clartg()

subroutine clartg	(	complex(wp)	f,
		complex(wp)	g,
		real(wp)	c,
		complex(wp)	s,
		complex(wp)	r
	)

CLARTG generates a plane rotation with real cosine and complex sine.

Purpose:

 CLARTG generates a plane rotation so that

    [  C         S  ] . [ F ]  =  [ R ]
    [ -conjg(S)  C  ]   [ G ]     [ 0 ]

 where C is real and C**2 + |S|**2 = 1.

 The mathematical formulas used for C and S are

    sgn(x) = {  x / |x|,   x != 0
             {  1,         x = 0

    R = sgn(F) * sqrt(|F|**2 + |G|**2)

    C = |F| / sqrt(|F|**2 + |G|**2)

    S = sgn(F) * conjg(G) / sqrt(|F|**2 + |G|**2)

 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
 identical to those returned by CLARTG.

 The algorithm used to compute these quantities incorporates scaling
 to avoid overflow or underflow in computing the square root of the
 sum of squares.

 This is a faster version of the BLAS1 routine CROTG, except for
 the following differences:
    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.

Parameters

[in]	F	F is COMPLEX(wp) The first component of vector to be rotated.
[in]	G	G is COMPLEX(wp) The second component of vector to be rotated.
[out]	C	C is REAL(wp) The cosine of the rotation.
[out]	S	S is COMPLEX(wp) The sine of the rotation.
[out]	R	R is COMPLEX(wp) The nonzero component of the rotated vector.

Author

Edward Anderson, Lockheed Martin

Date

August 2016

Contributors:

Weslley Pereira, University of Colorado Denver, USA

Further Details:

  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

Definition at line 117 of file clartg.f90.

   use la_constants, &
   only: wp=>sp, zero=>szero, one=>sone, two=>stwo, czero, &
         rtmin=>srtmin, rtmax=>srtmax, safmin=>ssafmin, safmax=>ssafmax
!
!  -- LAPACK auxiliary routine --
!  -- LAPACK is a software package provided by Univ. of Tennessee,    --
!  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
!     February 2021
!
!  .. Scalar Arguments ..
   real(wp)           c
   complex(wp)        f, g, r, s
!  ..
!  .. Local Scalars ..
   real(wp) :: d, f1, f2, g1, g2, h2, p, u, uu, v, vv, w
   complex(wp) :: fs, gs, 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 ..
!
   if( g == czero ) then
      c = one
      s = czero
      r = f
   else if( f == czero ) then
      c = zero
      g1 = max( abs(real(g)), abs(aimag(g)) )
      if( g1 > rtmin .and. g1 < rtmax ) then
!
!        Use unscaled algorithm
!
         g2 = abssq( g )
         d = sqrt( g2 )
         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 )
         s = conjg( gs ) / d
         r = d*u
      end if
   else
      f1 = max( abs(real(f)), abs(aimag(f)) )
      g1 = max( abs(real(g)), abs(aimag(g)) )
      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
         if( f2 > rtmin .and. h2 < rtmax ) then
            d = sqrt( f2*h2 )
         else
            d = sqrt( f2 )*sqrt( h2 )
         end if
         p = 1 / d
         c = f2*p
         s = conjg( g )*( f*p )
         r = f*( h2*p )
      else
!
!        Use scaled algorithm
!
         u = min( safmax, max( safmin, f1, g1 ) )
         uu = one / u
         gs = g*uu
         g2 = abssq( gs )
         if( f1*uu < 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
            f2 = abssq( fs )
            h2 = f2*w**2 + g2
         else
!
!           Otherwise use the same scaling for f and g.
!
            w = one
            fs = f*uu
            f2 = abssq( fs )
            h2 = f2 + g2
         end if
         if( f2 > rtmin .and. h2 < rtmax ) then
            d = sqrt( f2*h2 )
         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
      end if
   end if
   return

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