515 lines
16 KiB
Fortran
515 lines
16 KiB
Fortran
*> \brief \b ZLAQZ3
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*
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* =========== DOCUMENTATION ===========
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*
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* Online html documentation available at
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* http://www.netlib.org/lapack/explore-html/
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*
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*> \htmlonly
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*> Download ZLAQZ3 + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/ZLAQZ3.f">
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*> [TGZ]</a>
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/ZLAQZ3.f">
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*> [ZIP]</a>
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/ZLAQZ3.f">
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*> [TXT]</a>
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*> \endhtmlonly
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*
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* Definition:
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* ===========
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*
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* SUBROUTINE ZLAQZ3( ILSCHUR, ILQ, ILZ, N, ILO, IHI, NSHIFTS,
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* $ NBLOCK_DESIRED, ALPHA, BETA, A, LDA, B, LDB, Q, LDQ, Z, LDZ,
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* $ QC, LDQC, ZC, LDZC, WORK, LWORK, INFO )
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* IMPLICIT NONE
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*
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* Function arguments
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* LOGICAL, INTENT( IN ) :: ILSCHUR, ILQ, ILZ
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* INTEGER, INTENT( IN ) :: N, ILO, IHI, LDA, LDB, LDQ, LDZ, LWORK,
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* $ NSHIFTS, NBLOCK_DESIRED, LDQC, LDZC
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*
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* COMPLEX*16, INTENT( INOUT ) :: A( LDA, * ), B( LDB, * ), Q( LDQ,
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* $ * ), Z( LDZ, * ), QC( LDQC, * ), ZC( LDZC, * ), WORK( * ),
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* $ ALPHA( * ), BETA( * )
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*
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* INTEGER, INTENT( OUT ) :: INFO
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* ..
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*
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*
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*> \par Purpose:
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* =============
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*>
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*> \verbatim
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*>
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*> ZLAQZ3 Executes a single multishift QZ sweep
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*> \endverbatim
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*
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* Arguments:
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* ==========
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*
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*> \param[in] ILSCHUR
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*> \verbatim
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*> ILSCHUR is LOGICAL
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*> Determines whether or not to update the full Schur form
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*> \endverbatim
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*>
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*> \param[in] ILQ
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*> \verbatim
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*> ILQ is LOGICAL
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*> Determines whether or not to update the matrix Q
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*> \endverbatim
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*>
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*> \param[in] ILZ
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*> \verbatim
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*> ILZ is LOGICAL
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*> Determines whether or not to update the matrix Z
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*> \endverbatim
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*>
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*> \param[in] N
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*> \verbatim
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*> N is INTEGER
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*> The order of the matrices A, B, Q, and Z. N >= 0.
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*> \endverbatim
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*>
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*> \param[in] ILO
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*> \verbatim
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*> ILO is INTEGER
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*> \endverbatim
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*>
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*> \param[in] IHI
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*> \verbatim
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*> IHI is INTEGER
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*> \endverbatim
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*>
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*> \param[in] NSHIFTS
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*> \verbatim
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*> NSHIFTS is INTEGER
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*> The desired number of shifts to use
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*> \endverbatim
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*>
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*> \param[in] NBLOCK_DESIRED
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*> \verbatim
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*> NBLOCK_DESIRED is INTEGER
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*> The desired size of the computational windows
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*> \endverbatim
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*>
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*> \param[in] ALPHA
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*> \verbatim
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*> ALPHA is COMPLEX*16 array. SR contains
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*> the alpha parts of the shifts to use.
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*> \endverbatim
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*>
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*> \param[in] BETA
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*> \verbatim
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*> BETA is COMPLEX*16 array. SS contains
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*> the scale of the shifts to use.
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*> \endverbatim
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*>
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*> \param[in,out] A
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*> \verbatim
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*> A is COMPLEX*16 array, dimension (LDA, N)
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*> \endverbatim
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*>
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*> \param[in] LDA
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*> \verbatim
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*> LDA is INTEGER
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*> The leading dimension of the array A. LDA >= max( 1, N ).
