summary of the results. +* + CALL ALASUM( PATH, NOUT, NFAIL, NRUN, NERRS ) +* + 9999 FORMAT( 1X, A, ' M =', I5, ', N =', I5, ', NRHS =', I5, + $ ', KMAX =', I5, ', ABSTOL =', G12.5, + $ ', RELTOL =', G12.5, ', NB =', I4, ', NX =', I4, + $ ', type ', I2, ', test ', I2, ', ratio =', G12.5 ) +* +* End of SCHKQP3RK +* + END diff --git a/TESTING/LIN/slatb4.f b/TESTING/LIN/slatb4.f index 6bf236aaac..72a3107278 100644 --- a/TESTING/LIN/slatb4.f +++ b/TESTING/LIN/slatb4.f @@ -224,6 +224,110 @@ SUBROUTINE SLATB4( PATH, IMAT, M, N, TYPE, KL, KU, ANORM, MODE, ELSE ANORM = ONE END IF +* + ELSE IF( LSAMEN( 2, C2, 'QK' ) ) THEN +* +* xQK: truncated QR with pivoting. +* Set parameters to generate a general +* M x N matrix. +* +* Set TYPE, the type of matrix to be generated. 'N' is nonsymmetric. +* + TYPE = 'N' +* +* Set DIST, the type of distribution for the random +* number generator. 'S' is +* + DIST = 'S' +* +* Set the lower and upper bandwidths. +* + IF( IMAT.EQ.2 ) THEN +* +* 2. Random, Diagonal, CNDNUM = 2 +* + KL = 0 + KU = 0 + CNDNUM = TWO + ANORM = ONE + MODE = 3 + ELSE IF( IMAT.EQ.3 ) THEN +* +* 3. Random, Upper triangular, CNDNUM = 2 +* + KL = 0 + KU = MAX( N-1, 0 ) + CNDNUM = TWO + ANORM = ONE + MODE = 3 + ELSE IF( IMAT.EQ.4 ) THEN +* +* 4. Random, Lower triangular, CNDNUM = 2 +* + KL = MAX( M-1, 0 ) + KU = 0 + CNDNUM = TWO + ANORM = ONE + MODE = 3 + ELSE +* +* 5.-19. Rectangular matrix +* + KL = MAX( M-1, 0 ) + KU = MAX( N-1, 0 ) +* + IF( IMAT.GE.5 .AND. IMAT.LE.14 ) THEN +* +* 5.-14. Random, CNDNUM = 2. +* + CNDNUM = TWO + ANORM = ONE + MODE = 3 +* + ELSE IF( IMAT.EQ.15 ) THEN +* +* 15. Random, CNDNUM = sqrt(0.1/EPS) +* + CNDNUM = BADC1 + ANORM = ONE + MODE = 3 +* + ELSE IF( IMAT.EQ.16 ) THEN +* +* 16. Random, CNDNUM = 0.1/EPS +* + CNDNUM = BADC2 + ANORM = ONE + MODE = 3 +* + ELSE IF( IMAT.EQ.17 ) THEN +* +* 17. Random, CNDNUM = 0.1/EPS, +* one small singular value S(N)=1/CNDNUM +* + CNDNUM = BADC2 + ANORM = ONE + MODE = 2 +* + ELSE IF( IMAT.EQ.18 ) THEN +* +* 18. Random, scaled near underflow +* + CNDNUM = TWO + ANORM = SMALL + MODE = 3 +* + ELSE IF( IMAT.EQ.19 ) THEN +* +* 19. Random, scaled near overflow +* + CNDNUM = TWO + ANORM = LARGE + MODE = 3 +* + END IF +* + END IF * ELSE IF( LSAMEN( 2, C2, 'GE' ) ) THEN * diff --git a/TESTING/LIN/sqpt01.f b/TESTING/LIN/sqpt01.f index de0c80e53a..f53686a657 100644 --- a/TESTING/LIN/sqpt01.f +++ b/TESTING/LIN/sqpt01.f @@ -33,7 +33,8 @@ *> Householder vectors, and the rest of AF contains a partially updated *> matrix. *> -*> This function returns ||A*P - Q*R||/(||norm(A)||*eps*M) +*> This function returns ||A*P - Q*R|| / ( ||norm(A)||*eps*max(M,N) ) +*> where || . || is matrix one norm. *> \endverbatim * * Arguments: @@ -172,28 +173,28 @@ REAL FUNCTION SQPT01( M, N, K, A, AF, LDA, TAU, JPVT, * NORMA = SLANGE( 'One-norm', M, N, A, LDA, RWORK ) * - DO 30 J = 1, K - DO 10 I = 1, MIN( J, M ) + DO J = 1, K + DO I = 1, MIN( J, M ) WORK( ( J-1 )*M+I ) = AF( I, J ) - 10 CONTINUE - DO 20 I = J + 1, M + END DO + DO I = J + 1, M WORK( ( J-1 )*M+I ) = ZERO - 20 CONTINUE - 30 CONTINUE - DO 40 J = K + 1, N + END DO + END DO + DO J = K + 1, N CALL SCOPY( M, AF( 1, J ), 1, WORK( ( J-1 )*M+1 ), 1 ) - 40 CONTINUE + END DO * CALL SORMQR( 'Left', 'No transpose', M, N, K, AF, LDA, TAU, WORK, $ M, WORK( M*N+1 ), LWORK-M*N, INFO ) * - DO 50 J = 1, N + DO J = 1, N * * Compare i-th column of QR and jpvt(i)-th column of A * CALL SAXPY( M, -ONE, A( 1, JPVT( J ) ), 1, WORK( ( J-1 )*M+1 ), $ 1 ) - 50 CONTINUE + END DO * SQPT01 = SLANGE( 'One-norm', M, N, WORK, M, RWORK ) / $ ( REAL( MAX( M, N ) )*SLAMCH( 'Epsilon' ) ) diff --git a/TESTING/LIN/sqrt11.f b/TESTING/LIN/sqrt11.f index d4422dacbf..a3753adcf9 100644 --- a/TESTING/LIN/sqrt11.f +++ b/TESTING/LIN/sqrt11.f @@ -157,9 +157,9 @@ REAL FUNCTION SQRT11( M, K, A, LDA, TAU, WORK, LWORK ) CALL SORM2R( 'Left', 'Transpose', M, M, K, A, LDA, TAU, WORK, M, $ WORK( M*M+1 ), INFO ) * - DO 10 J = 1, M + DO J = 1, M WORK( ( J-1 )*M+J ) = WORK( ( J-1 )*M+J ) - ONE - 10 CONTINUE + END DO * SQRT11 = SLANGE( 'One-norm', M, M, WORK, M, RDUMMY ) / $ ( REAL( M )*SLAMCH( 'Epsilon' ) ) diff --git a/TESTING/LIN/sqrt12.f b/TESTING/LIN/sqrt12.f index 23fc94c63d..46b359e07b 100644 --- a/TESTING/LIN/sqrt12.f +++ b/TESTING/LIN/sqrt12.f @@ -26,7 +26,7 @@ *> SQRT12 computes the singular values `svlues' of the upper trapezoid *> of A(1:M,1:N) and returns the ratio *> -*> || s - svlues||/(||svlues||*eps*max(M,N)) +*> || svlues - s ||/(||s||*eps*max(M,N)) *> \endverbatim * * Arguments: @@ -144,11 +144,11 @@ REAL FUNCTION SQRT12( M, N, A, LDA, S, WORK, LWORK ) * Copy upper triangle of A