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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 ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download ZLAQR1 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zlaqr1.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zlaqr1.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zlaqr1.f">
*> [TXT]</a>
*> \endhtmlonly
*
* 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 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] 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 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 December 2016
*
*> \ingroup complex16OTHERauxiliary
*
*> \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 (version 3.7.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* December 2016
*
* .. 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 ..
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
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