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Merge pull request #3834 from martin-frbg/lapack631 

Use new algorithms for computing Givens rotations (Reference-LAPACK PR631)
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Martin Kroeker 2022-11-21 08:30:14 +01:00 committed by GitHub
parent 9343499256 7ae4269add
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4 changed files with 247 additions and 133 deletions
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127
lapack-netlib/SRC/clartg.f90
View File

@ -38,19 +38,20 @@
!> !>
!> S = sgn(F) * conjg(G) / sqrt(|F|**2 + |G|**2) !> S = sgn(F) * conjg(G) / sqrt(|F|**2 + |G|**2)
!> !>
!> Special conditions:
!> If G=0, then C=1 and S=0.
!> If F=0, then C=0 and S is chosen so that R is real.
!>
!> When F and G are real, the formulas simplify to C = F/R and !> When F and G are real, the formulas simplify to C = F/R and
!> S = G/R, and the returned values of C, S, and R should be !> S = G/R, and the returned values of C, S, and R should be
!> identical to those returned by CLARTG. !> identical to those returned by SLARTG.
!> !>
!> The algorithm used to compute these quantities incorporates scaling !> The algorithm used to compute these quantities incorporates scaling
!> to avoid overflow or underflow in computing the square root of the !> to avoid overflow or underflow in computing the square root of the
!> sum of squares. !> sum of squares.
!> !>
!> This is a faster version of the BLAS1 routine CROTG, except for !> This is the same routine CROTG fom BLAS1, except that
!> the following differences:
!> F and G are unchanged on return. !> F and G are unchanged on return.
!> If G=0, then C=1 and S=0.
!> If F=0, then C=0 and S is chosen so that R is real.
!> !>
!> Below, wp=>sp stands for single precision from LA_CONSTANTS module. !> Below, wp=>sp stands for single precision from LA_CONSTANTS module.
!> \endverbatim !> \endverbatim
@ -91,22 +92,19 @@
! Authors: ! Authors:
! ======== ! ========
! !
!> \author Edward Anderson, Lockheed Martin !> \author Weslley Pereira, University of Colorado Denver, USA
! !
!> \date August 2016 !> \date December 2021
! !
!> \ingroup OTHERauxiliary !> \ingroup OTHERauxiliary
! !
!> \par Contributors:
! ==================
!>
!> Weslley Pereira, University of Colorado Denver, USA
!
!> \par Further Details: !> \par Further Details:
! ===================== ! =====================
!> !>
!> \verbatim !> \verbatim
!> !>
!> Based on the algorithm from
!>
!> Anderson E. (2017) !> Anderson E. (2017)
!> Algorithm 978: Safe Scaling in the Level 1 BLAS !> Algorithm 978: Safe Scaling in the Level 1 BLAS
!> ACM Trans Math Softw 44:1--28 !> ACM Trans Math Softw 44:1--28
@ -117,7 +115,7 @@
subroutine CLARTG( f, g, c, s, r ) subroutine CLARTG( f, g, c, s, r )
use LA_CONSTANTS, & use LA_CONSTANTS, &
only: wp=>sp, zero=>szero, one=>sone, two=>stwo, czero, & only: wp=>sp, zero=>szero, one=>sone, two=>stwo, czero, &
rtmin=>srtmin, rtmax=>srtmax, safmin=>ssafmin, safmax=>ssafmax safmin=>ssafmin, safmax=>ssafmax
! !
! -- LAPACK auxiliary routine -- ! -- LAPACK auxiliary routine --
! -- LAPACK is a software package provided by Univ. of Tennessee, -- ! -- LAPACK is a software package provided by Univ. of Tennessee, --
@ -129,7 +127,7 @@ subroutine CLARTG( f, g, c, s, r )
complex(wp) f, g, r, s complex(wp) f, g, r, s
! .. ! ..
! .. Local Scalars .. ! .. 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, rtmin, rtmax
complex(wp) :: fs, gs, t complex(wp) :: fs, gs, t
! .. ! ..
! .. Intrinsic Functions .. ! .. Intrinsic Functions ..
@ -141,6 +139,9 @@ subroutine CLARTG( f, g, c, s, r )
! .. Statement Function definitions .. ! .. Statement Function definitions ..
ABSSQ( t ) = real( t )**2 + aimag( t )**2 ABSSQ( t ) = real( t )**2 + aimag( t )**2
! .. ! ..