into work * CALL SLASET( 'Full', M, N, ZERO, ZERO, WORK, M ) - DO 20 J = 1, N - DO 10 I = 1, MIN( J, M ) + DO J = 1, N + DO I = 1, MIN( J, M ) WORK( ( J-1 )*M+I ) = A( I, J ) - 10 CONTINUE - 20 CONTINUE + END DO + END DO * * Get machine parameters * @@ -197,9 +197,9 @@ REAL FUNCTION SQRT12( M, N, A, LDA, S, WORK, LWORK ) * ELSE * - DO 30 I = 1, MN + DO I = 1, MN WORK( M*N+I ) = ZERO - 30 CONTINUE + END DO END IF * * Compare s and singular values of work diff --git a/TESTING/LIN/zchkaa.F b/TESTING/LIN/zchkaa.F index a118515a5d..57d71833f9 100644 --- a/TESTING/LIN/zchkaa.F +++ b/TESTING/LIN/zchkaa.F @@ -69,6 +69,7 @@ *> ZLQ 8 List types on next line if 0 < NTYPES < 8 *> ZQL 8 List types on next line if 0 < NTYPES < 8 *> ZQP 6 List types on next line if 0 < NTYPES < 6 +*> ZQK 19 List types on next line if 0 < NTYPES < 19 *> ZTZ 3 List types on next line if 0 < NTYPES < 3 *> ZLS 6 List types on next line if 0 < NTYPES < 6 *> ZEQ @@ -153,12 +154,11 @@ PROGRAM ZCHKAA $ NBVAL( MAXIN ), NBVAL2( MAXIN ), $ NSVAL( MAXIN ), NVAL( MAXIN ), NXVAL( MAXIN ), $ RANKVAL( MAXIN ), PIV( NMAX ) - DOUBLE PRECISION S( 2*NMAX ) - COMPLEX*16 E( NMAX ) -* -* .. Allocatable Arrays .. +* .. +* .. Allocatable Arrays .. INTEGER AllocateStatus - DOUBLE PRECISION, DIMENSION(:), ALLOCATABLE:: RWORK + DOUBLE PRECISION, DIMENSION(:), ALLOCATABLE:: RWORK, S + COMPLEX*16, DIMENSION(:), ALLOCATABLE :: E COMPLEX*16, DIMENSION(:,:), ALLOCATABLE:: A, B, WORK * .. * .. External Functions .. @@ -170,15 +170,16 @@ PROGRAM ZCHKAA EXTERNAL ALAREQ, ZCHKEQ, ZCHKGB, ZCHKGE, ZCHKGT, ZCHKHE, $ ZCHKHE_ROOK, ZCHKHE_RK, ZCHKHE_AA, ZCHKHP, $ ZCHKLQ, ZCHKUNHR_COL, ZCHKPB, ZCHKPO, ZCHKPS, - $ ZCHKPP, ZCHKPT, ZCHKQ3, ZCHKQL, ZCHKQR, ZCHKRQ, - $ ZCHKSP, ZCHKSY, ZCHKSY_ROOK, ZCHKSY_RK, - $ ZCHKSY_AA, ZCHKTB, ZCHKTP, ZCHKTR, ZCHKTZ, - $ ZDRVGB, ZDRVGE, ZDRVGT, ZDRVHE, ZDRVHE_ROOK, - $ ZDRVHE_RK, ZDRVHE_AA, ZDRVHE_AA_2STAGE, ZDRVHP, - $ ZDRVLS, ZDRVPB, ZDRVPO, ZDRVPP, ZDRVPT, - $ ZDRVSP, ZDRVSY, ZDRVSY_ROOK, ZDRVSY_RK, - $ ZDRVSY_AA, ZDRVSY_AA_2STAGE, ILAVER, ZCHKQRT, - $ ZCHKQRTP, ZCHKLQT, ZCHKLQTP, ZCHKTSQR + $ ZCHKPP, ZCHKPT, ZCHKQ3, ZCHKQP3RK, ZCHKQL, + $ ZCHKQR, ZCHKRQ, ZCHKSP, ZCHKSY, ZCHKSY_ROOK, + $ ZCHKSY_RK, ZCHKSY_AA, ZCHKTB, ZCHKTP, ZCHKTR, + $ ZCHKTZ, ZDRVGB, ZDRVGE, ZDRVGT, ZDRVHE, + $ ZDRVHE_ROOK, ZDRVHE_RK, ZDRVHE_AA, + $ ZDRVHE_AA_2STAGE, ZDRVHP, ZDRVLS, ZDRVPB, + $ ZDRVPO, ZDRVPP, ZDRVPT, ZDRVSP, ZDRVSY, + $ ZDRVSY_ROOK, ZDRVSY_RK, ZDRVSY_AA, + $ ZDRVSY_AA_2STAGE, ILAVER, ZCHKQRT, ZCHKQRTP, + $ ZCHKLQT, ZCHKLQTP, ZCHKTSQR * .. * .. Scalars in Common .. LOGICAL LERR, OK @@ -197,13 +198,18 @@ PROGRAM ZCHKAA DATA THREQ / 2.0D0 / , INTSTR / '0123456789' / * * .. Allocate memory dynamically .. - ALLOCATE (RWORK( 150*NMAX+2*MAXRHS ), STAT = AllocateStatus) +* + ALLOCATE ( A ( (KDMAX+1) * NMAX, 7 ), STAT = AllocateStatus) + IF (AllocateStatus /= 0) STOP "*** Not enough memory ***" + ALLOCATE ( B ( NMAX * MAXRHS, 4 ), STAT = AllocateStatus) + IF (AllocateStatus /= 0 ) STOP "*** Not enough memory ***" + ALLOCATE ( WORK ( NMAX, NMAX+MAXRHS+10 ), STAT = AllocateStatus) IF (AllocateStatus /= 0) STOP "*** Not enough memory ***" - ALLOCATE (A ((KDMAX+1) * NMAX, 7), STAT = AllocateStatus) + ALLOCATE ( E( NMAX ), STAT = AllocateStatus ) IF (AllocateStatus /= 0) STOP "*** Not enough memory ***" - ALLOCATE (B (NMAX * MAXRHS, 4), STAT = AllocateStatus) + ALLOCATE ( S( 2*NMAX ), STAT = AllocateStatus) IF (AllocateStatus /= 0) STOP "*** Not enough memory ***" - ALLOCATE (WORK (NMAX, NMAX+MAXRHS+10), STAT = AllocateStatus) + ALLOCATE ( RWORK( 150*NMAX+2*MAXRHS ), STAT = AllocateStatus) IF (AllocateStatus /= 0) STOP "*** Not enough memory ***" * .. * .. Executable Statements .. @@ -1109,6 +1115,23 @@ PROGRAM ZCHKAA ELSE WRITE( NOUT, FMT = 9989 )PATH END IF +* + ELSE IF( LSAMEN( 2, C2, 'QK' ) ) THEN +* +* QK: truncated QR factorization with pivoting +* + NTYPES = 19 + CALL ALAREQ( PATH, NMATS, DOTYPE, NTYPES, NIN, NOUT ) +* + IF( TSTCHK ) THEN + CALL ZCHKQP3RK( DOTYPE, NM, MVAL, NN, NVAL, NNS, NSVAL, + $ NNB, NBVAL, NXVAL, THRESH, A( 1, 1 ), + $ A( 1, 2 ), B( 1, 1 ), B( 1, 2 ), + $ S( 1 ), B( 1, 4 ), + $ WORK, RWORK, IWORK, NOUT ) + ELSE + WRITE( NOUT, FMT = 9989 )PATH + END IF * ELSE IF( LSAMEN( 2, C2, 'LS' ) ) THEN * diff --git a/TESTING/LIN/zchkqp3rk.f b/TESTING/LIN/zchkqp3rk.f new file mode 100644 index 0000000000..302c7b1a87 --- /dev/null +++ b/TESTING/LIN/zchkqp3rk.f @@ -0,0 +1,836 @@ +*> \brief \b ZCHKQP3RK +* +* =========== DOCUMENTATION =========== +* +* Online html documentation available at +* http://www.netlib.org/lapack/explore-html/ +* +* Definition: +* =========== +* +* SUBROUTINE