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*> \endverbatim
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*>
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*> \param[in,out] B
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*> \verbatim
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*> B is COMPLEX*16 array, dimension (LDB, N)
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*> \endverbatim
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*>
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*> \param[in] LDB
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*> \verbatim
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*> LDB is INTEGER
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*> The leading dimension of the array B. LDB >= max( 1, N ).
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*> \endverbatim
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*>
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*> \param[in,out] Q
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*> \verbatim
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*> Q is COMPLEX*16 array, dimension (LDQ, N)
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*> \endverbatim
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*>
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*> \param[in] LDQ
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*> \verbatim
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*> LDQ is INTEGER
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*> \endverbatim
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*>
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*> \param[in,out] Z
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*> \verbatim
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*> Z is COMPLEX*16 array, dimension (LDZ, N)
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*> \endverbatim
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*>
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*> \param[in] LDZ
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*> \verbatim
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*> LDZ is INTEGER
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*> \endverbatim
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*>
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*> \param[in,out] QC
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*> \verbatim
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*> QC is COMPLEX*16 array, dimension (LDQC, NBLOCK_DESIRED)
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*> \endverbatim
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*>
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*> \param[in] LDQC
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*> \verbatim
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*> LDQC is INTEGER
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*> \endverbatim
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*>
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*> \param[in,out] ZC
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*> \verbatim
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*> ZC is COMPLEX*16 array, dimension (LDZC, NBLOCK_DESIRED)
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*> \endverbatim
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*>
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*> \param[in] LDZC
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*> \verbatim
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*> LDZ is INTEGER
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*> \endverbatim
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*>
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*> \param[out] WORK
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*> \verbatim
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*> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
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*> On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.
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*> \endverbatim
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*>
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*> \param[in] LWORK
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*> \verbatim
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*> LWORK is INTEGER
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*> The dimension of the array WORK. LWORK >= max(1,N).
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*>
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*> If LWORK = -1, then a workspace query is assumed; the routine
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*> only calculates the optimal size of the WORK array, returns
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*> this value as the first entry of the WORK array, and no error
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*> message related to LWORK is issued by XERBLA.
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*> \endverbatim
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*>
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*> \param[out] INFO
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*> \verbatim
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*> INFO is INTEGER
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*> = 0: successful exit
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*> < 0: if INFO = -i, the i-th argument had an illegal value
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*> \endverbatim
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*
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* Authors:
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* ========
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*
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*> \author Thijs Steel, KU Leuven
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*
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*> \date May 2020
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*
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*> \ingroup complex16GEcomputational
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*>
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* =====================================================================
