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Merged
merged 10 commits into from
Nov 9, 2022
Merged

New algorithms for computing Givens rotations #631

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2495f1c
Solves a precision bug in clartg
weslleyspereira Oct 15, 2021
b89b15b
Removes one line from clartg
weslleyspereira Oct 18, 2021
ac11f62
Several changes to reduce the computation error
weslleyspereira Oct 26, 2021
4320882
Starting to modify zlartg
weslleyspereira Oct 28, 2021
37a1a1e
Fix all other Givens rotation routines
weslleyspereira Nov 3, 2021
a49a659
Merge branch 'fix-precision-in-clartgf90-2' of github.com:weslleysper…
weslleyspereira Nov 3, 2021
2904d87
Algorithm precise and with no bias in the error
weslleyspereira Nov 23, 2021
95b6e84
Updates Givens rotations with preciser algorithms
weslleyspereira Dec 10, 2021
cdc8f33
Fix documentation thanks to @vladimir-ch
weslleyspereira Dec 13, 2021
c362fff
Minor changes
weslleyspereira Dec 13, 2021
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Fix all other Givens rotation routines
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@weslleyspereira
weslleyspereira committed Nov 3, 2021
commit 37a1a1e6896a4037fdb1f287ee643f8a57ce679e
37 changes: 15 additions & 22 deletions BLAS/SRC/crotg.f90
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Expand Up @@ -122,7 +122,7 @@ subroutine CROTG( a, b, c, s )
complex(wp) :: a, b, s
! ..
! .. Local Scalars ..
real(wp) :: d, f1, f2, g1, g2, h2, p, u, uu, v, vv, w
real(wp) :: d, f1, f2, g1, g2, h2, u, v, w
complex(wp) :: f, fs, g, gs, r, t
! ..
! .. Intrinsic Functions ..
Expand All @@ -149,19 +149,16 @@ subroutine CROTG( a, b, c, s )
!
! Use unscaled algorithm
!
g2 = ABSSQ( g )
d = sqrt( g2 )
d = abs( g )
s = conjg( g ) / d
r = d
else
!
! Use scaled algorithm
!
u = min( safmax, max( safmin, g1 ) )
uu = one / u
gs = g*uu
g2 = ABSSQ( gs )
d = sqrt( g2 )
gs = g / u
d = abs( gs )
s = conjg( gs ) / d
r = d*u
end if
Expand All @@ -181,35 +178,32 @@ subroutine CROTG( a, b, c, s )
else
d = sqrt( f2 )*sqrt( h2 )
end if
p = 1 / d
c = f2*p
s = conjg( g )*( f*p )
r = f*( h2*p )
c = f2 / d
s = conjg( g )*( f / d )
r = f*( h2 / d )
else
!
! Use scaled algorithm
!
u = min( safmax, max( safmin, f1, g1 ) )
uu = one / u
gs = g*uu
gs = g / u
g2 = ABSSQ( gs )
if( f1*uu < rtmin ) then
if( f1 / u < rtmin ) then
!
! f is not well-scaled when scaled by g1.
! Use a different scaling for f.
!
v = min( safmax, max( safmin, f1 ) )
vv = one / v
w = v * uu
fs = f*vv
w = v / u
fs = f / v
f2 = ABSSQ( fs )
h2 = f2*w**2 + g2
else
!
! Otherwise use the same scaling for f and g.
!
w = one
fs = f*uu
fs = f / u
f2 = ABSSQ( fs )
h2 = f2 + g2
end if
Expand All @@ -218,10 +212,9 @@ subroutine CROTG( a, b, c, s )
else
d = sqrt( f2 )*sqrt( h2 )
end if
p = 1 / d
c = ( f2*p )*w
s = conjg( gs )*( fs*p )
r = ( fs*( h2*p ) )*u
c = ( f2 / d )*w
s = conjg( gs )*( fs / d )
r = ( fs*( h2 / d ) )*u
end if
end if
a = r
Expand Down
37 changes: 15 additions & 22 deletions BLAS/SRC/zrotg.f90
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Expand Up @@ -122,7 +122,7 @@ subroutine ZROTG( a, b, c, s )
complex(wp) :: a, b, s
! ..