ZCHKQP3RK( DOTYPE, NM, MVAL, NN, NVAL, NNS, NSVAL, +* $ NNB, NBVAL, NXVAL, THRESH, A, COPYA, +* $ B, COPYB, S, TAU, +* $ WORK, RWORK, IWORK, NOUT ) +* IMPLICIT NONE +* +* .. Scalar Arguments .. +* INTEGER NM, NN, NNB, NOUT +* DOUBLE PRECISION THRESH +* .. +* .. Array Arguments .. +* LOGICAL DOTYPE( * ) +* INTEGER IWORK( * ), MVAL( * ), NBVAL( * ), NVAL( * ), +* $ NXVAL( * ) +* DOUBLE PRECISION S( * ), RWORK( * ) +* COMPLEX*16 A( * ), COPYA( * ), TAU( * ), WORK( * ) +* .. +* +* +*> \par Purpose: +* ============= +*> +*> \verbatim +*> +*> ZCHKQP3RK tests ZGEQP3RK. +*> \endverbatim +* +* Arguments: +* ========== +* +*> \param[in] DOTYPE +*> \verbatim +*> DOTYPE is LOGICAL array, dimension (NTYPES) +*> The matrix types to be used for testing. Matrices of type j +*> (for 1 <= j <= NTYPES) are used for testing if DOTYPE(j) = +*> .TRUE.; if DOTYPE(j) = .FALSE., then type j is not used. +*> \endverbatim +*> +*> \param[in] NM +*> \verbatim +*> NM is INTEGER +*> The number of values of M contained in the vector MVAL. +*> \endverbatim +*> +*> \param[in] MVAL +*> \verbatim +*> MVAL is INTEGER array, dimension (NM) +*> The values of the matrix row dimension M. +*> \endverbatim +*> +*> \param[in] NN +*> \verbatim +*> NN is INTEGER +*> The number of values of N contained in the vector NVAL. +*> \endverbatim +*> +*> \param[in] NVAL +*> \verbatim +*> NVAL is INTEGER array, dimension (NN) +*> The values of the matrix column dimension N. +*> \endverbatim +*> +*> \param[in] NNS +*> \verbatim +*> NNS is INTEGER +*> The number of values of NRHS contained in the vector NSVAL. +*> \endverbatim +*> +*> \param[in] NSVAL +*> \verbatim +*> NSVAL is INTEGER array, dimension (NNS) +*> The values of the number of right hand sides NRHS. +*> \endverbatim +*> \param[in] NNB +*> \verbatim +*> NNB is INTEGER +*> The number of values of NB and NX contained in the +*> vectors NBVAL and NXVAL. The blocking parameters are used +*> in pairs (NB,NX). +*> \endverbatim +*> +*> \param[in] NBVAL +*> \verbatim +*> NBVAL is INTEGER array, dimension (NNB) +*> The values of the blocksize NB. +*> \endverbatim +*> +*> \param[in] NXVAL +*> \verbatim +*> NXVAL is INTEGER array, dimension (NNB) +*> The values of the crossover point NX. +*> \endverbatim +*> +*> \param[in] THRESH +*> \verbatim +*> THRESH is DOUBLE PRECISION +*> The threshold value for the test ratios. A result is +*> included in the output file if RESULT >= THRESH. To have +*> every test ratio printed, use THRESH = 0. +*> \endverbatim +*> +*> \param[out] A +*> \verbatim +*> A is COMPLEX*16 array, dimension (MMAX*NMAX) +*> where MMAX is the maximum value of M in MVAL and NMAX is the +*> maximum value of N in NVAL. +*> \endverbatim +*> +*> \param[out] COPYA +*> \verbatim +*> COPYA is COMPLEX*16 array, dimension (MMAX*NMAX) +*> \endverbatim +*> +*> \param[out] B +*> \verbatim +*> B is COMPLEX*16 array, dimension (MMAX*NSMAX) +*> where MMAX is the maximum value of M in MVAL and NSMAX is the +*> maximum value of NRHS in NSVAL. +*> \endverbatim +*> +*> \param[out] COPYB +*> \verbatim +*> COPYB is COMPLEX*16 array, dimension (MMAX*NSMAX) +*> \endverbatim +*> +*> \param[out] S +*> \verbatim +*> S is DOUBLE PRECISION array, dimension +*> (min(MMAX,NMAX)) +*> \endverbatim +*> +*> \param[out] TAU +*> \verbatim +*> TAU is COMPLEX*16 array, dimension (MMAX) +*> \endverbatim +*> +*> \param[out] WORK +*> \verbatim +*> WORK is COMPLEX*16 array, dimension +*> (max(M*max(M,N) + 4*min(M,N) + max(M,N))) +*> \endverbatim +*> +*> \param[out] RWORK +*> \verbatim +*> RWORK is DOUBLE PRECISION array, dimension (4*NMAX) +*> \endverbatim +*> +*> \param[out] IWORK +*> \verbatim +*> IWORK is INTEGER array, dimension (2*NMAX) +*> \endverbatim +*> +*> \param[in] NOUT +*> \verbatim +*> NOUT is INTEGER +*> The unit number for output. +*> \endverbatim +* +* Authors: +* ======== +* +*> \author Univ. of Tennessee +*> \author Univ. of California Berkeley +*> \author Univ. of Colorado Denver +*> \author NAG Ltd. +* +*> \ingroup complex16_lin +* +* ===================================================================== + SUBROUTINE ZCHKQP3RK( DOTYPE, NM, MVAL, NN, NVAL, NNS, NSVAL, + $ NNB, NBVAL, NXVAL, THRESH, A, COPYA, + $ B, COPYB, S, TAU, + $ WORK, RWORK, IWORK, NOUT ) + IMPLICIT NONE +* +* -- LAPACK test routine -- +* -- LAPACK is a software package provided by Univ. of Tennessee, -- +* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- +* +* .. Scalar Arguments .. + INTEGER NM, NN, NNB, NNS, NOUT + DOUBLE PRECISION THRESH +* .. +* .. Array Arguments .. + LOGICAL DOTYPE( * ) + INTEGER IWORK( * ), NBVAL( * ), MVAL( * ), NVAL( * ), + $ NSVAL( * ), NXVAL( * ) + DOUBLE PRECISION S( * ), RWORK( * ) + COMPLEX*16 A( * ), COPYA( * ), B( * ), COPYB( * ), + $ TAU( * ), WORK( * ) +* .. +* +* ===================================================================== +* +* .. Parameters .. + INTEGER NTYPES + PARAMETER ( NTYPES = 19 ) + INTEGER NTESTS + PARAMETER ( NTESTS = 5 ) + DOUBLE PRECISION ONE, ZERO, BIGNUM + COMPLEX*16 CONE, CZERO + PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0, + $ CZERO = ( 0.0D+0, 0.0D+0 ), + $ CONE = ( 1.0D+0, 0.0D+0 ), + $ BIGNUM = 1.0D+38 ) +* .. +* .. Local Scalars .. + CHARACTER DIST, TYPE + CHARACTER*3 PATH + INTEGER I, IHIGH, ILOW, IM, IMAT, IN, INC_ZERO, + $ INB, IND_OFFSET_GEN, + $ IND_IN, IND_OUT, INS, INFO, + $ ISTEP, J, J_INC, J_FIRST_NZ, JB_ZERO, + $ KFACT, KL, KMAX, KU, LDA, LW, LWORK, + $ LWORK_MQR, M, MINMN, MINMNB_GEN, MODE, N, + $ NB, NB_ZERO, NERRS, NFAIL, NB_GEN, NRHS, + $ NRUN, NX, T + DOUBLE PRECISION ANORM, CNDNUM, EPS, ABSTOL, RELTOL, + $ DTEMP, MAXC2NRMK, RELMAXC2NRMK +* .. +* .. Local Arrays .. + INTEGER ISEED( 4 ), ISEEDY( 4 ) + DOUBLE PRECISION RESULT( NTESTS ), RDUMMY( 1 ) +* .. +* .. External Functions .. + DOUBLE PRECISION DLAMCH, ZQPT01, ZQRT11, ZQRT12, ZLANGE + EXTERNAL DLAMCH, ZQPT01, ZQRT11, ZQRT12, ZLANGE +* .. +* .. External Subroutines .. + EXTERNAL ALAERH, ALAHD, ALASUM, DLAORD, ICOPY, ZAXPY, + $ XLAENV, ZGEQP3RK, ZLACPY, ZLASET, ZLATB4, + $ ZLATMS, ZUNMQR, ZSWAP +* .. +* .. Intrinsic Functions .. + INTRINSIC ABS, DBLE, MAX, MIN, MOD +* .. +* .. Scalars in Common .. + LOGICAL LERR, OK + CHARACTER*32 SRNAMT + INTEGER INFOT, IOUNIT, ZUNMQR_LWORK +* .. +* .. Common blocks .. + COMMON / INFOC / INFOT, IOUNIT, OK, LERR + COMMON / SRNAMC / SRNAMT +* .. +* .. Data statements .. + DATA ISEEDY / 1988, 1989, 1990, 1991 / +* .. +* .. Executable Statements .. +* +* Initialize constants and the random number seed. +* + PATH( 1: 1 ) = 'Zomplex precision' + PATH( 2: 3 ) = 'QK' + NRUN = 0 + NFAIL = 0 + NERRS = 0 + DO I = 1, 4 + ISEED( I ) = ISEEDY( I ) + END DO + EPS = DLAMCH( 'Epsilon' ) + INFOT = 0 +* + DO IM = 1, NM +* +* Do for each value of M in MVAL. +* + M = MVAL( IM ) + LDA = MAX( 1, M ) +* + DO IN = 1, NN +* +* Do for each value of N in NVAL. +* + N = NVAL( IN ) + MINMN = MIN( M, N ) + LWORK = MAX( 1, M*MAX( M, N )+4*MINMN+MAX( M, N ), + $ M*N + 2*MINMN + 4*N ) +* + DO INS = 1, NNS + NRHS = NSVAL( INS ) +* +* Set up parameters with ZLATB4 and generate +* M-by-NRHS B matrix with ZLATMS. +* IMAT = 14: +* Random matrix, CNDNUM = 2, NORM = ONE, +* MODE = 3 (geometric distribution of singular values). +* + CALL ZLATB4( PATH, 14, M, NRHS, TYPE, KL, KU, ANORM, + $ MODE, CNDNUM, DIST ) +* + SRNAMT = 'ZLATMS' + CALL ZLATMS( M, NRHS, DIST, ISEED, TYPE, S, MODE, + $ CNDNUM, ANORM, KL, KU, 'No packing', + $ COPYB, LDA, WORK, INFO ) +* +* Check error code from ZLATMS. +* + IF( INFO.NE.0 ) THEN + CALL ALAERH( PATH, 'ZLATMS', INFO, 0, ' ', M, + $ NRHS, -1, -1, -1, 6, NFAIL, NERRS, + $ NOUT ) + CYCLE + END IF +* + DO IMAT = 1, NTYPES +* +* Do the tests only if DOTYPE( IMAT ) is true. +* + IF( .NOT.DOTYPE( IMAT ) ) + $ CYCLE +* +* The type of distribution used to generate the random +* eigen-/singular values: +* ( 'S' for symmetric distribution ) => UNIFORM( -1, 1 ) +* +* Do for each type of NON-SYMMETRIC matrix: CNDNUM NORM MODE +* 1. Zero matrix +* 2. Random, Diagonal, CNDNUM = 2 CNDNUM = 2 ONE 3 ( geometric distribution of singular values ) +* 3. Random, Upper triangular, CNDNUM = 2 CNDNUM = 2 ONE 3 ( geometric distribution of singular values ) +* 4. Random, Lower triangular, CNDNUM = 2 CNDNUM = 2 ONE 3 ( geometric distribution of singular values ) +* 5. Random, First column is zero, CNDNUM = 2 CNDNUM = 2 ONE 3 ( geometric distribution of singular values ) +* 6. Random, Last MINMN column is zero, CNDNUM = 2 CNDNUM = 2 ONE 3 ( geometric distribution of singular values ) +* 7. Random, Last N column is zero, CNDNUM = 2 CNDNUM = 2 ONE 3 ( geometric distribution of singular values ) +* 8. Random, Middle column in MINMN is zero, CNDNUM = 2 CNDNUM = 2 ONE 3 ( geometric distribution of singular values ) +* 9. Random, First half of MINMN columns are zero, CNDNUM = 2 CNDNUM = 2 ONE 3 ( geometric distribution of singular values ) +* 10. Random, Last columns are zero starting from MINMN/2+1, CNDNUM = 2 CNDNUM = 2 ONE 3 ( geometric distribution of singular values ) +* 11. Random, Half MINMN columns in the middle are zero starting +* from MINMN/2-(MINMN/2)/2+1, CNDNUM = 2 CNDNUM = 2 ONE 3 ( geometric distribution of singular values ) +* 12. Random, Odd columns are ZERO, CNDNUM = 2 CNDNUM = 2 ONE 3 ( geometric