added new routines [S,D,C,Z]GELST and test code for the least squares

[scr2016]

Description
added new routines [S,D,C,Z]GELST and test code for the least squares
squares algorithm that solves overdetermined or underdetermined systems for GE matrices using QR or LQ factorization with compact WY representation of Q inside the routine.

*GELST is similar to *GELS, but unlike *GELS does not compute triangular blocks twice, i.e. runs faster.

modified: SRC/Makefile
new file: SRC/cgelst.f
new file: SRC/dgelst.f
new file: SRC/sgelst.f
new file: SRC/zgelst.f
modified: TESTING/LIN/alahd.f
modified: TESTING/LIN/cdrvls.f
modified: TESTING/LIN/cerrls.f
modified: TESTING/LIN/ddrvls.f
modified: TESTING/LIN/derrls.f
modified: TESTING/LIN/sdrvls.f
modified: TESTING/LIN/serrls.f
modified: TESTING/LIN/zdrvls.f

[angsch]

@scr2016 Is there a particular reason why your changes are a separate routine and not an upgrade of GELS? Your changes appear fully compatible with GELS and could be a great modernization that could be widely picked up by, say, octave or Julia, which likely call GELS as the de facto standard LS solver.

Edit: I mixed up geqrt and geqrt3. Sorry. Have you considered interleaving the QR factorization and the transform of B, which reduces the workspace to a single triangular factor? Sketched for QR, $m \geq n$:

do i = 1, nb
  ib = min(nb,n-i+1) ! No special treatment of the last block
  ! Compute QR factorization of panel A(i:end, i:i+ib-1)
  call geqrt()
  ! Trailing matrix update H**T * A(i:m,i+ib:n)
  call larfb()
  ! Transform H**T * B(i:m,1:nrhs)
  call larfb()

Then your change is really fully compatible with GELS

[dbielich]

I see edits to all TESTING/LIN/Xerrls.f files except TESTING/LIN/zerrls.f.

So there is no way to check any errors from the double complex code currently.

[abiiget765]

verify-elf: ERROR: ./usr/lib64/liblapack.so.3.12.0: undefined symbol: cdotc_64_
verify-elf: ERROR: ./usr/lib64/liblapack.so.3.12.0: undefined symbol: zherk_64_
verify-elf: ERROR: ./usr/lib64/liblapack.so.3.12.0: undefined symbol: ctpmv_64_
verify-elf: ERROR: ./usr/lib64/liblapack.so.3.12.0: undefined symbol: sgemv_64_
verify-elf: ERROR: ./usr/lib64/liblapack.so.3.12.0: undefined symbol: strsm_64_
verify-elf: ERROR: ./usr/lib64/liblapack.so.3.12.0: undefined symbol: cdotu_64_
verify-elf: ERROR: ./usr/lib64/liblapack.so.3.12.0: undefined symbol: cgbmv_64_
verify-elf: ERROR: ./usr/lib64/liblapack.so.3.12.0: undefined symbol: ssbmv_64_
verify-elf: ERROR: ./usr/lib64/liblapack.so.3.12.0: undefined symbol: zher2_64_
verify-elf: ERROR: ./usr/lib64/liblapack.so.3.12.0: undefined symbol: zgerc_64_
verify-elf: ERROR: ./usr/lib64/liblapack.so.3.12.0: undefined symbol: ctpsv_64_
verify-elf: ERROR: ./usr/lib64/liblapack.so.3.12.0: undefined symbol: dgemv_64_
verify-elf: ERROR: ./usr/lib64/liblapack.so.3.12.0: undefined symbol: dtbsv_64_
verify-elf: ERROR: ./usr/lib64/liblapack.so.3.12.0: undefined symbol: ztrmm_64_
verify-elf: ERROR: ./usr/lib64/liblapack.so.3.12.0: undefined symbol: ztbsv_64_
verify-elf: ERROR: ./usr/lib64/liblapack.so.3.12.0: undefined symbol: cherk_64_
verify-elf: ERROR: ./usr/lib64/liblapack.so.3.12.0: undefined symbol: zaxpy_64_

[martin-frbg]

probably same as #1147 - if you're building with a non-default BLAS you need to make sure that it has the 64bit integer versions of the functions