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SUBROUTINE ZLAQZ3( ILSCHUR, ILQ, ILZ, N, ILO, IHI, NSHIFTS,
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$ NBLOCK_DESIRED, ALPHA, BETA, A, LDA, B, LDB,
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$ Q, LDQ, Z, LDZ, QC, LDQC, ZC, LDZC, WORK,
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$ LWORK, INFO )
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IMPLICIT NONE
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* Function arguments
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LOGICAL, INTENT( IN ) :: ILSCHUR, ILQ, ILZ
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INTEGER, INTENT( IN ) :: N, ILO, IHI, LDA, LDB, LDQ, LDZ, LWORK,
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$ NSHIFTS, NBLOCK_DESIRED, LDQC, LDZC
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COMPLEX*16, INTENT( INOUT ) :: A( LDA, * ), B( LDB, * ), Q( LDQ,
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$ * ), Z( LDZ, * ), QC( LDQC, * ), ZC( LDZC, * ), WORK( * ),
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$ ALPHA( * ), BETA( * )
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INTEGER, INTENT( OUT ) :: INFO
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* Parameters
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COMPLEX*16 CZERO, CONE
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PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ), CONE = ( 1.0D+0,
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$ 0.0D+0 ) )
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DOUBLE PRECISION :: ZERO, ONE, HALF
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PARAMETER( ZERO = 0.0D0, ONE = 1.0D0, HALF = 0.5D0 )
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* Local scalars
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INTEGER :: I, J, NS, ISTARTM, ISTOPM, SHEIGHT, SWIDTH, K, NP,
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$ ISTARTB, ISTOPB, ISHIFT, NBLOCK, NPOS
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DOUBLE PRECISION :: SAFMIN, SAFMAX, C, SCALE
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COMPLEX*16 :: TEMP, TEMP2, TEMP3, S
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* External Functions
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EXTERNAL :: XERBLA, DLABAD, ZLASET, ZLARTG, ZROT, ZLAQZ1, ZGEMM,
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$ ZLACPY
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DOUBLE PRECISION, EXTERNAL :: DLAMCH
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INFO = 0
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IF ( NBLOCK_DESIRED .LT. NSHIFTS+1 ) THEN
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INFO = -8
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END IF
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IF ( LWORK .EQ.-1 ) THEN
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* workspace query, quick return
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WORK( 1 ) = N*NBLOCK_DESIRED
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RETURN
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ELSE IF ( LWORK .LT. N*NBLOCK_DESIRED ) THEN
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INFO = -25
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END IF
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'ZLAQZ3', -INFO )
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RETURN
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END IF
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*
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* Executable statements
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*
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* Get machine constants
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SAFMIN = DLAMCH( 'SAFE MINIMUM' )
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SAFMAX = ONE/SAFMIN
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CALL DLABAD( SAFMIN, SAFMAX )
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IF ( ILO .GE. IHI ) THEN
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RETURN
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END IF
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IF ( ILSCHUR ) THEN
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ISTARTM = 1
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ISTOPM = N
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ELSE
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ISTARTM = ILO
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ISTOPM = IHI
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END IF
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NS = NSHIFTS
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NPOS = MAX( NBLOCK_DESIRED-NS, 1 )
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* The following block introduces the shifts and chases
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* them down one by one just enough to make space for
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* the other shifts. The near-the-diagonal block is
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* of size (ns+1) x ns.
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CALL ZLASET( 'FULL', NS+1, NS+1, CZERO, CONE, QC, LDQC )
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CALL ZLASET( 'FULL', NS, NS, CZERO, CONE, ZC, LDZC )
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DO I = 1, NS
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* Introduce the shift
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SCALE = SQRT( ABS( ALPHA( I ) ) ) * SQRT( ABS( BETA( I ) ) )
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IF( SCALE .GE. SAFMIN .AND. SCALE .LE. SAFMAX ) THEN
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ALPHA( I ) = ALPHA( I )/SCALE
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BETA( I ) = BETA( I )/SCALE
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END IF
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TEMP2 = BETA( I )*A( ILO, ILO )-ALPHA( I )*B( ILO, ILO )
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TEMP3 = BETA( I )*A( ILO+1, ILO )
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IF ( ABS( TEMP2 ) .GT. SAFMAX .OR.