! .. Constants ..
rtmin = sqrt( safmin )
! ..
! .. Executable Statements .. ! .. Executable Statements ..
! !
if( g == czero ) then if( g == czero ) then
@ -149,11 +150,21 @@ subroutine CLARTG( f, g, c, s, r )
r = f r = f
else if( f == czero ) then else if( f == czero ) then
c = zero c = zero
if( real(g) == zero ) then
r = abs(aimag(g))
s = conjg( g ) / r
elseif( aimag(g) == zero ) then
r = abs(real(g))
s = conjg( g ) / r
else
g1 = max( abs(real(g)), abs(aimag(g)) ) g1 = max( abs(real(g)), abs(aimag(g)) )
rtmax = sqrt( safmax/2 )
if( g1 > rtmin .and. g1 < rtmax ) then if( g1 > rtmin .and. g1 < rtmax ) then
! !
! Use unscaled algorithm ! Use unscaled algorithm
! !
! The following two lines can be replaced by `d = abs( g )`.
! This algorithm do not use the intrinsic complex abs.
g2 = ABSSQ( g ) g2 = ABSSQ( g )
d = sqrt( g2 ) d = sqrt( g2 )
s = conjg( g ) / d s = conjg( g ) / d
@ -163,16 +174,19 @@ subroutine CLARTG( f, g, c, s, r )
! Use scaled algorithm ! Use scaled algorithm
! !
u = min( safmax, max( safmin, g1 ) ) u = min( safmax, max( safmin, g1 ) )
uu = one / u gs = g / u
gs = g*uu ! The following two lines can be replaced by `d = abs( gs )`.
! This algorithm do not use the intrinsic complex abs.
g2 = ABSSQ( gs ) g2 = ABSSQ( gs )
d = sqrt( g2 ) d = sqrt( g2 )
s = conjg( gs ) / d s = conjg( gs ) / d
r = d*u r = d*u
end if end if
end if
else else
f1 = max( abs(real(f)), abs(aimag(f)) ) f1 = max( abs(real(f)), abs(aimag(f)) )
g1 = max( abs(real(g)), abs(aimag(g)) ) g1 = max( abs(real(g)), abs(aimag(g)) )
rtmax = sqrt( safmax/4 )
if( f1 > rtmin .and. f1 < rtmax .and. & if( f1 > rtmin .and. f1 < rtmax .and. &
g1 > rtmin .and. g1 < rtmax ) then g1 > rtmin .and. g1 < rtmax ) then
! !
@ -181,32 +195,51 @@ subroutine CLARTG( f, g, c, s, r )
f2 = ABSSQ( f ) f2 = ABSSQ( f )
g2 = ABSSQ( g ) g2 = ABSSQ( g )
h2 = f2 + g2 h2 = f2 + g2
! safmin <= f2 <= h2 <= safmax
if( f2 >= h2 * safmin ) then
! safmin <= f2/h2 <= 1, and h2/f2 is finite
c = sqrt( f2 / h2 )
r = f / c
rtmax = rtmax * 2
if( f2 > rtmin .and. h2 < rtmax ) then if( f2 > rtmin .and. h2 < rtmax ) then
d = sqrt( f2*h2 ) ! safmin <= sqrt( f2*h2 ) <= safmax
s = conjg( g ) * ( f / sqrt( f2*h2 ) )
else else
d = sqrt( f2 )*sqrt( h2 ) s = conjg( g ) * ( r / h2 )
end if
else
! f2/h2 <= safmin may be subnormal, and h2/f2 may overflow.
! Moreover,
! safmin <= f2*f2 * safmax < f2 * h2 < h2*h2 * safmin <= safmax,
! sqrt(safmin) <= sqrt(f2 * h2) <= sqrt(safmax).
! Also,
! g2 >> f2, which means that h2 = g2.
d = sqrt( f2 * h2 )
c = f2 / d
if( c >= safmin ) then
r = f / c
else
! f2 / sqrt(f2 * h2) < safmin, then
! sqrt(safmin) <= f2 * sqrt(safmax) <= h2 / sqrt(f2 * h2) <= h2 * (safmin / f2) <= h2 <= safmax
r = f * ( h2 / d )
end if
s = conjg( g ) * ( f / d )
end if end if
p = 1 / d
c = f2*p
s = conjg( g )*( f*p )
r = f*( h2*p )
else else
! !