distribution of singular values ) +* 13. Random, Even columns are ZERO, CNDNUM = 2 CNDNUM = 2 ONE 3 ( geometric distribution of singular values ) +* 14. Random, CNDNUM = 2 CNDNUM = 2 ONE 3 ( geometric distribution of singular values ) +* 15. Random, CNDNUM = sqrt(0.1/EPS) CNDNUM = BADC1 = sqrt(0.1/EPS) ONE 3 ( geometric distribution of singular values ) +* 16. Random, CNDNUM = 0.1/EPS CNDNUM = BADC2 = 0.1/EPS ONE 3 ( geometric distribution of singular values ) +* 17. Random, CNDNUM = 0.1/EPS, CNDNUM = BADC2 = 0.1/EPS ONE 2 ( one small singular value, S(N)=1/CNDNUM ) +* one small singular value S(N)=1/CNDNUM +* 18. Random, CNDNUM = 2, scaled near underflow CNDNUM = 2 SMALL = SAFMIN +* 19. Random, CNDNUM = 2, scaled near overflow CNDNUM = 2 LARGE = 1.0/( 0.25 * ( SAFMIN / EPS ) ) 3 ( geometric distribution of singular values ) +* + IF( IMAT.EQ.1 ) THEN +* +* Matrix 1: Zero matrix +* + CALL ZLASET( 'Full', M, N, CZERO, CZERO, COPYA, LDA ) + DO I = 1, MINMN + S( I ) = ZERO + END DO +* + ELSE IF( (IMAT.GE.2 .AND. IMAT.LE.4 ) + $ .OR. (IMAT.GE.14 .AND. IMAT.LE.19 ) ) THEN +* +* Matrices 2-5. +* +* Set up parameters with DLATB4 and generate a test +* matrix with ZLATMS. +* + CALL ZLATB4( PATH, IMAT, M, N, TYPE, KL, KU, ANORM, + $ MODE, CNDNUM, DIST ) +* + SRNAMT = 'ZLATMS' + CALL ZLATMS( M, N, DIST, ISEED, TYPE, S, MODE, + $ CNDNUM, ANORM, KL, KU, 'No packing', + $ COPYA, LDA, WORK, INFO ) +* +* Check error code from ZLATMS. +* + IF( INFO.NE.0 ) THEN + CALL ALAERH( PATH, 'ZLATMS', INFO, 0, ' ', M, N, + $ -1, -1, -1, IMAT, NFAIL, NERRS, + $ NOUT ) + CYCLE + END IF +* + CALL DLAORD( 'Decreasing', MINMN, S, 1 ) +* + ELSE IF( MINMN.GE.2 + $ .AND. IMAT.GE.5 .AND. IMAT.LE.13 ) THEN +* +* Rectangular matrices 5-13 that contain zero columns, +* only for matrices MINMN >=2. +* +* JB_ZERO is the column index of ZERO block. +* NB_ZERO is the column block size of ZERO block. +* NB_GEN is the column blcok size of the +* generated block. +* J_INC in the non_zero column index increment +* for matrix 12 and 13. +* J_FIRS_NZ is the index of the first non-zero +* column. +* + IF( IMAT.EQ.5 ) THEN +* +* First column is zero. +* + JB_ZERO = 1 + NB_ZERO = 1 + NB_GEN = N - NB_ZERO +* + ELSE IF( IMAT.EQ.6 ) THEN +* +* Last column MINMN is zero. +* + JB_ZERO = MINMN + NB_ZERO = 1 + NB_GEN = N - NB_ZERO +* + ELSE IF( IMAT.EQ.7 ) THEN +* +* Last column N is zero. +* + JB_ZERO = N + NB_ZERO = 1 + NB_GEN = N - NB_ZERO +* + ELSE IF( IMAT.EQ.8 ) THEN +* +* Middle column in MINMN is zero. +* + JB_ZERO = MINMN / 2 + 1 + NB_ZERO = 1 + NB_GEN = N - NB_ZERO +* + ELSE IF( IMAT.EQ.9 ) THEN +* +* First half of MINMN columns is zero. +* + JB_ZERO = 1 + NB_ZERO = MINMN / 2 + NB_GEN = N - NB_ZERO +* + ELSE IF( IMAT.EQ.10 ) THEN +* +* Last columns are zero columns, +* starting from (MINMN / 2 + 1) column. +* + JB_ZERO = MINMN / 2 + 1 + NB_ZERO = N - JB_ZERO + 1 + NB_GEN = N - NB_ZERO +* + ELSE IF( IMAT.EQ.11 ) THEN +* +* Half of the columns in the middle of MINMN +* columns is zero, starting from +* MINMN/2 - (MINMN/2)/2 + 1 column. +* + JB_ZERO = MINMN / 2 - (MINMN / 2) / 2 + 1 + NB_ZERO = MINMN / 2 + NB_GEN = N - NB_ZERO +* + ELSE IF( IMAT.EQ.12 ) THEN +* +* Odd-numbered columns are zero, +* + NB_GEN = N / 2 + NB_ZERO = N - NB_GEN + J_INC = 2 + J_FIRST_NZ = 2 +* + ELSE IF( IMAT.EQ.13 ) THEN +* +* Even-numbered columns are zero. +* + NB_ZERO = N / 2 + NB_GEN = N - NB_ZERO + J_INC = 2 + J_FIRST_NZ = 1 +* + END IF +* +* +* 1) Set the first NB_ZERO columns in COPYA(1:M,1:N) +* to zero. +* + CALL ZLASET( 'Full', M, NB_ZERO, CZERO, CZERO, + $ COPYA, LDA ) +* +* 2) Generate an M-by-(N-NB_ZERO) matrix with the +* chosen singular value distribution +* in COPYA(1:M,NB_ZERO+1:N). +* + CALL ZLATB4( PATH, IMAT, M, NB_GEN, TYPE, KL, KU, + $ ANORM, MODE, CNDNUM, DIST ) +* + SRNAMT = 'ZLATMS' +* + IND_OFFSET_GEN = NB_ZERO * LDA +* + CALL ZLATMS( M, NB_GEN, DIST, ISEED, TYPE, S, MODE, + $ CNDNUM, ANORM, KL, KU, 'No packing', + $ COPYA( IND_OFFSET_GEN + 1 ), LDA, + $ WORK, INFO ) +* +* Check error code from ZLATMS. +* + IF( INFO.NE.0 ) THEN + CALL ALAERH( PATH, 'ZLATMS', INFO, 0, ' ', M, + $ NB_GEN, -1, -1, -1, IMAT, NFAIL, + $ NERRS, NOUT ) + CYCLE + END IF +* +* 3) Swap the gererated colums from the right side +* NB_GEN-size block in COPYA into correct column +* positions. +* + IF( IMAT.EQ.6 + $ .OR. IMAT.EQ.7 + $ .OR. IMAT.EQ.8 + $ .OR. IMAT.EQ.10 + $ .OR. IMAT.EQ.11 ) THEN +* +* Move by swapping the generated columns +* from the right NB_GEN-size block from +* (NB_ZERO+1:NB_ZERO+JB_ZERO) +* into columns (1:JB_ZERO-1). +* + DO J = 1, JB_ZERO-1, 1 + CALL ZSWAP( M, + $ COPYA( ( NB_ZERO+J-1)*LDA+1), 1, + $ COPYA( (J-1)*LDA + 1 ), 1 ) + END DO +* + ELSE IF( IMAT.EQ.12 .OR. IMAT.EQ.13 ) THEN +* +* ( IMAT = 12, Odd-numbered ZERO columns. ) +* Swap the generated columns from the right +* NB_GEN-size block into the even zero colums in the +* left NB_ZERO-size block. +* +* ( IMAT = 13, Even-numbered ZERO columns. ) +* Swap the generated columns from the right +* NB_GEN-size block into the odd zero colums in the +* left NB_ZERO-size block. +* + DO J = 1, NB_GEN, 1 + IND_OUT = ( NB_ZERO+J-1 )*LDA + 1 + IND_IN = ( J_INC*(J-1)+(J_FIRST_NZ-1) )*LDA + $ + 1 + CALL ZSWAP( M, + $ COPYA( IND_OUT ), 1, + $ COPYA( IND_IN), 1 ) + END DO +* + END IF +* +* 5) Order the singular values generated by +* DLAMTS in decreasing order and add trailing zeros +* that correspond to zero columns. +* The total number of singular values is MINMN. +* + MINMNB_GEN = MIN( M, NB_GEN ) +* + CALL DLAORD( 'Decreasing', MINMNB_GEN, S, 1 ) + + DO I = MINMNB_GEN+1, MINMN + S( I ) = ZERO + END DO +* + ELSE +* +* IF(MINMN.LT.2) skip this size for this matrix type. +* + CYCLE + END IF +* +* Initialize a copy array for a pivot array for DGEQP3RK. +* + DO I = 1, N + IWORK( I ) = 0 + END DO +* + DO INB = 1, NNB +* +* Do for each pair of values (NB,NX) in NBVAL and NXVAL. +* + NB = NBVAL( INB ) + CALL XLAENV( 1, NB ) + NX = NXVAL( INB ) + CALL XLAENV( 3, NX ) +* +* We do MIN(M,N)+1 because we need a test for KMAX > N, +* when KMAX is larger than MIN(M,N), KMAX should be +* KMAX = MIN(M,N) +* + DO KMAX = 0, MIN(M,N)+1 +* +* Get a working copy of COPYA into A( 1:M,1:N ). +* Get a working copy of COPYB into A( 1:M, (N+1):NRHS ). +* Get a working copy of COPYB into into B( 1:M, 1:NRHS ). +* Get a working copy of IWORK(1:N) awith zeroes into +* which is going to be used as pivot array IWORK( N+1:2N ). +* NOTE: IWORK(2N+1:3N) is going to be used as a WORK array +* for the routine. +* + CALL ZLACPY( 'All', M, N, COPYA, LDA, A, LDA ) + CALL ZLACPY( 'All', M, NRHS, COPYB, LDA, + $ A( LDA*N + 1 ), LDA ) + CALL ZLACPY( 'All', M, NRHS, COPYB, LDA, + $ B, LDA ) + CALL ICOPY( N, IWORK( 1 ), 1, IWORK( N+1 ), 1 ) +* + ABSTOL = -1.0 + RELTOl = -1.0 +* +* Compute the QR factorization with pivoting of A +* + LW = MAX( 1, MAX( 2*N + NB*( N+NRHS+1 ), + $ 3*N + NRHS - 1 ) ) +* +* Compute ZGEQP3RK factorization of A. +* + SRNAMT = 'ZGEQP3RK' + CALL ZGEQP3RK( M, N, NRHS, KMAX, ABSTOL, RELTOL, + $ A, LDA, KFACT, MAXC2NRMK, + $ RELMAXC2NRMK, IWORK( N+1 ), TAU, + $ WORK, LW, RWORK, IWORK( 2*N+1 ), + $ INFO ) +* +* Check error code from ZGEQP3RK. +* + IF( INFO.LT.0 ) + $ CALL ALAERH( PATH, 'ZGEQP3RK', INFO, 0, ' ', + $ M, N, NX, -1, NB, IMAT, + $ NFAIL, NERRS, NOUT ) +* + IF( KFACT.EQ.MINMN ) THEN +* +* Compute test 1: +* +* This test in only for the full rank factorization of +* the matrix A. +* +* Array S(1:min(M,N)) contains svd(A) the sigular values +* of the original matrix A in decreasing absolute value +* order. The test computes svd(R), the vector sigular +* values of the upper trapezoid of A(1:M,1:N) that +* contains the factor R, in decreasing order. The test +* returns the ratio: +* +* 2-norm(svd(R) - svd(A)) / ( max(M,N) * 2-norm(svd(A)) * EPS ) +* + RESULT( 1 ) = ZQRT12( M, N, A, LDA, S, WORK, + $ LWORK , RWORK ) +* + DO T = 1, 1 + IF( RESULT( T ).GE.THRESH ) THEN + IF( NFAIL.EQ.0 .AND. NERRS.EQ.0 ) + $ CALL ALAHD( NOUT, PATH ) + WRITE( NOUT, FMT = 9999 ) 'ZGEQP3RK', M, N, + $ NRHS, KMAX, ABSTOL, RELTOL, NB, NX, + $ IMAT, T, RESULT( T ) + NFAIL = NFAIL + 1 + END IF + END DO + NRUN = NRUN + 1 +* +* End test 1 +* + END IF + +* Compute test 2: +* +* The test returns the ratio: +* +* 1-norm( A*P - Q*R ) / ( max(M,N) * 1-norm(A) * EPS ) +* + RESULT( 2 ) = ZQPT01( M, N, KFACT, COPYA, A, LDA, TAU, + $ IWORK( N+1 ), WORK, LWORK ) +* +* Compute test 3: +* +* The test returns the ratio: +* +* 1-norm( Q**T * Q - I ) / ( M * EPS ) +* + RESULT( 3 ) = ZQRT11( M, KFACT, A, LDA, TAU, WORK, + $ LWORK ) +* +* Print information about the tests that did not pass +* the threshold. +* + DO T = 2, 3 + IF( RESULT( T ).GE.THRESH ) THEN + IF( NFAIL.EQ.0 .AND. NERRS.EQ.0 ) + $ CALL ALAHD( NOUT, PATH ) + WRITE( NOUT, FMT = 9999 ) 'ZGEQP3RK', M, N, + $ NRHS, KMAX, ABSTOL, RELTOL, + $ NB, NX, IMAT, T, RESULT( T ) + NFAIL = NFAIL + 1 + END IF + END DO + NRUN = NRUN + 2 +* +* Compute test 4: +* +* This test is only for the factorizations with the +* rank greater than 2. +* The elements on the diagonal of R should be non- +* increasing. +* +* The test returns the ratio: +* +* Returns 1.0D+100 if abs(R(K+1,K+1)) > abs(R(K,K)), +* K=1:KFACT-1 +* + IF( MIN(KFACT, MINMN).GE.2 ) THEN +* + DO J = 