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$ ABS( TEMP3 ) .GT. SAFMAX ) THEN
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TEMP2 = CONE
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TEMP3 = CZERO
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END IF
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CALL ZLARTG( TEMP2, TEMP3, C, S, TEMP )
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CALL ZROT( NS, A( ILO, ILO ), LDA, A( ILO+1, ILO ), LDA, C,
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$ S )
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CALL ZROT( NS, B( ILO, ILO ), LDB, B( ILO+1, ILO ), LDB, C,
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$ S )
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CALL ZROT( NS+1, QC( 1, 1 ), 1, QC( 1, 2 ), 1, C,
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$ DCONJG( S ) )
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* Chase the shift down
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DO J = 1, NS-I
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CALL ZLAQZ1( .TRUE., .TRUE., J, 1, NS, IHI-ILO+1, A( ILO,
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$ ILO ), LDA, B( ILO, ILO ), LDB, NS+1, 1, QC,
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$ LDQC, NS, 1, ZC, LDZC )
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END DO
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END DO
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* Update the rest of the pencil
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* Update A(ilo:ilo+ns,ilo+ns:istopm) and B(ilo:ilo+ns,ilo+ns:istopm)
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* from the left with Qc(1:ns+1,1:ns+1)'
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SHEIGHT = NS+1
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SWIDTH = ISTOPM-( ILO+NS )+1
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IF ( SWIDTH > 0 ) THEN
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CALL ZGEMM( 'C', 'N', SHEIGHT, SWIDTH, SHEIGHT, CONE, QC, LDQC,
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$ A( ILO, ILO+NS ), LDA, CZERO, WORK, SHEIGHT )
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CALL ZLACPY( 'ALL', SHEIGHT, SWIDTH, WORK, SHEIGHT, A( ILO,
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$ ILO+NS ), LDA )
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CALL ZGEMM( 'C', 'N', SHEIGHT, SWIDTH, SHEIGHT, CONE, QC, LDQC,
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$ B( ILO, ILO+NS ), LDB, CZERO, WORK, SHEIGHT )
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CALL ZLACPY( 'ALL', SHEIGHT, SWIDTH, WORK, SHEIGHT, B( ILO,
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$ ILO+NS ), LDB )
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END IF
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IF ( ILQ ) THEN
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CALL ZGEMM( 'N', 'N', N, SHEIGHT, SHEIGHT, CONE, Q( 1, ILO ),
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$ LDQ, QC, LDQC, CZERO, WORK, N )
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CALL ZLACPY( 'ALL', N, SHEIGHT, WORK, N, Q( 1, ILO ), LDQ )
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END IF
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* Update A(istartm:ilo-1,ilo:ilo+ns-1) and B(istartm:ilo-1,ilo:ilo+ns-1)
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* from the right with Zc(1:ns,1:ns)
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SHEIGHT = ILO-1-ISTARTM+1
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SWIDTH = NS
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IF ( SHEIGHT > 0 ) THEN
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CALL ZGEMM( 'N', 'N', SHEIGHT, SWIDTH, SWIDTH, CONE,
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$ A( ISTARTM, ILO ), LDA, ZC, LDZC, CZERO, WORK,
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$ SHEIGHT )
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CALL ZLACPY( 'ALL', SHEIGHT, SWIDTH, WORK, SHEIGHT, A( ISTARTM,
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$ ILO ), LDA )
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CALL ZGEMM( 'N', 'N', SHEIGHT, SWIDTH, SWIDTH, CONE,
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$ B( ISTARTM, ILO ), LDB, ZC, LDZC, CZERO, WORK,
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$ SHEIGHT )
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CALL ZLACPY( 'ALL', SHEIGHT, SWIDTH, WORK, SHEIGHT, B( ISTARTM,
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$ ILO ), LDB )
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END IF
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IF ( ILZ ) THEN
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CALL ZGEMM( 'N', 'N', N, SWIDTH, SWIDTH, CONE, Z( 1, ILO ),
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$ LDZ, ZC, LDZC, CZERO, WORK, N )
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CALL ZLACPY( 'ALL', N, SWIDTH, WORK, N, Z( 1, ILO ), LDZ )
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END IF
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* The following block chases the shifts down to the bottom
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* right block. If possible, a shift is moved down npos
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* positions at a time
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K = ILO
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DO WHILE ( K < IHI-NS )
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NP = MIN( IHI-NS-K, NPOS )
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* Size of the near-the-diagonal block