! Use scaled algorithm ! Use scaled algorithm
! !
u = min( safmax, max( safmin, f1, g1 ) ) u = min( safmax, max( safmin, f1, g1 ) )
uu = one / u gs = g / u
gs = g*uu
g2 = ABSSQ( gs ) g2 = ABSSQ( gs )
if( f1*uu < rtmin ) then if( f1 / u < rtmin ) then
! !
! f is not well-scaled when scaled by g1. ! f is not well-scaled when scaled by g1.
! Use a different scaling for f. ! Use a different scaling for f.
! !
v = min( safmax, max( safmin, f1 ) ) v = min( safmax, max( safmin, f1 ) )
vv = one / v w = v / u
w = v * uu fs = f / v
fs = f*vv
f2 = ABSSQ( fs ) f2 = ABSSQ( fs )
h2 = f2*w**2 + g2 h2 = f2*w**2 + g2
else else
@ -214,19 +247,43 @@ subroutine CLARTG( f, g, c, s, r )
! Otherwise use the same scaling for f and g. ! Otherwise use the same scaling for f and g.
! !
w = one w = one
fs = f*uu fs = f / u
f2 = ABSSQ( fs ) f2 = ABSSQ( fs )
h2 = f2 + g2 h2 = f2 + g2
end if end if
! safmin <= f2 <= h2 <= safmax
if( f2 >= h2 * safmin ) then
! safmin <= f2/h2 <= 1, and h2/f2 is finite
c = sqrt( f2 / h2 )
r = fs / c
rtmax = rtmax * 2
if( f2 > rtmin .and. h2 < rtmax ) then if( f2 > rtmin .and. h2 < rtmax ) then
d = sqrt( f2*h2 ) ! safmin <= sqrt( f2*h2 ) <= safmax
s = conjg( gs ) * ( fs / sqrt( f2*h2 ) )
else else
d = sqrt( f2 )*sqrt( h2 ) s = conjg( gs ) * ( r / h2 )
end if end if
p = 1 / d else
c = ( f2*p )*w ! f2/h2 <= safmin may be subnormal, and h2/f2 may overflow.
s = conjg( gs )*( fs*p ) ! Moreover,
r = ( fs*( h2*p ) )*u ! safmin <= f2*f2 * safmax < f2 * h2 < h2*h2 * safmin <= safmax,
! sqrt(safmin) <= sqrt(f2 * h2) <= sqrt(safmax).
! Also,
! g2 >> f2, which means that h2 = g2.
d = sqrt( f2 * h2 )
c = f2 / d
if( c >= safmin ) then
r = fs / c
else
! f2 / sqrt(f2 * h2) < safmin, then
! sqrt(safmin) <= f2 * sqrt(safmax) <= h2 / sqrt(f2 * h2) <= h2 * (safmin / f2) <= h2 <= safmax
r = fs * ( h2 / d )
end if
s = conjg( gs ) * ( fs / d )
end if
! Rescale c and r
c = c * w
r = r * u
end if end if
end if end if
return return

28
lapack-netlib/SRC/dlartg.f90
View File

@ -45,8 +45,6 @@
!> floating point operations (saves work in DBDSQR when !> floating point operations (saves work in DBDSQR when
!> there are zeros on the diagonal). !> there are zeros on the diagonal).
!> !>
!> If F exceeds G in magnitude, C will be positive.
!>
!> Below, wp=>dp stands for double precision from LA_CONSTANTS module. !> Below, wp=>dp stands for double precision from LA_CONSTANTS module.
!> \endverbatim !> \endverbatim
! !
@ -112,7 +110,7 @@
subroutine DLARTG( f, g, c, s, r ) subroutine DLARTG( f, g, c, s, r )
use LA_CONSTANTS, & use LA_CONSTANTS, &
only: wp=>dp, zero=>dzero, half=>dhalf, one=>done, & only: wp=>dp, zero=>dzero, half=>dhalf, one=>done, &
rtmin=>drtmin, rtmax=>drtmax, safmin=>dsafmin, safmax=>dsafmax safmin=>dsafmin, safmax=>dsafmax
! !
