clargv.f

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*> \brief \b CLARGV generates a vector of plane rotations with real cosines and complex sines.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE CLARGV( N, X, INCX, Y, INCY, C, INCC )
*
*       .. Scalar Arguments ..
*       INTEGER            INCC, INCX, INCY, N
*       ..
*       .. Array Arguments ..
*       REAL               C( * )
*       COMPLEX            X( * ), Y( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> CLARGV generates a vector of complex plane rotations with real
*> cosines, determined by elements of the complex vectors x and y.
*> For i = 1,2,...,n
*>
*>    (        c(i)   s(i) ) ( x(i) ) = ( r(i) )
*>    ( -conjg(s(i))  c(i) ) ( y(i) ) = (   0  )
*>
*>    where c(i)**2 + ABS(s(i))**2 = 1
*>
*> The following conventions are used (these are the same as in CLARTG,
*> but differ from the BLAS1 routine CROTG):
*>    If y(i)=0, then c(i)=1 and s(i)=0.
*>    If x(i)=0, then c(i)=0 and s(i) is chosen so that r(i) is real.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of plane rotations to be generated.
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*>          X is COMPLEX array, dimension (1+(N-1)*INCX)
*>          On entry, the vector x.
*>          On exit, x(i) is overwritten by r(i), for i = 1,...,n.
*> \endverbatim
*>
*> \param[in] INCX
*> \verbatim
*>          INCX is INTEGER
*>          The increment between elements of X. INCX > 0.
*> \endverbatim
*>
*> \param[in,out] Y
*> \verbatim
*>          Y is COMPLEX array, dimension (1+(N-1)*INCY)
*>          On entry, the vector y.
*>          On exit, the sines of the plane rotations.
*> \endverbatim
*>
*> \param[in] INCY
*> \verbatim
*>          INCY is INTEGER
*>          The increment between elements of Y. INCY > 0.
*> \endverbatim
*>
*> \param[out] C
*> \verbatim
*>          C is REAL array, dimension (1+(N-1)*INCC)
*>          The cosines of the plane rotations.
*> \endverbatim
*>
*> \param[in] INCC
*> \verbatim
*>          INCC is INTEGER
*>          The increment between elements of C. INCC > 0.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup largv
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  6-6-96 - Modified with a new algorithm by W. Kahan and J. Demmel
*>
*>  This version has a few statements commented out for thread safety
*>  (machine parameters are computed on each entry). 10 feb 03, SJH.
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE clargv( N, X, INCX, Y, INCY, C, INCC )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INCC, INCX, INCY, N
*     ..
*     .. Array Arguments ..
      REAL               C( * )
      COMPLEX            X( * ), Y( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               TWO, ONE, ZERO
      parameter( two = 2.0e+0, one = 1.0e+0, zero = 0.0e+0 )
      COMPLEX            CZERO
      parameter( czero = ( 0.0e+0, 0.0e+0 ) )
*     ..
*     .. Local Scalars ..
*     LOGICAL            FIRST
      INTEGER            COUNT, I, IC, IX, IY, J
      REAL               CS, D, DI, DR, EPS, F2, F2S, G2, G2S, SAFMIN,
     $                   SAFMN2, SAFMX2, SCALE
      COMPLEX            F, FF, FS, G, GS, R, SN
*     ..
*     .. External Functions ..
      REAL               SLAMCH, SLAPY2
      EXTERNAL           slamch, slapy2
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, aimag, cmplx, conjg, int, log, max, real,
     $                   sqrt
*     ..
*     .. Statement Functions ..
      REAL               ABS1, ABSSQ
*     ..
*     .. Save statement ..
*     SAVE               FIRST, SAFMX2, SAFMIN, SAFMN2
*     ..
*     .. Data statements ..
*     DATA               FIRST / .TRUE. /
*     ..
*     .. Statement Function definitions ..
      abs1( ff ) = max( abs( real( ff ) ), abs( aimag( ff ) ) )
      abssq( ff ) = real( ff )**2 + aimag( ff )**2
*     ..
*     .. Executable Statements ..