! .. Local Scalars ..
real(wp) :: d, f1, f2, g1, g2, h2, p, u, uu, v, vv, w
real(wp) :: d, f1, f2, g1, g2, h2, u, v, w
complex(wp) :: f, fs, g, gs, r, t
! ..
! .. Intrinsic Functions ..
Expand All @@ -149,19 +149,16 @@ subroutine ZROTG( a, b, c, s )
!
! Use unscaled algorithm
!
g2 = ABSSQ( g )
d = sqrt( g2 )
d = abs( g )
s = conjg( g ) / d
r = d
else
!
! Use scaled algorithm
!
u = min( safmax, max( safmin, g1 ) )
uu = one / u
gs = g*uu
g2 = ABSSQ( gs )
d = sqrt( g2 )
gs = g / u
d = abs( gs )
s = conjg( gs ) / d
r = d*u
end if
Expand All @@ -181,35 +178,32 @@ subroutine ZROTG( a, b, c, s )
else
d = sqrt( f2 )*sqrt( h2 )
end if
p = 1 / d
c = f2*p
s = conjg( g )*( f*p )
r = f*( h2*p )
c = f2 / d
s = conjg( g )*( f / d )
r = f*( h2 / d )
else
!
! Use scaled algorithm
!
u = min( safmax, max( safmin, f1, g1 ) )
uu = one / u
gs = g*uu
gs = g / u
g2 = ABSSQ( gs )
if( f1*uu < rtmin ) then
if( f1 / u < rtmin ) then
!
! f is not well-scaled when scaled by g1.
! Use a different scaling for f.
!
v = min( safmax, max( safmin, f1 ) )
vv = one / v
w = v * uu
fs = f*vv
w = v / u
fs = f / v
f2 = ABSSQ( fs )
h2 = f2*w**2 + g2
else
!
! Otherwise use the same scaling for f and g.
!
w = one
fs = f*uu
fs = f / u
f2 = ABSSQ( fs )
h2 = f2 + g2
end if
Expand All @@ -218,10 +212,9 @@ subroutine ZROTG( a, b, c, s )
else
d = sqrt( f2 )*sqrt( h2 )
end if
p = 1 / d
c = ( f2*p )*w
s = conjg( gs )*( fs*p )
r = ( fs*( h2*p ) )*u
c = ( f2 / d )*w
s = conjg( gs )*( fs / d )
r = ( fs*( h2 / d ) )*u
end if
end if
a = r
Expand Down
49 changes: 13 additions & 36 deletions SRC/clartg.f90
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Expand Up @@ -178,33 +178,22 @@ subroutine CLARTG( f, g, c, s, r )
f2 = ABSSQ( f )
g2 = ABSSQ( g )
h2 = f2 + g2
if( f2 > safmin * g2 ) then
d = sqrt( one + g2/f2 )
c = one / d
if( f2 > rtmin .and. h2 < rtmax ) then
s = conjg( g )*( f / sqrt( f2*h2 ) )
else
s = conjg( g )*( f /( f2*d ) )
end if
r = f * d
if( f2 > rtmin .and. h2 < rtmax ) then
d = sqrt( f2*h2 )
else
if( f2 > rtmin .and. h2 < rtmax ) then
d = sqrt( f2*h2 )
else
d = sqrt( f2 )*sqrt( h2 )
end if
c = f2 / d
s = conjg( g )*( f / d )
r = f*( h2 / d )
d = sqrt( f2 )*sqrt( h2 )
end if
c = f2 / d
s = conjg( g )*( f / d )
r = f*( h2 / d )
else
!
! Use scaled algorithm
!
u = min( safmax, max( safmin, f1, g1 ) )
gs = g / u
g2 = ABSSQ( gs )
if( f1 < rtmin * u ) then
if( f1 / u < rtmin ) then
!
