numeric-linalg

Educational material on the SciPy implementation of numerical linear algebra algorithms

Name	Size	Mode
..
lapack/SRC/zlargv.f	8961B	-rw-r--r--

*> \brief \b ZLARGV 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/
*
*> \htmlonly
*> Download ZLARGV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zlargv.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zlargv.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zlargv.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE ZLARGV( N, X, INCX, Y, INCY, C, INCC )
*
*       .. Scalar Arguments ..
*       INTEGER            INCC, INCX, INCY, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   C( * )
*       COMPLEX*16         X( * ), Y( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> ZLARGV 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 ZLARTG,
*> but differ from the BLAS1 routine ZROTG):
*>    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*16 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*16 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 DOUBLE PRECISION 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 ZLARGV( 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 ..
      DOUBLE PRECISION   C( * )
      COMPLEX*16         X( * ), Y( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   TWO, ONE, ZERO
      PARAMETER          ( TWO = 2.0D+0, ONE = 1.0D+0, ZERO = 0.0D+0 )
      COMPLEX*16         CZERO
      PARAMETER          ( CZERO = ( 0.0D+0, 0.0D+0 ) )
*     ..
*     .. Local Scalars ..
*     LOGICAL            FIRST

      INTEGER            COUNT, I, IC, IX, IY, J
      DOUBLE PRECISION   CS, D, DI, DR, EPS, F2, F2S, G2, G2S, SAFMIN,
     $                   SAFMN2, SAFMX2, SCALE
      COMPLEX*16         F, FF, FS, G, GS, R, SN
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH, DLAPY2
      EXTERNAL           DLAMCH, DLAPY2
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, DBLE, DCMPLX, DCONJG, DIMAG, INT, LOG,
     $                   MAX, SQRT
*     ..
*     .. Statement Functions ..
      DOUBLE PRECISION   ABS1, ABSSQ
*     ..
*     .. Save statement ..
*     SAVE               FIRST, SAFMX2, SAFMIN, SAFMN2
*     ..
*     .. Data statements ..
*     DATA               FIRST / .TRUE. /
*     ..
*     .. Statement Function definitions ..
      ABS1( FF ) = MAX( ABS( DBLE( FF ) ), ABS( DIMAG( FF ) ) )
      ABSSQ( FF ) = DBLE( FF )**2 + DIMAG( FF )**2
*     ..
*     .. Executable Statements ..
*
*     IF( FIRST ) THEN
*        FIRST = .FALSE.
         SAFMIN = DLAMCH( 'S' )
         EPS = DLAMCH( 'E' )
         SAFMN2 = DLAMCH( 'B' )**INT( LOG( SAFMIN / EPS ) /
     $            LOG( DLAMCH( '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 ZLARTG
*
         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 = DLAPY2( DBLE( G ), DIMAG( G ) )
*              Do complex/real division explicitly with two real
*              divisions
               D = DLAPY2( DBLE( GS ), DIMAG( GS ) )
               SN = DCMPLX( DBLE( GS ) / D, -DIMAG( GS ) / D )
               GO TO 50
            END IF
            F2S = DLAPY2( DBLE( FS ), DIMAG( 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 = DLAPY2( DBLE( F ), DIMAG( F ) )
               FF = DCMPLX( DBLE( F ) / D, DIMAG( F ) / D )
            ELSE
               DR = SAFMX2*DBLE( F )
               DI = SAFMX2*DIMAG( F )
               D = DLAPY2( DR, DI )
               FF = DCMPLX( DR / D, DI / D )
            END IF
            SN = FF*DCMPLX( DBLE( GS ) / G2S, -DIMAG( 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 = DCMPLX( F2S*DBLE( FS ), F2S*DIMAG( FS ) )
            CS = ONE / F2S
            D = F2 + G2
*           Do complex/real division explicitly with two real divisions
            SN = DCMPLX( DBLE( R ) / D, DIMAG( R ) / D )
            SN = SN*DCONJG( 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 ZLARGV
*
      END