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NBLOCK = NS+NP
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* istartb points to the first row we will be updating
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ISTARTB = K+1
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* istopb points to the last column we will be updating
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ISTOPB = K+NBLOCK-1
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CALL ZLASET( 'FULL', NS+NP, NS+NP, CZERO, CONE, QC, LDQC )
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CALL ZLASET( 'FULL', NS+NP, NS+NP, CZERO, CONE, ZC, LDZC )
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* Near the diagonal shift chase
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DO I = NS-1, 0, -1
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DO J = 0, NP-1
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* Move down the block with index k+i+j, updating
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* the (ns+np x ns+np) block:
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* (k:k+ns+np,k:k+ns+np-1)
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CALL ZLAQZ1( .TRUE., .TRUE., K+I+J, ISTARTB, ISTOPB, IHI,
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$ A, LDA, B, LDB, NBLOCK, K+1, QC, LDQC,
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$ NBLOCK, K, ZC, LDZC )
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END DO
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END DO
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* Update rest of the pencil
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* Update A(k+1:k+ns+np, k+ns+np:istopm) and
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* B(k+1:k+ns+np, k+ns+np:istopm)
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* from the left with Qc(1:ns+np,1:ns+np)'
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SHEIGHT = NS+NP
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SWIDTH = ISTOPM-( K+NS+NP )+1
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IF ( SWIDTH > 0 ) THEN
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CALL ZGEMM( 'C', 'N', SHEIGHT, SWIDTH, SHEIGHT, CONE, QC,
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$ LDQC, A( K+1, K+NS+NP ), LDA, CZERO, WORK,
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$ SHEIGHT )
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CALL ZLACPY( 'ALL', SHEIGHT, SWIDTH, WORK, SHEIGHT, A( K+1,
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$ K+NS+NP ), LDA )
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CALL ZGEMM( 'C', 'N', SHEIGHT, SWIDTH, SHEIGHT, CONE, QC,
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$ LDQC, B( K+1, K+NS+NP ), LDB, CZERO, WORK,
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$ SHEIGHT )
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CALL ZLACPY( 'ALL', SHEIGHT, SWIDTH, WORK, SHEIGHT, B( K+1,
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$ K+NS+NP ), LDB )
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END IF
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IF ( ILQ ) THEN
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CALL ZGEMM( 'N', 'N', N, NBLOCK, NBLOCK, CONE, Q( 1, K+1 ),
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$ LDQ, QC, LDQC, CZERO, WORK, N )
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CALL ZLACPY( 'ALL', N, NBLOCK, WORK, N, Q( 1, K+1 ), LDQ )
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END IF
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* Update A(istartm:k,k:k+ns+npos-1) and B(istartm:k,k:k+ns+npos-1)
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* from the right with Zc(1:ns+np,1:ns+np)
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SHEIGHT = K-ISTARTM+1
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SWIDTH = NBLOCK
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IF ( SHEIGHT > 0 ) THEN
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CALL ZGEMM( 'N', 'N', SHEIGHT, SWIDTH, SWIDTH, CONE,
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$ A( ISTARTM, K ), LDA, ZC, LDZC, CZERO, WORK,
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$ SHEIGHT )
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CALL ZLACPY( 'ALL', SHEIGHT, SWIDTH, WORK, SHEIGHT,
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$ A( ISTARTM, K ), LDA )
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CALL ZGEMM( 'N', 'N', SHEIGHT, SWIDTH, SWIDTH, CONE,
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$ B( ISTARTM, K ), LDB, ZC, LDZC, CZERO, WORK,
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$ SHEIGHT )
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CALL ZLACPY( 'ALL', SHEIGHT, SWIDTH, WORK, SHEIGHT,
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$ B( ISTARTM, K ), LDB )
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END IF
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IF ( ILZ ) THEN
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CALL ZGEMM( 'N', 'N', N, NBLOCK, NBLOCK, CONE, Z( 1, K ),
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$ LDZ, ZC, LDZC, CZERO, WORK, N )
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CALL ZLACPY( 'ALL', N, NBLOCK, WORK, N, Z( 1, K ), LDZ )
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END IF
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K = K+NP
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END DO
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* The following block removes the shifts from the bottom right corner
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* one by one. Updates are initially applied to A(ihi-ns+1:ihi,ihi-ns:ihi).
