slaqr1.f

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*> \brief \b SLAQR1 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 ===========
*
* Online html documentation available at 
*            http://www.netlib.org/lapack/explore-html/ 
*
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*> \endhtmlonly 
*
*  Definition:
*  ===========
*
*       SUBROUTINE SLAQR1( N, H, LDH, SR1, SI1, SR2, SI2, V )
* 
*       .. Scalar Arguments ..
*       REAL               SI1, SI2, SR1, SR2
*       INTEGER            LDH, N
*       ..
*       .. Array Arguments ..
*       REAL               H( LDH, * ), V( * )
*       ..
*  
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*>      Given a 2-by-2 or 3-by-3 matrix H, SLAQR1 sets v to a
*>      scalar multiple of the first column of the product
*>
*>      (*)  K = (H - (sr1 + i*si1)*I)*(H - (sr2 + i*si2)*I)
*>
*>      scaling to avoid overflows and most underflows. It
*>      is assumed that either
*>
*>              1) sr1 = sr2 and si1 = -si2
*>          or
*>              2) si1 = si2 = 0.
*>
*>      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 REAL array of 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.GE.N
*> \endverbatim
*>
*> \param[in] SR1
*> \verbatim
*>          SR1 is REAL
*> \endverbatim
*>
*> \param[in] SI1
*> \verbatim
*>          SI1 is REAL
*> \endverbatim
*>
*> \param[in] SR2
*> \verbatim
*>          SR2 is REAL
*> \endverbatim
*>
*> \param[in] SI2
*> \verbatim
*>          SI2 is REAL
*>              The shifts in (*).
*> \endverbatim
*>
*> \param[out] V
*> \verbatim
*>          V is REAL array of 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. 
*
*> \date September 2012
*
*> \ingroup realOTHERauxiliary
*
*> \par Contributors:
*  ==================
*>
*>       Karen Braman and Ralph Byers, Department of Mathematics,
*>       University of Kansas, USA
*>
*  =====================================================================
      SUBROUTINE slaqr1( N, H, LDH, SR1, SI1, SR2, SI2, V )
*
*  -- LAPACK auxiliary routine (version 3.4.2) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     September 2012
*
*     .. Scalar Arguments ..
      REAL               SI1, SI2, SR1, SR2
      INTEGER            LDH, N
*     ..
*     .. Array Arguments ..
      REAL               H( ldh, * ), V( * )
*     ..
*
*  ================================================================
*
*     .. Parameters ..
      REAL               ZERO
      parameter                ( zero = 0.0e0 )
*     ..
*     .. Local Scalars ..
      REAL               H21S, H31S, S
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs
*     ..
*     .. Executable Statements ..
      IF( n.EQ.2 ) THEN
         s = abs( h( 1, 1 )-sr2 ) + abs( si2 ) + abs( h( 2, 1 ) )
         IF( s.EQ.zero ) THEN
            v( 1 ) = zero
            v( 2 ) = zero
         ELSE
            h21s = h( 2, 1 ) / s
            v( 1 ) = h21s*h( 1, 2 ) + ( h( 1, 1 )-sr1 )*
     $               ( ( h( 1, 1 )-sr2 ) / s ) - si1*( si2 / s )
            v( 2 ) = h21s*( h( 1, 1 )+h( 2, 2 )-sr1-sr2 )
         END IF
      ELSE
         s = abs( h( 1, 1 )-sr2 ) + abs( si2 ) + abs( h( 2, 1 ) ) +
     $       abs( 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 )-sr1 )*( ( h( 1, 1 )-sr2 ) / s ) -
     $               si1*( si2 / s ) + h( 1, 2 )*h21s + h( 1, 3 )*h31s
            v( 2 ) = h21s*( h( 1, 1 )+h( 2, 2 )-sr1-sr2 ) +
     $               h( 2, 3 )*h31s
            v( 3 ) = h31s*( h( 1, 1 )+h( 3, 3 )-sr1-sr2 ) +
     $               h21s*h( 3, 2 )
         END IF
      END IF
      END