zlaqr1.f

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*> \brief \b ZLAQR1 sets a scalar multiple of the first column of the product of 2-by-2 or 3-by-3 matrix H and specified shifts.
*
*  =========== DOCUMENTATION ===========
*
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*            http://www.netlib.org/lapack/explore-html/
*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE ZLAQR1( N, H, LDH, S1, S2, V )
*
*       .. Scalar Arguments ..
*       COMPLEX*16         S1, S2
*       INTEGER            LDH, N
*       ..
*       .. Array Arguments ..
*       COMPLEX*16         H( LDH, * ), V( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*>      Given a 2-by-2 or 3-by-3 matrix H, ZLAQR1 sets v to a
*>      scalar multiple of the first column of the product
*>
*>      (*)  K = (H - s1*I)*(H - s2*I)
*>
*>      scaling to avoid overflows and most underflows.
*>
*>      This is useful for starting double implicit shift bulges
*>      in the QR algorithm.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>              Order of the matrix H. N must be either 2 or 3.
*> \endverbatim
*>
*> \param[in] H
*> \verbatim
*>          H is COMPLEX*16 array, dimension (LDH,N)
*>              The 2-by-2 or 3-by-3 matrix H in (*).
*> \endverbatim
*>
*> \param[in] LDH
*> \verbatim
*>          LDH is INTEGER
*>              The leading dimension of H as declared in
*>              the calling procedure.  LDH >= N
*> \endverbatim
*>
*> \param[in] S1
*> \verbatim
*>          S1 is COMPLEX*16
*> \endverbatim
*>
*> \param[in] S2
*> \verbatim
*>          S2 is COMPLEX*16
*>
*>          S1 and S2 are the shifts defining K in (*) above.
*> \endverbatim
*>
*> \param[out] V
*> \verbatim
*>          V is COMPLEX*16 array, dimension (N)
*>              A scalar multiple of the first column of the
*>              matrix K in (*).
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laqr1
*
*> \par Contributors:
*  ==================
*>
*>       Karen Braman and Ralph Byers, Department of Mathematics,
*>       University of Kansas, USA
*>
*  =====================================================================
      SUBROUTINE zlaqr1( N, H, LDH, S1, S2, V )
*
*  -- 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 ..
      COMPLEX*16         S1, S2
      INTEGER            LDH, N
*     ..
*     .. Array Arguments ..
      COMPLEX*16         H( LDH, * ), V( * )
*     ..
*
*  ================================================================
*
*     .. Parameters ..
      COMPLEX*16         ZERO
      parameter( zero = ( 0.0d0, 0.0d0 ) )
      DOUBLE PRECISION   RZERO
      parameter( rzero = 0.0d0 )
*     ..
*     .. Local Scalars ..
      COMPLEX*16         CDUM, H21S, H31S
      DOUBLE PRECISION   S
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, dble, dimag
*     ..
*     .. Statement Functions ..
      DOUBLE PRECISION   CABS1
*     ..
*     .. Statement Function definitions ..
      cabs1( cdum ) = abs( dble( cdum ) ) + abs( dimag( cdum ) )
*     ..
*     .. Executable Statements ..
*
*     Quick return if possible
*
      IF( n.NE.2 .AND. n.NE.3 ) THEN
         RETURN
      END IF
*
      IF( n.EQ.2 ) THEN
         s = cabs1( h( 1, 1 )-s2 ) + cabs1( h( 2, 1 ) )
         IF( s.EQ.rzero ) THEN
            v( 1 ) = zero
            v( 2 ) = zero
         ELSE
            h21s = h( 2, 1 ) / s
            v( 1 ) = h21s*h( 1, 2 ) + ( h( 1, 1 )-s1 )*
     $               ( ( h( 1, 1 )-s2 ) / s )
            v( 2 ) = h21s*( h( 1, 1 )+h( 2, 2 )-s1-s2 )
         END IF
      ELSE
         s = cabs1( h( 1, 1 )-s2 ) + cabs1( h( 2, 1 ) ) +
     $       cabs1( h( 3, 1 ) )
         IF( s.EQ.zero ) THEN
            v( 1 ) = zero
            v( 2 ) = zero
            v( 3 ) = zero
         ELSE
            h21s = h( 2, 1 ) / s
            h31s = h( 3, 1 ) / s
            v( 1 ) = ( h( 1, 1 )-s1 )*( ( h( 1, 1 )-s2 ) / s ) +
     $               h( 1, 2 )*h21s + h( 1, 3 )*h31s
            v( 2 ) = h21s*( h( 1, 1 )+h( 2, 2 )-s1-s2 ) + h( 2, 3 )*h31s
            v( 3 ) = h31s*( h( 1, 1 )+h( 3, 3 )-s1-s2 ) + h21s*h( 3, 2 )
         END IF
      END IF
      END