564 lines
18 KiB
Fortran
564 lines
18 KiB
Fortran
*> \brief \b SLAQZ3
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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 SLAQZ3 + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/slaqz3.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/slaqz3.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/slaqz3.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 SLAQZ3( ILSCHUR, ILQ, ILZ, N, ILO, IHI, NW, A, LDA, B,
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* $ LDB, Q, LDQ, Z, LDZ, NS, ND, ALPHAR, ALPHAI, BETA, QC, LDQC,
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* $ ZC, LDZC, WORK, LWORK, REC, INFO )
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* IMPLICIT NONE
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*
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* Arguments
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* LOGICAL, INTENT( IN ) :: ILSCHUR, ILQ, ILZ
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* INTEGER, INTENT( IN ) :: N, ILO, IHI, NW, LDA, LDB, LDQ, LDZ,
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* $ LDQC, LDZC, LWORK, REC
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*
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* REAL, INTENT( INOUT ) :: A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
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* $ Z( LDZ, * ), ALPHAR( * ), ALPHAI( * ), BETA( * )
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* INTEGER, INTENT( OUT ) :: NS, ND, INFO
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* REAL :: QC( LDQC, * ), ZC( LDZC, * ), WORK( * )
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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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*> SLAQZ3 performs AED
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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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*> \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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*> ILO and IHI mark the rows and columns of (A,B) which
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*> are to be normalized
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*> \endverbatim
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*>
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*> \param[in] NW
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*> \verbatim
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*> NW is INTEGER
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*> The desired size of the deflation window.
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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 REAL 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 REAL 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 REAL 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 REAL 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[out] NS
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*> \verbatim
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*> NS is INTEGER
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*> The number of unconverged eigenvalues available to
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*> use as shifts.
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*> \endverbatim
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*>
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*> \param[out] ND
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*> \verbatim
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*> ND is INTEGER
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*> The number of converged eigenvalues found.
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*> \endverbatim
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*>
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*> \param[out] ALPHAR
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*> \verbatim
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*> ALPHAR is REAL array, dimension (N)
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*> The real parts of each scalar alpha defining an eigenvalue
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*> of GNEP.
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*> \endverbatim
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*>
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*> \param[out] ALPHAI
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*> \verbatim
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*> ALPHAI is REAL array, dimension (N)
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*> The imaginary parts of each scalar alpha defining an
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*> eigenvalue of GNEP.
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*> If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
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*> positive, then the j-th and (j+1)-st eigenvalues are a
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*> complex conjugate pair, with ALPHAI(j+1) = -ALPHAI(j).
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*> \endverbatim
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*>
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*> \param[out] BETA
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*> \verbatim
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*> BETA is REAL array, dimension (N)
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*> The scalars beta that define the eigenvalues of GNEP.
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*> Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
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*> beta = BETA(j) represent the j-th eigenvalue of the matrix
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*> pair (A,B), in one of the forms lambda = alpha/beta or
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*> mu = beta/alpha. Since either lambda or mu may overflow,
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*> they should not, in general, be computed.
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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 REAL array, dimension (LDQC, NW)
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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 REAL array, dimension (LDZC, NW)
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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 REAL 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[in] REC
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*> \verbatim
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*> REC is INTEGER
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*> REC indicates the current recursion level. Should be set
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*> to 0 on first call.
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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 doubleGEcomputational
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*>
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* =====================================================================
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RECURSIVE SUBROUTINE SLAQZ3( ILSCHUR, ILQ, ILZ, N, ILO, IHI, NW,
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$ A, LDA, B, LDB, Q, LDQ, Z, LDZ, NS,
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$ ND, ALPHAR, ALPHAI, BETA, QC, LDQC,
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$ ZC, LDZC, WORK, LWORK, REC, INFO )
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IMPLICIT NONE
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* Arguments
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LOGICAL, INTENT( IN ) :: ILSCHUR, ILQ, ILZ
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INTEGER, INTENT( IN ) :: N, ILO, IHI, NW, LDA, LDB, LDQ, LDZ,
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$ LDQC, LDZC, LWORK, REC
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REAL, INTENT( INOUT ) :: A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
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$ Z( LDZ, * ), ALPHAR( * ), ALPHAI( * ), BETA( * )
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INTEGER, INTENT( OUT ) :: NS, ND, INFO
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REAL :: QC( LDQC, * ), ZC( LDZC, * ), WORK( * )
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* Parameters
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REAL :: ZERO, ONE, HALF
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PARAMETER( ZERO = 0.0, ONE = 1.0, HALF = 0.5 )
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* Local Scalars
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LOGICAL :: BULGE