! f is not well-scaled when scaled by g1.
! Use a different scaling for f.
Expand All @@ -223,26 +212,14 @@ subroutine CLARTG( f, g, c, s, r )
f2 = ABSSQ( fs )
h2 = f2 + g2
end if
if( f2 > safmin * g2 ) then
! Use a precise algorithm
d = sqrt( w**2 + g2/f2 )
c = w / d
if( f2 > rtmin .and. h2 < rtmax ) then
s = conjg( gs )*( fs / sqrt( f2*h2 ) )
else
s = conjg( gs )*( fs / ( f2*d ) )
end if
r = ( fs * d ) * u
if( f2 > rtmin .and. h2 < rtmax ) then
d = sqrt( f2*h2 )
else
if( f2 > rtmin .and. h2 < rtmax ) then
d = sqrt( f2*h2 )
else
d = sqrt( f2 )*sqrt( h2 )
end if
c = ( f2 / d )*w
s = conjg( gs )*( fs / d )
r = ( fs*( h2 / d ) )*u
d = sqrt( f2 )*sqrt( h2 )
end if
c = ( f2 / d )*w
s = conjg( gs )*( fs / d )
r = ( fs*( h2 / d ) )*u
end if
end if
return
Expand Down
20 changes: 9 additions & 11 deletions SRC/dlartg.f90
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Expand Up @@ -123,7 +123,7 @@ subroutine DLARTG( f, g, c, s, r )
real(wp) :: c, f, g, r, s
! ..
! .. Local Scalars ..
real(wp) :: d, f1, fs, g1, gs, p, u, uu
real(wp) :: d, f1, fs, g1, gs, u
! ..
! .. Intrinsic Functions ..
intrinsic :: abs, sign, sqrt
Expand All @@ -143,20 +143,18 @@ subroutine DLARTG( f, g, c, s, r )
else if( f1 > rtmin .and. f1 < rtmax .and. &
g1 > rtmin .and. g1 < rtmax ) then
d = sqrt( f*f + g*g )
p = one / d
c = f1*p
s = g*sign( p, f )
c = f1 / d
r = sign( d, f )
s = g / r
else
u = min( safmax, max( safmin, f1, g1 ) )
uu = one / u
fs = f*uu
gs = g*uu
fs = f / u
gs = g / u
d = sqrt( fs*fs + gs*gs )
p = one / d
c = abs( fs )*p
s = gs*sign( p, f )
r = sign( d, f )*u
c = abs( fs ) / d
r = sign( d, f )
s = gs / r
r = r * u
end if
return
end subroutine
20 changes: 9 additions & 11 deletions SRC/slartg.f90
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Expand Up @@ -123,7 +123,7 @@ subroutine SLARTG( f, g, c, s, r )
real(wp) :: c, f, g, r, s
! ..
! .. Local Scalars ..
real(wp) :: d, f1, fs, g1, gs, p, u, uu
real(wp) :: d, f1, fs, g1, gs, u
! ..
! .. Intrinsic Functions ..
intrinsic :: abs, sign, sqrt
Expand All @@ -143,20 +143,18 @@ subroutine SLARTG( f, g, c, s, r )
else if( f1 > rtmin .and. f1 < rtmax .and. &
g1 > rtmin .and. g1 < rtmax ) then
d = sqrt( f*f + g*g )
p = one / d
c = f1*p
s = g*sign( p, f )
c = f1 / d
r = sign( d, f )
s = g / r
else
u = min( safmax, max( safmin, f1, g1 ) )
uu = one / u
fs = f*uu
gs = g*uu
fs = f / u
gs = g / u
d = sqrt( fs*fs + gs*gs )
p = one / d
c = abs( fs )*p
s = gs*sign( p, f )
r = sign( d, f )*u
c = abs( fs ) / d
r = sign( d, f )
s = gs / r
r = r*u
end if
return
end subroutine
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