*> \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 *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \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, 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. * *> \date June 2017 * *> \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 3.7.1) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * June 2017 * * .. 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