*
*     IF( FIRST ) THEN
*        FIRST = .FALSE.
         safmin = slamch( 'S' )
         eps = slamch( 'E' )
         safmn2 = slamch( 'B' )**int( log( safmin / eps ) /
     $            log( slamch( 'B' ) ) / two )
         safmx2 = one / safmn2
*     END IF
      ix = 1
      iy = 1
      ic = 1
      DO 60 i = 1, n
         f = x( ix )
         g = y( iy )
*
*        Use identical algorithm as in CLARTG
*
         scale = max( abs1( f ), abs1( g ) )
         fs = f
         gs = g
         count = 0
         IF( scale.GE.safmx2 ) THEN
   10       CONTINUE
            count = count + 1
            fs = fs*safmn2
            gs = gs*safmn2
            scale = scale*safmn2
            IF( scale.GE.safmx2 .AND. count .LT. 20 )
     $         GO TO 10
         ELSE IF( scale.LE.safmn2 ) THEN
            IF( g.EQ.czero ) THEN
               cs = one
               sn = czero
               r = f
               GO TO 50
            END IF
   20       CONTINUE
            count = count - 1
            fs = fs*safmx2
            gs = gs*safmx2
            scale = scale*safmx2
            IF( scale.LE.safmn2 )
     $         GO TO 20
         END IF
         f2 = abssq( fs )
         g2 = abssq( gs )
         IF( f2.LE.max( g2, one )*safmin ) THEN
*
*           This is a rare case: F is very small.
*
            IF( f.EQ.czero ) THEN
               cs = zero
               r = slapy2( real( g ), aimag( g ) )
*              Do complex/real division explicitly with two real
*              divisions
               d = slapy2( real( gs ), aimag( gs ) )
               sn = cmplx( real( gs ) / d, -aimag( gs ) / d )
               GO TO 50
            END IF
            f2s = slapy2( real( fs ), aimag( fs ) )
*           G2 and G2S are accurate
*           G2 is at least SAFMIN, and G2S is at least SAFMN2
            g2s = sqrt( g2 )
*           Error in CS from underflow in F2S is at most
*           UNFL / SAFMN2 .lt. sqrt(UNFL*EPS) .lt. EPS
*           If MAX(G2,ONE)=G2, then F2 .lt. G2*SAFMIN,
*           and so CS .lt. sqrt(SAFMIN)
*           If MAX(G2,ONE)=ONE, then F2 .lt. SAFMIN
*           and so CS .lt. sqrt(SAFMIN)/SAFMN2 = sqrt(EPS)
*           Therefore, CS = F2S/G2S / sqrt( 1 + (F2S/G2S)**2 ) = F2S/G2S
            cs = f2s / g2s
*           Make sure abs(FF) = 1
*           Do complex/real division explicitly with 2 real divisions
            IF( abs1( f ).GT.one ) THEN
               d = slapy2( real( f ), aimag( f ) )
               ff = cmplx( real( f ) / d, aimag( f ) / d )
            ELSE
               dr = safmx2*real( f )
               di = safmx2*aimag( f )
               d = slapy2( dr, di )
               ff = cmplx( dr / d, di / d )
            END IF
            sn = ff*cmplx( real( gs ) / g2s, -aimag( gs ) / g2s )
            r = cs*f + sn*g
         ELSE
*
*           This is the most common case.
*           Neither F2 nor F2/G2 are less than SAFMIN
*           F2S cannot overflow, and it is accurate
*
            f2s = sqrt( one+g2 / f2 )
*           Do the F2S(real)*FS(complex) multiply with two real
*           multiplies
            r = cmplx( f2s*real( fs ), f2s*aimag( fs ) )
            cs = one / f2s
            d = f2 + g2
*           Do complex/real division explicitly with two real divisions
            sn = cmplx( real( r ) / d, aimag( r ) / d )
            sn = sn*conjg( gs )
            IF( count.NE.0 ) THEN
               IF( count.GT.0 ) THEN
                  DO 30 j = 1, count
                     r = r*safmx2
   30             CONTINUE
               ELSE
                  DO 40 j = 1, -count
                     r = r*safmn2
   40             CONTINUE
               END IF
            END IF
         END IF
   50    CONTINUE
         c( ic ) = cs
         y( iy ) = sn
         x( ix ) = r
         ic = ic + incc
         iy = iy + incy
         ix = ix + incx
   60 CONTINUE
      RETURN
*
*     End of CLARGV
*
      SUBROUTINE clargv( N, X, INCX, Y, INCY, C, INCC ) …
      END