!> \brief \b ZROTG generates a Givens rotation with real cosine and complex sine.
!
! =========== DOCUMENTATION ===========
!
! Online html documentation available at
! http://www.netlib.org/lapack/explore-html/
!
! Definition:
! ===========
!
! ZROTG constructs a plane rotation
! [ c s ] [ a ] = [ r ]
! [ -conjg(s) c ] [ b ] [ 0 ]
! where c is real, s is complex, and c**2 + conjg(s)*s = 1.
!
!> \par Purpose:
! =============
!>
!> \verbatim
!>
!> The computation uses the formulas
!> |x| = sqrt( Re(x)**2 + Im(x)**2 )
!> sgn(x) = x / |x| if x /= 0
!> = 1 if x = 0
!> c = |a| / sqrt(|a|**2 + |b|**2)
!> s = sgn(a) * conjg(b) / sqrt(|a|**2 + |b|**2)
!> r = sgn(a)*sqrt(|a|**2 + |b|**2)
!> When a and b are real and r /= 0, the formulas simplify to
!> c = a / r
!> s = b / r
!> the same as in DROTG when |a| > |b|. When |b| >= |a|, the
!> sign of c and s will be different from those computed by DROTG
!> if the signs of a and b are not the same.
!>
!> \endverbatim
!
! Arguments:
! ==========
!
!> \param[in,out] A
!> \verbatim
!> A is DOUBLE COMPLEX
!> On entry, the scalar a.
!> On exit, the scalar r.
!> \endverbatim
!>
!> \param[in] B
!> \verbatim
!> B is DOUBLE COMPLEX
!> The scalar b.
!> \endverbatim
!>
!> \param[out] C
!> \verbatim
!> C is DOUBLE PRECISION
!> The scalar c.
!> \endverbatim
!>
!> \param[out] S
!> \verbatim
!> S is DOUBLE COMPLEX
!> The scalar s.
!> \endverbatim
!
! Authors:
! ========
!
!> \author Weslley Pereira, University of Colorado Denver, USA
!
!> \date December 2021
!
!> \ingroup single_blas_level1
!
!> \par Further Details:
! =====================
!>
!> \verbatim
!>
!> Based on the algorithm from
!>
!> Anderson E. (2017)
!> Algorithm 978: Safe Scaling in the Level 1 BLAS
!> ACM Trans Math Softw 44:1--28
!> https://doi.org/10.1145/3061665
!>
!> \endverbatim
!
! =====================================================================
subroutine ZROTG( a, b, c, s )
integer, parameter :: wp = kind(1.d0)
!
! -- Reference BLAS level1 routine --
! -- Reference BLAS is a software package provided by Univ. of Tennessee, --
! -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
!
! .. Constants ..
real(wp), parameter :: zero = 0.0_wp
real(wp), parameter :: one = 1.0_wp
complex(wp), parameter :: czero = 0.0_wp
! ..
! .. Scaling constants ..
real(wp), parameter :: safmin = real(radix(0._wp),wp)**max( &
minexponent(0._wp)-1, &
1-maxexponent(0._wp) &
)
real(wp), parameter :: safmax = real(radix(0._wp),wp)**max( &
1-minexponent(0._wp), &
maxexponent(0._wp)-1 &
)
real(wp), parameter :: rtmin = sqrt( safmin )
! ..
! .. Scalar Arguments ..
real(wp) :: c
complex(wp) :: a, b, s
! ..
! .. Local Scalars ..
real(wp) :: d, f1, f2, g1, g2, h2, u, v, w, rtmax
complex(wp) :: f, fs, g, gs, r, t
! ..
! .. Intrinsic Functions ..
intrinsic :: abs, aimag, conjg, max, min, real, sqrt
! ..
! .. Statement Functions ..
real(wp) :: ABSSQ
! ..
! .. Statement Function definitions ..
ABSSQ( t ) = real( t )**2 + aimag( t )**2
! ..
! .. Executable Statements ..
!
f = a
g = b
if( g == czero ) then
c = one
s = czero
r = f
else if( f == czero ) then
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)) )
rtmax = sqrt( safmax/2 )
if( g1 > rtmin .and. g1 < rtmax ) then
!
! 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 )
d = sqrt( g2 )
s = conjg( g ) / d
r = d
else
!
! Use scaled algorithm
!
u = min( safmax, max( safmin, g1 ) )
gs = g / u
! The following two lines can be replaced by `d = abs( gs )`.
! This algorithm do not use the intrinsic complex abs.
g2 = ABSSQ( gs )
d = sqrt( g2 )
s = conjg( gs ) / d
r = d*u
end if
end if
else
f1 = max( abs(real(f)), abs(aimag(f)) )
g1 = max( abs(real(g)), abs(aimag(g)) )
rtmax = sqrt( safmax/4 )
if( f1 > rtmin .and. f1 < rtmax .and. &
g1 > rtmin .and. g1 < rtmax ) then
!
! Use unscaled algorithm
!
f2 = ABSSQ( f )
g2 = ABSSQ( g )
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
! safmin <= sqrt( f2*h2 ) <= safmax
s = conjg( g ) * ( f / sqrt( f2*h2 ) )
else
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
else
!
! Use scaled algorithm
!
u = min( safmax, max( safmin, f1, g1 ) )
gs = g / u
g2 = ABSSQ( gs )
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 ) )
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 / u
f2 = ABSSQ( fs )
h2 = f2 + g2
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
! safmin <= sqrt( f2*h2 ) <= safmax
s = conjg( gs ) * ( fs / sqrt( f2*h2 ) )
else
s = conjg( gs ) * ( 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 = 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
a = r
return
end subroutine