! -- LAPACK auxiliary routine -- ! -- LAPACK auxiliary routine --
! -- LAPACK is a software package provided by Univ. of Tennessee, -- ! -- LAPACK is a software package provided by Univ. of Tennessee, --
@ -123,11 +121,15 @@ subroutine DLARTG( f, g, c, s, r )
real(wp) :: c, f, g, r, s real(wp) :: c, f, g, r, s
! .. ! ..
! .. Local Scalars .. ! .. Local Scalars ..
real(wp) :: d, f1, fs, g1, gs, p, u, uu real(wp) :: d, f1, fs, g1, gs, u, rtmin, rtmax
! .. ! ..
! .. Intrinsic Functions .. ! .. Intrinsic Functions ..
intrinsic :: abs, sign, sqrt intrinsic :: abs, sign, sqrt
! .. ! ..
! .. Constants ..
rtmin = sqrt( safmin )
rtmax = sqrt( safmax/2 )
! ..
! .. Executable Statements .. ! .. Executable Statements ..
! !
f1 = abs( f ) f1 = abs( f )
@ -143,20 +145,18 @@ subroutine DLARTG( f, g, c, s, r )
else if( f1 > rtmin .and. f1 < rtmax .and. & else if( f1 > rtmin .and. f1 < rtmax .and. &
g1 > rtmin .and. g1 < rtmax ) then g1 > rtmin .and. g1 < rtmax ) then
d = sqrt( f*f + g*g ) d = sqrt( f*f + g*g )
p = one / d c = f1 / d
c = f1*p
s = g*sign( p, f )
r = sign( d, f ) r = sign( d, f )
s = g / r
else else
u = min( safmax, max( safmin, f1, g1 ) ) u = min( safmax, max( safmin, f1, g1 ) )
uu = one / u fs = f / u
fs = f*uu gs = g / u
gs = g*uu
d = sqrt( fs*fs + gs*gs ) d = sqrt( fs*fs + gs*gs )
p = one / d c = abs( fs ) / d
c = abs( fs )*p r = sign( d, f )
s = gs*sign( p, f ) s = gs / r
r = sign( d, f )*u r = r*u
end if end if
return return
end subroutine end subroutine

30
lapack-netlib/SRC/slartg.f90
View File

@ -35,7 +35,7 @@
!> square root of the sum of squares. !> square root of the sum of squares.
!> !>
!> This version is discontinuous in R at F = 0 but it returns the same !> This version is discontinuous in R at F = 0 but it returns the same
!> C and S as SLARTG for complex inputs (F,0) and (G,0). !> C and S as CLARTG for complex inputs (F,0) and (G,0).
!> !>
!> This is a more accurate version of the BLAS1 routine SROTG, !> This is a more accurate version of the BLAS1 routine SROTG,
!> with the following other differences: !> with the following other differences:
@ -45,8 +45,6 @@
!> floating point operations (saves work in SBDSQR when !> floating point operations (saves work in SBDSQR when
!> there are zeros on the diagonal). !> there are zeros on the diagonal).
!> !>
!> If F exceeds G in magnitude, C will be positive.
!>
!> Below, wp=>sp stands for single precision from LA_CONSTANTS module. !> Below, wp=>sp stands for single precision from LA_CONSTANTS module.
!> \endverbatim !> \endverbatim
! !
@ -112,7 +110,7 @@
subroutine SLARTG( f, g, c, s, r ) subroutine SLARTG( f, g, c, s, r )
use LA_CONSTANTS, & use LA_CONSTANTS, &
only: wp=>sp, zero=>szero, half=>shalf, one=>sone, & only: wp=>sp, zero=>szero, half=>shalf, one=>sone, &
rtmin=>srtmin, rtmax=>srtmax, safmin=>ssafmin, safmax=>ssafmax safmin=>ssafmin, safmax=>ssafmax
! !
! -- LAPACK auxiliary routine -- ! -- LAPACK auxiliary routine --
! -- LAPACK is a software package provided by Univ. of Tennessee, -- ! -- LAPACK is a software package provided by Univ. of Tennessee, --
@ -123,11 +121,15 @@ subroutine SLARTG( f, g, c, s, r )
real(wp) :: c, f, g, r, s real(wp) :: c, f, g, r, s
! .. ! ..
! .. Local Scalars .. ! .. Local Scalars ..
real(wp) :: d, f1, fs, g1, gs, p, u, uu real(wp) :: d, f1, fs, g1, gs, u, rtmin, rtmax
! .. ! ..