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CALL ZLASET( 'FULL', NS, NS, CZERO, CONE, QC, LDQC )
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CALL ZLASET( 'FULL', NS+1, NS+1, CZERO, CONE, ZC, LDZC )
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* istartb points to the first row we will be updating
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ISTARTB = IHI-NS+1
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|
* istopb points to the last column we will be updating
|
|
ISTOPB = IHI
|
|
|
|
DO I = 1, NS
|
|
* Chase the shift down to the bottom right corner
|
|
DO ISHIFT = IHI-I, IHI-1
|
|
CALL ZLAQZ1( .TRUE., .TRUE., ISHIFT, ISTARTB, ISTOPB, IHI,
|
|
$ A, LDA, B, LDB, NS, IHI-NS+1, QC, LDQC, NS+1,
|
|
$ IHI-NS, ZC, LDZC )
|
|
END DO
|
|
|
|
END DO
|
|
|
|
* Update rest of the pencil
|
|
|
|
* Update A(ihi-ns+1:ihi, ihi+1:istopm)
|
|
* from the left with Qc(1:ns,1:ns)'
|
|
SHEIGHT = NS
|
|
SWIDTH = ISTOPM-( IHI+1 )+1
|
|
IF ( SWIDTH > 0 ) THEN
|
|
CALL ZGEMM( 'C', 'N', SHEIGHT, SWIDTH, SHEIGHT, CONE, QC, LDQC,
|
|
$ A( IHI-NS+1, IHI+1 ), LDA, CZERO, WORK, SHEIGHT )
|
|
CALL ZLACPY( 'ALL', SHEIGHT, SWIDTH, WORK, SHEIGHT,
|
|
$ A( IHI-NS+1, IHI+1 ), LDA )
|
|
CALL ZGEMM( 'C', 'N', SHEIGHT, SWIDTH, SHEIGHT, CONE, QC, LDQC,
|
|
$ B( IHI-NS+1, IHI+1 ), LDB, CZERO, WORK, SHEIGHT )
|
|
CALL ZLACPY( 'ALL', SHEIGHT, SWIDTH, WORK, SHEIGHT,
|
|
$ B( IHI-NS+1, IHI+1 ), LDB )
|
|
END IF
|
|
IF ( ILQ ) THEN
|
|
CALL ZGEMM( 'N', 'N', N, NS, NS, CONE, Q( 1, IHI-NS+1 ), LDQ,
|
|
$ QC, LDQC, CZERO, WORK, N )
|
|
CALL ZLACPY( 'ALL', N, NS, WORK, N, Q( 1, IHI-NS+1 ), LDQ )
|
|
END IF
|
|
|
|
* Update A(istartm:ihi-ns,ihi-ns:ihi)
|
|
* from the right with Zc(1:ns+1,1:ns+1)
|
|
SHEIGHT = IHI-NS-ISTARTM+1
|
|
SWIDTH = NS+1
|
|
IF ( SHEIGHT > 0 ) THEN
|
|
CALL ZGEMM( 'N', 'N', SHEIGHT, SWIDTH, SWIDTH, CONE,
|
|
$ A( ISTARTM, IHI-NS ), LDA, ZC, LDZC, CZERO, WORK,
|
|
$ SHEIGHT )
|
|
CALL ZLACPY( 'ALL', SHEIGHT, SWIDTH, WORK, SHEIGHT, A( ISTARTM,
|
|
$ IHI-NS ), LDA )
|
|
CALL ZGEMM( 'N', 'N', SHEIGHT, SWIDTH, SWIDTH, CONE,
|
|
$ B( ISTARTM, IHI-NS ), LDB, ZC, LDZC, CZERO, WORK,
|
|
$ SHEIGHT )
|
|
CALL ZLACPY( 'ALL', SHEIGHT, SWIDTH, WORK, SHEIGHT, B( ISTARTM,
|
|
$ IHI-NS ), LDB )
|
|
END IF
|
|
IF ( ILZ ) THEN
|
|
CALL ZGEMM( 'N', 'N', N, NS+1, NS+1, CONE, Z( 1, IHI-NS ), LDZ,
|
|
$ ZC, LDZC, CZERO, WORK, N )
|
|
CALL ZLACPY( 'ALL', N, NS+1, WORK, N, Z( 1, IHI-NS ), LDZ )
|
|
END IF
|
|
|
|
END SUBROUTINE
|