1, KFACT-1, 1 +* + DTEMP = (( ABS( A( (J-1)*M+J ) ) - + $ ABS( A( (J)*M+J+1 ) ) ) / + $ ABS( A(1) ) ) +* + IF( DTEMP.LT.ZERO ) THEN + RESULT( 4 ) = BIGNUM + END IF +* + END DO +* +* Print information about the tests that did not +* pass the threshold. +* + DO T = 4, 4 + IF( RESULT( T ).GE.THRESH ) THEN + IF( NFAIL.EQ.0 .AND. NERRS.EQ.0 ) + $ CALL ALAHD( NOUT, PATH ) + WRITE( NOUT, FMT = 9999 ) 'ZGEQP3RK', + $ M, N, NRHS, KMAX, ABSTOL, RELTOL, + $ NB, NX, IMAT, T, + $ RESULT( T ) + NFAIL = NFAIL + 1 + END IF + END DO + NRUN = NRUN + 1 +* +* End test 4. +* + END IF +* +* Compute test 5: +* +* This test in only for matrix A with min(M,N) > 0. +* +* The test returns the ratio: +* +* 1-norm(Q**T * B - Q**T * B ) / +* ( M * EPS ) +* +* (1) Compute B:=Q**T * B in the matrix B. +* + IF( MINMN.GT.0 ) THEN +* + LWORK_MQR = MAX(1, NRHS) + CALL ZUNMQR( 'Left', 'Conjugate transpose', + $ M, NRHS, KFACT, A, LDA, TAU, B, LDA, + $ WORK, LWORK_MQR, INFO ) +* + DO I = 1, NRHS +* +* Compare N+J-th column of A and J-column of B. +* + CALL ZAXPY( M, -CONE, A( ( N+I-1 )*LDA+1 ), 1, + $ B( ( I-1 )*LDA+1 ), 1 ) + END DO +* + RESULT( 5 ) = + $ ABS( + $ ZLANGE( 'One-norm', M, NRHS, B, LDA, RDUMMY ) / + $ ( DBLE( M )*DLAMCH( 'Epsilon' ) ) + $ ) +* +* Print information about the tests that did not pass +* the threshold. +* + DO T = 5, 5 + IF( RESULT( T ).GE.THRESH ) THEN + IF( NFAIL.EQ.0 .AND. NERRS.EQ.0 ) + $ CALL ALAHD( NOUT, PATH ) + WRITE( NOUT, FMT = 9999 ) 'ZGEQP3RK', M, N, + $ NRHS, KMAX, ABSTOL, RELTOL, + $ NB, NX, IMAT, T, RESULT( T ) + NFAIL = NFAIL + 1 + END IF + END DO + NRUN = NRUN + 1 +* +* End compute test 5. +* + END IF +* +* END DO KMAX = 1, MIN(M,N)+1 +* + END DO +* +* END DO for INB = 1, NNB +* + END DO +* +* END DO for IMAT = 1, NTYPES +* + END DO +* +* END DO for INS = 1, NNS +* + END DO +* +* END DO for IN = 1, NN +* + END DO +* +* END DO for IM = 1, NM +* + END DO +* +* Print a summary of the results. +* + CALL ALASUM( PATH, NOUT, NFAIL, NRUN, NERRS ) +* + 9999 FORMAT( 1X, A, ' M =', I5, ', N =', I5, ', NRHS =', I5, + $ ', KMAX =', I5, ', ABSTOL =', G12.5, + $ ', RELTOL =', G12.5, ', NB =', I4, ', NX =', I4, + $ ', type ', I2, ', test ', I2, ', ratio =', G12.5 ) +* +* End of ZCHKQP3RK +* + END diff --git a/TESTING/LIN/zlatb4.f b/TESTING/LIN/zlatb4.f index 5001774dbf..a2b19f83d5 100644 --- a/TESTING/LIN/zlatb4.f +++ b/TESTING/LIN/zlatb4.f @@ -225,6 +225,110 @@ SUBROUTINE ZLATB4( PATH, IMAT, M, N, TYPE, KL, KU, ANORM, MODE, ELSE ANORM = ONE END IF +* + ELSE IF( LSAMEN( 2, C2, 'QK' ) ) THEN +* +* xQK: truncated QR with pivoting. +* Set parameters to generate a general +* M x N matrix. +* +* Set TYPE, the type of matrix to be generated. 'N' is nonsymmetric. +* + TYPE = 'N' +* +* Set DIST, the type of distribution for the random +* number generator. 'S' is +* + DIST = 'S' +* +* Set the lower and upper bandwidths. +* + IF( IMAT.EQ.2 ) THEN +* +* 2. Random, Diagonal, CNDNUM = 2 +* + KL = 0 + KU = 0 + CNDNUM = TWO + ANORM = ONE + MODE = 3 + ELSE IF( IMAT.EQ.3 ) THEN +* +* 3. Random, Upper triangular, CNDNUM = 2 +* + KL = 0 + KU = MAX( N-1, 0 ) + CNDNUM = TWO + ANORM = ONE + MODE = 3 + ELSE IF( IMAT.EQ.4 ) THEN +* +* 4. Random, Lower triangular, CNDNUM = 2 +* + KL = MAX( M-1, 0 ) + KU = 0 + CNDNUM = TWO + ANORM = ONE + MODE = 3 + ELSE +* +* 5.-19. Rectangular matrix +* + KL = MAX( M-1, 0 ) + KU = MAX( N-1, 0 ) +* + IF( IMAT.GE.5 .AND. IMAT.LE.14 ) THEN +* +* 5.-14. Random, CNDNUM = 2. +* + CNDNUM = TWO + ANORM = ONE + MODE = 3 +* + ELSE IF( IMAT.EQ.15 ) THEN +* +* 15. Random, CNDNUM = sqrt(0.1/EPS) +* + CNDNUM = BADC1 + ANORM = ONE + MODE = 3 +* + ELSE IF( IMAT.EQ.16 ) THEN +* +* 16. Random, CNDNUM = 0.1/EPS +* + CNDNUM = BADC2 + ANORM = ONE + MODE = 3 +* + ELSE IF( IMAT.EQ.17 ) THEN +* +* 17. Random, CNDNUM = 0.1/EPS, +* one small singular value S(N)=1/CNDNUM +* + CNDNUM = BADC2 + ANORM = ONE + MODE = 2 +* + ELSE IF( IMAT.EQ.18 ) THEN +* +* 18. Random, scaled near underflow +* + CNDNUM = TWO + ANORM = SMALL + MODE = 3 +* + ELSE IF( IMAT.EQ.19 ) THEN +* +* 19. Random, scaled near overflow +* + CNDNUM = TWO + ANORM = LARGE + MODE = 3 +* + END IF +* + END IF * ELSE IF( LSAMEN( 2, C2, 'GE' ) ) THEN * diff --git a/TESTING/LIN/zqpt01.f b/TESTING/LIN/zqpt01.f index 4e53f92c84..c69eb658fd 100644 --- a/TESTING/LIN/zqpt01.f +++ b/TESTING/LIN/zqpt01.f @@ -33,7 +33,7 @@ *> Householder vectors, and the rest of AF contains a partially updated *> matrix. *> -*> This function returns ||A*P - Q*R||/(||norm(A)||*eps*M) +*> This function returns ||A*P - Q*R|| / ( ||norm(A)||*eps*max(M,N) ) *> \endverbatim * * Arguments: @@ -172,28 +172,28 @@ DOUBLE