! .. Intrinsic Functions .. ! .. Intrinsic Functions ..
intrinsic :: abs, sign, sqrt intrinsic :: abs, sign, sqrt
! .. ! ..
! .. Constants ..
rtmin = sqrt( safmin )
rtmax = sqrt( safmax/2 )
! ..
! .. Executable Statements .. ! .. Executable Statements ..
! !
f1 = abs( f ) f1 = abs( f )
@ -143,20 +145,18 @@ subroutine SLARTG( f, g, c, s, r )
else if( f1 > rtmin .and. f1 < rtmax .and. & else if( f1 > rtmin .and. f1 < rtmax .and. &
g1 > rtmin .and. g1 < rtmax ) then g1 > rtmin .and. g1 < rtmax ) then
d = sqrt( f*f + g*g ) d = sqrt( f*f + g*g )
p = one / d c = f1 / d
c = f1*p
s = g*sign( p, f )
r = sign( d, f ) r = sign( d, f )
s = g / r
else else
u = min( safmax, max( safmin, f1, g1 ) ) u = min( safmax, max( safmin, f1, g1 ) )
uu = one / u fs = f / u
fs = f*uu gs = g / u
gs = g*uu
d = sqrt( fs*fs + gs*gs ) d = sqrt( fs*fs + gs*gs )
p = one / d c = abs( fs ) / d
c = abs( fs )*p r = sign( d, f )
s = gs*sign( p, f ) s = gs / r
r = sign( d, f )*u r = r*u
end if end if
return return
end subroutine end subroutine

125
lapack-netlib/SRC/zlartg.f90
View File

@ -38,6 +38,10 @@
!> !>
!> S = sgn(F) * conjg(G) / sqrt(|F|**2 + |G|**2) !> S = sgn(F) * conjg(G) / sqrt(|F|**2 + |G|**2)
!> !>
!> Special conditions:
!> If G=0, then C=1 and S=0.
!> If F=0, then C=0 and S is chosen so that R is real.
!>
!> When F and G are real, the formulas simplify to C = F/R and !> When F and G are real, the formulas simplify to C = F/R and
!> S = G/R, and the returned values of C, S, and R should be !> S = G/R, and the returned values of C, S, and R should be
!> identical to those returned by DLARTG. !> identical to those returned by DLARTG.
@ -46,11 +50,8 @@
!> to avoid overflow or underflow in computing the square root of the !> to avoid overflow or underflow in computing the square root of the
!> sum of squares. !> sum of squares.
!> !>
!> This is a faster version of the BLAS1 routine ZROTG, except for !> This is the same routine ZROTG fom BLAS1, except that
!> the following differences:
!> F and G are unchanged on return. !> F and G are unchanged on return.
!> If G=0, then C=1 and S=0.
!> If F=0, then C=0 and S is chosen so that R is real.
!> !>
!> Below, wp=>dp stands for double precision from LA_CONSTANTS module. !> Below, wp=>dp stands for double precision from LA_CONSTANTS module.
!> \endverbatim !> \endverbatim
@ -91,22 +92,19 @@
! Authors: ! Authors:
! ======== ! ========
! !
!> \author Edward Anderson, Lockheed Martin !> \author Weslley Pereira, University of Colorado Denver, USA
! !
!> \date August 2016 !> \date December 2021
! !
!> \ingroup OTHERauxiliary !> \ingroup OTHERauxiliary
! !
!> \par Contributors:
! ==================
!>
!> Weslley Pereira, University of Colorado Denver, USA
!
!> \par Further Details: !> \par Further Details:
! ===================== ! =====================
!> !>
!> \verbatim !> \verbatim
!> !>
!> Based on the algorithm from
!>
!> Anderson E. (2017) !> Anderson E. (2017)
!> Algorithm 978: Safe Scaling in the Level 1 BLAS !> Algorithm 978: Safe Scaling in the Level 1 BLAS
!> ACM Trans Math Softw 44:1--28 !> ACM Trans Math Softw 44:1--28
@ -117,7 +115,7 @@
subroutine ZLARTG( f, g, c, s, r ) subroutine ZLARTG( f, g, c, s, r )
use LA_CONSTANTS, & use LA_CONSTANTS, &
only: wp=>dp, zero=>dzero, one=>done, two=>dtwo, czero=>zzero, & only: wp=>dp, zero=>dzero, one=>done, two=>dtwo, czero=>zzero, &
rtmin=>drtmin, rtmax=>drtmax, safmin=>dsafmin, safmax=>dsafmax safmin=>dsafmin, safmax=>dsafmax
! !
! -- LAPACK auxiliary routine -- ! -- LAPACK auxiliary routine --
! -- LAPACK is a software package provided by Univ. of Tennessee, -- ! -- LAPACK is a software package provided by Univ. of Tennessee, --
@ -129,7 +127,7 @@ subroutine ZLARTG( f, g, c, s, r )
complex(wp) f, g, r, s complex(wp) f, g, r, s
! .. ! ..
! .. Local Scalars .. ! .. 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, rtmin, rtmax
complex(wp) :: fs, gs, t complex(wp) :: fs, gs, t
! .. ! ..
! .. Intrinsic Functions .. ! .. Intrinsic Functions ..
@ -141,6 +139,9 @@ subroutine ZLARTG( f, g, c, s, r )
! .. Statement Function definitions .. ! .. Statement Function definitions ..
ABSSQ( t ) = real( t )**2 + aimag( t )**2 ABSSQ( t ) = real( t )**2 + aimag( t )**2
! .. ! ..
! .. Constants ..
rtmin = sqrt( safmin )
! ..
! .. Executable Statements .. ! .. Executable Statements ..
! !
if( g == czero ) then if( g == czero ) then
@ -149,11 +150,21 @@ subroutine ZLARTG( f, g, c, s, r )
r = f r = f
else if( f == czero ) then else if( f == czero ) then
c = zero c = zero
if( real(g) == zero ) then
r = abs(aimag(g))
s = conjg( g ) / r
elseif( aimag(g) == zero ) then
r = abs(real(g))
s = conjg( g ) / r
else
g1 = max( abs(real(g)), abs(aimag(g)) ) g1 = max( abs(real(g)), abs(aimag(g)) )
rtmax = sqrt( safmax/2 )
if( g1 > rtmin .and. g1 < rtmax ) then if( g1 > rtmin .and. g1 < rtmax ) then
! !
! Use unscaled algorithm ! Use unscaled algorithm
! !
! The following two lines can be replaced by `d = abs( g )`.
! This algorithm do not use the intrinsic complex abs.
g2 = ABSSQ( g ) g2 = ABSSQ( g )
d = sqrt( g2 ) d = sqrt( g2 )
s = conjg( g ) / d s = conjg( g ) / d
@ -163,16 +174,19 @@ subroutine ZLARTG( f, g, c, s, r )
! Use scaled algorithm ! Use scaled algorithm
! !
u = min( safmax, max( safmin, g1 ) ) u = min( safmax, max( safmin, g1 ) )
uu = one / u gs = g / u
gs = g*uu ! The following two lines can be replaced by `d = abs( gs )`.
! This algorithm do not use the intrinsic complex abs.
g2 = ABSSQ( gs ) g2 = ABSSQ( gs )
d = sqrt( g2 ) d = sqrt( g2 )
s = conjg( gs ) / d s = conjg( gs ) / d
r = d*u r = d*u
end if end if
end if
else else
f1 = max( abs(real(f)), abs(aimag(f)) ) f1 = max( abs(real(f)), abs(aimag(f)) )
g1 = max( abs(real(g)), abs(aimag(g)) ) g1 = max( abs(real(g)), abs(aimag(g)) )
rtmax = sqrt( safmax/4 )
if( f1 > rtmin .and. f1 < rtmax .and. & if( f1 > rtmin .and. f1 < rtmax .and. &
g1 > rtmin .and. g1 < rtmax ) then g1 > rtmin .and. g1 < rtmax ) then
! !
@ -181,32 +195,51 @@ subroutine ZLARTG( f, g, c, s, r )
f2 = ABSSQ( f ) f2 = ABSSQ( f )
g2 = ABSSQ( g ) g2 = ABSSQ( g )
h2 = f2 + g2 h2 = f2 + g2
! safmin <= f2 <= h2 <= safmax
if( f2 >= h2 * safmin ) then
! safmin <= f2/h2 <= 1, and h2/f2 is finite
c = sqrt( f2 / h2 )
r = f / c
rtmax = rtmax * 2
if( f2 > rtmin .and. h2 < rtmax ) then if( f2 > rtmin .and. h2 < rtmax ) then
d = sqrt( f2*h2 ) ! safmin <= sqrt( f2*h2 ) <= safmax
s = conjg( g ) * ( f / sqrt( f2*h2 ) )
else else
d = sqrt( f2 )*sqrt( h2 ) s = conjg( g ) * ( r / h2 )
end if
else
! f2/h2 <= safmin may be subnormal, and h2/f2 may overflow.