PRECISION FUNCTION ZQPT01( M, N, K, A, AF, LDA, TAU, JPVT, * NORMA = ZLANGE( 'One-norm', M, N, A, LDA, RWORK ) * - DO 30 J = 1, K - DO 10 I = 1, MIN( J, M ) + DO J = 1, K + DO I = 1, MIN( J, M ) WORK( ( J-1 )*M+I ) = AF( I, J ) - 10 CONTINUE - DO 20 I = J + 1, M + END DO + DO I = J + 1, M WORK( ( J-1 )*M+I ) = ZERO - 20 CONTINUE - 30 CONTINUE - DO 40 J = K + 1, N + END DO + END DO + DO J = K + 1, N CALL ZCOPY( M, AF( 1, J ), 1, WORK( ( J-1 )*M+1 ), 1 ) - 40 CONTINUE + END DO * CALL ZUNMQR( 'Left', 'No transpose', M, N, K, AF, LDA, TAU, WORK, $ M, WORK( M*N+1 ), LWORK-M*N, INFO ) * - DO 50 J = 1, N + DO J = 1, N * * Compare i-th column of QR and jpvt(i)-th column of A * CALL ZAXPY( M, DCMPLX( -ONE ), A( 1, JPVT( J ) ), 1, $ WORK( ( J-1 )*M+1 ), 1 ) - 50 CONTINUE + END DO * ZQPT01 = ZLANGE( 'One-norm', M, N, WORK, M, RWORK ) / $ ( DBLE( MAX( M, N ) )*DLAMCH( 'Epsilon' ) ) diff --git a/TESTING/LIN/zqrt11.f b/TESTING/LIN/zqrt11.f index c3be59c365..dc4af744f6 100644 --- a/TESTING/LIN/zqrt11.f +++ b/TESTING/LIN/zqrt11.f @@ -158,9 +158,9 @@ DOUBLE PRECISION FUNCTION ZQRT11( M, K, A, LDA, TAU, WORK, LWORK ) CALL ZUNM2R( 'Left', 'Conjugate transpose', M, M, K, A, LDA, TAU, $ WORK, M, WORK( M*M+1 ), INFO ) * - DO 10 J = 1, M + DO J = 1, M WORK( ( J-1 )*M+J ) = WORK( ( J-1 )*M+J ) - ONE - 10 CONTINUE + END DO * ZQRT11 = ZLANGE( 'One-norm', M, M, WORK, M, RDUMMY ) / $ ( DBLE( M )*DLAMCH( 'Epsilon' ) ) diff --git a/TESTING/LIN/zqrt12.f b/TESTING/LIN/zqrt12.f index b128579288..91477b5ea2 100644 --- a/TESTING/LIN/zqrt12.f +++ b/TESTING/LIN/zqrt12.f @@ -28,7 +28,7 @@ *> ZQRT12 computes the singular values `svlues' of the upper trapezoid *> of A(1:M,1:N) and returns the ratio *> -*> || s - svlues||/(||svlues||*eps*max(M,N)) +*> || svlues - s||/(||s||*eps*max(M,N)) *> \endverbatim * * Arguments: @@ -154,11 +154,11 @@ DOUBLE PRECISION FUNCTION ZQRT12( M, N, A, LDA, S, WORK, LWORK, * CALL ZLASET( 'Full', M, N, DCMPLX( ZERO ), DCMPLX( ZERO ), WORK, $ M ) - DO 20 J = 1, N - DO 10 I = 1, MIN( J, M ) + DO J = 1, N + DO I = 1, MIN( J, M ) WORK( ( J-1 )*M+I ) = A( I, J ) - 10 CONTINUE - 20 CONTINUE + END DO + END DO * * Get machine parameters * @@ -207,9 +207,9 @@ DOUBLE PRECISION FUNCTION ZQRT12( M, N, A, LDA, S, WORK, LWORK, * ELSE * - DO 30 I = 1, MN + DO I = 1, MN RWORK( I ) = ZERO - 30 CONTINUE + END DO END IF * * Compare s and singular values of work @@ -217,6 +217,7 @@ DOUBLE PRECISION FUNCTION ZQRT12( M, N, A, LDA, S, WORK, LWORK, CALL DAXPY( MN, -ONE, S, 1, RWORK( 1 ), 1 ) ZQRT12 = DASUM( MN, RWORK( 1 ), 1 ) / $ ( DLAMCH( 'Epsilon' )*DBLE( MAX( M, N ) ) ) +* IF( NRMSVL.NE.ZERO ) $ ZQRT12 = ZQRT12 / NRMSVL * diff --git a/TESTING/ctest.in b/TESTING/ctest.in index a3588b4a12..74ff31ab8d 100644 --- a/TESTING/ctest.in +++ b/TESTING/ctest.in @@ -42,6 +42,7 @@ CRQ 8 List types on next line if 0 < NTYPES < 8 CLQ 8 List types on next line if 0 < NTYPES < 8 CQL 8 List types on next line if 0 < NTYPES < 8 CQP 6 List types on next line if 0 < NTYPES < 6 +CQK 19 List types on next line if 0 < NTYPES < 19 CTZ 3 List types on next line if 0 < NTYPES < 3 CLS 6 List types on next line if 0 < NTYPES < 6 CEQ diff --git a/TESTING/dtest.in b/TESTING/dtest.in index 29bb8b92eb..1b6c7bd4a8 100644 --- a/TESTING/dtest.in +++ b/TESTING/dtest.in @@ -36,6 +36,7 @@ DRQ 8 List types on next line if 0 < NTYPES < 8 DLQ 8 List types on next line if 0 < NTYPES < 8 DQL 8 List types on next line if 0 < NTYPES < 8 DQP 6 List types on next line if 0 < NTYPES < 6 +DQK 19 LIst types on next line if 0 < NTYPES < 19 DTZ 3 List types on next line if 0 < NTYPES < 3 DLS 6 List types on next line if 0 < NTYPES < 6 DEQ diff --git a/TESTING/stest.in b/TESTING/stest.in index 27ac30040f..7faa8b7a11 100644 --- a/TESTING/stest.in +++ b/TESTING/stest.in @@ -36,6 +36,7 @@ SRQ 8 List types on next line if 0 < NTYPES < 8 SLQ 8 List types on next line if 0 < NTYPES < 8 SQL 8 List types on next line if 0 < NTYPES < 8 SQP 6 List types on next line if 0 < NTYPES < 6 +SQK 19 List types on next line if 0 < NTYPES < 19 STZ 3 List types on next line if 0 < NTYPES < 3 SLS 6 List types on next line if 0 < NTYPES < 6 SEQ diff --git a/TESTING/ztest.in b/TESTING/ztest.in index 58da33d605..c83e82e456 100644 --- a/TESTING/ztest.in +++ b/TESTING/ztest.in @@ -42,6 +42,7 @@ ZRQ 8 List types on next line if 0 < NTYPES < 8 ZLQ 8 List types on next line if 0 < NTYPES < 8 ZQL 8 List types on next line if 0 < NTYPES < 8 ZQP 6 List types on next line if 0 < NTYPES < 6 +ZQK 19 List types on next line if 0 < NTYPES < 19 ZTZ 3 List types on next line if 0 < NTYPES < 3 ZLS 6 List types on next line if 0 < NTYPES < 6 ZEQ