! Moreover,
! safmin <= f2*f2 * safmax < f2 * h2 < h2*h2 * safmin <= safmax,
! sqrt(safmin) <= sqrt(f2 * h2) <= sqrt(safmax).
! Also,
! g2 >> f2, which means that h2 = g2.
d = sqrt( f2 * h2 )
c = f2 / d
if( c >= safmin ) then
r = f / c
else
! f2 / sqrt(f2 * h2) < safmin, then
! sqrt(safmin) <= f2 * sqrt(safmax) <= h2 / sqrt(f2 * h2) <= h2 * (safmin / f2) <= h2 <= safmax
r = f * ( h2 / d )
end if
s = conjg( g ) * ( f / d )
end if end if
p = 1 / d
c = f2*p
s = conjg( g )*( f*p )
r = f*( h2*p )
else else
! !
! Use scaled algorithm ! Use scaled algorithm
! !
u = min( safmax, max( safmin, f1, g1 ) ) u = min( safmax, max( safmin, f1, g1 ) )
uu = one / u gs = g / u
gs = g*uu
g2 = ABSSQ( gs ) g2 = ABSSQ( gs )
if( f1*uu < rtmin ) then if( f1 / u < rtmin ) then
! !
! f is not well-scaled when scaled by g1. ! f is not well-scaled when scaled by g1.
! Use a different scaling for f. ! Use a different scaling for f.
! !
v = min( safmax, max( safmin, f1 ) ) v = min( safmax, max( safmin, f1 ) )
vv = one / v w = v / u
w = v * uu fs = f / v
fs = f*vv
f2 = ABSSQ( fs ) f2 = ABSSQ( fs )
h2 = f2*w**2 + g2 h2 = f2*w**2 + g2
else else
@ -214,19 +247,43 @@ subroutine ZLARTG( f, g, c, s, r )
! Otherwise use the same scaling for f and g. ! Otherwise use the same scaling for f and g.
! !
w = one w = one
fs = f*uu fs = f / u
f2 = ABSSQ( fs ) f2 = ABSSQ( fs )
h2 = f2 + g2 h2 = f2 + g2
end if end if
! safmin <= f2 <= h2 <= safmax
if( f2 >= h2 * safmin ) then
! safmin <= f2/h2 <= 1, and h2/f2 is finite
c = sqrt( f2 / h2 )
r = fs / c
rtmax = rtmax * 2
if( f2 > rtmin .and. h2 < rtmax ) then if( f2 > rtmin .and. h2 < rtmax ) then
d = sqrt( f2*h2 ) ! safmin <= sqrt( f2*h2 ) <= safmax
s = conjg( gs ) * ( fs / sqrt( f2*h2 ) )
else else
d = sqrt( f2 )*sqrt( h2 ) s = conjg( gs ) * ( r / h2 )
end if end if
p = 1 / d else
c = ( f2*p )*w ! f2/h2 <= safmin may be subnormal, and h2/f2 may overflow.
s = conjg( gs )*( fs*p ) ! Moreover,
r = ( fs*( h2*p ) )*u ! safmin <= f2*f2 * safmax < f2 * h2 < h2*h2 * safmin <= safmax,
! sqrt(safmin) <= sqrt(f2 * h2) <= sqrt(safmax).
! Also,
! g2 >> f2, which means that h2 = g2.
d = sqrt( f2 * h2 )
c = f2 / d
if( c >= safmin ) then
r = fs / c
else
! f2 / sqrt(f2 * h2) < safmin, then
! sqrt(safmin) <= f2 * sqrt(safmax) <= h2 / sqrt(f2 * h2) <= h2 * (safmin / f2) <= h2 <= safmax
r = fs * ( h2 / d )
end if
s = conjg( gs ) * ( fs / d )
end if
! Rescale c and r
c = c * w
r = r * u
end if end if
end if end if
return return