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INTEGER :: JW, KWTOP, KWBOT, ISTOPM, ISTARTM, K, K2, STGEXC_INFO,
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$ IFST, ILST, LWORKREQ, QZ_SMALL_INFO
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REAL :: S, SMLNUM, ULP, SAFMIN, SAFMAX, C1, S1, TEMP
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* External Functions
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EXTERNAL :: XERBLA, STGEXC, SLABAD, SLAQZ0, SLACPY, SLASET,
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$ SLAQZ2, SROT, SLARTG, SLAG2, SGEMM
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REAL, EXTERNAL :: SLAMCH
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INFO = 0
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* Set up deflation window
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JW = MIN( NW, IHI-ILO+1 )
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KWTOP = IHI-JW+1
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IF ( KWTOP .EQ. ILO ) THEN
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S = ZERO
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ELSE
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S = A( KWTOP, KWTOP-1 )
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END IF
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* Determine required workspace
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IFST = 1
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ILST = JW
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CALL STGEXC( .TRUE., .TRUE., JW, A, LDA, B, LDB, QC, LDQC, ZC,
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$ LDZC, IFST, ILST, WORK, -1, STGEXC_INFO )
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LWORKREQ = INT( WORK( 1 ) )
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CALL SLAQZ0( 'S', 'V', 'V', JW, 1, JW, A( KWTOP, KWTOP ), LDA,
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$ B( KWTOP, KWTOP ), LDB, ALPHAR, ALPHAI, BETA, QC,
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$ LDQC, ZC, LDZC, WORK, -1, REC+1, QZ_SMALL_INFO )
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LWORKREQ = MAX( LWORKREQ, INT( WORK( 1 ) )+2*JW**2 )
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LWORKREQ = MAX( LWORKREQ, N*NW, 2*NW**2+N )
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IF ( LWORK .EQ.-1 ) THEN
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* workspace query, quick return
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WORK( 1 ) = LWORKREQ
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RETURN
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ELSE IF ( LWORK .LT. LWORKREQ ) THEN
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INFO = -26
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END IF
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'SLAQZ3', -INFO )
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RETURN
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END IF
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* Get machine constants
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SAFMIN = SLAMCH( 'SAFE MINIMUM' )
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SAFMAX = ONE/SAFMIN
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CALL SLABAD( SAFMIN, SAFMAX )
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ULP = SLAMCH( 'PRECISION' )
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SMLNUM = SAFMIN*( REAL( N )/ULP )
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IF ( IHI .EQ. KWTOP ) THEN
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* 1 by 1 deflation window, just try a regular deflation
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ALPHAR( KWTOP ) = A( KWTOP, KWTOP )
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ALPHAI( KWTOP ) = ZERO
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BETA( KWTOP ) = B( KWTOP, KWTOP )
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NS = 1
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ND = 0
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IF ( ABS( S ) .LE. MAX( SMLNUM, ULP*ABS( A( KWTOP,
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$ KWTOP ) ) ) ) THEN
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NS = 0
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ND = 1
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IF ( KWTOP .GT. ILO ) THEN
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A( KWTOP, KWTOP-1 ) = ZERO
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END IF
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END IF
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END IF
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* Store window in case of convergence failure
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CALL SLACPY( 'ALL', JW, JW, A( KWTOP, KWTOP ), LDA, WORK, JW )
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CALL SLACPY( 'ALL', JW, JW, B( KWTOP, KWTOP ), LDB, WORK( JW**2+
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$ 1 ), JW )
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* Transform window to real schur form
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CALL SLASET( 'FULL', JW, JW, ZERO, ONE, QC, LDQC )
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CALL SLASET( 'FULL', JW, JW, ZERO, ONE, ZC, LDZC )
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CALL SLAQZ0( 'S', 'V', 'V', JW, 1, JW, A( KWTOP, KWTOP ), LDA,
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$ B( KWTOP, KWTOP ), LDB, ALPHAR, ALPHAI, BETA, QC,
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$ LDQC, ZC, LDZC, WORK( 2*JW**2+1 ), LWORK-2*JW**2,
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$ REC+1, QZ_SMALL_INFO )
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IF( QZ_SMALL_INFO .NE. 0 ) THEN
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* Convergence failure, restore the window and exit
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ND = 0
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NS = JW-QZ_SMALL_INFO
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CALL SLACPY( 'ALL', JW, JW, WORK, JW, A( KWTOP, KWTOP ), LDA )
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CALL SLACPY( 'ALL', JW, JW, WORK( JW**2+1 ), JW, B( KWTOP,
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$ KWTOP ), LDB )
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RETURN
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END IF
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* Deflation detection loop
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IF ( KWTOP .EQ. ILO .OR. S .EQ. ZERO ) THEN
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KWBOT = KWTOP-1
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ELSE
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KWBOT = IHI
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K = 1
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K2 = 1
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DO WHILE ( K .LE. JW )
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BULGE = .FALSE.
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IF ( KWBOT-KWTOP+1 .GE. 2 ) THEN
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BULGE = A( KWBOT, KWBOT-1 ) .NE. ZERO
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END IF
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IF ( BULGE ) THEN
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* Try to deflate complex conjugate eigenvalue pair
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TEMP = ABS( A( KWBOT, KWBOT ) )+SQRT( ABS( A( KWBOT,
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$ KWBOT-1 ) ) )*SQRT( ABS( A( KWBOT-1, KWBOT ) ) )
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IF( TEMP .EQ. ZERO )THEN
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TEMP = ABS( S )
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END IF
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IF ( MAX( ABS( S*QC( 1, KWBOT-KWTOP ) ), ABS( S*QC( 1,
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$ KWBOT-KWTOP+1 ) ) ) .LE. MAX( SMLNUM,
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$ ULP*TEMP ) ) THEN
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* Deflatable
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KWBOT = KWBOT-2
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ELSE
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* Not deflatable, move out of the way
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IFST = KWBOT-KWTOP+1
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ILST = K2
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CALL STGEXC( .TRUE., .TRUE., JW, A( KWTOP, KWTOP ),
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$ LDA, B( KWTOP, KWTOP ), LDB, QC, LDQC,
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$ ZC, LDZC, IFST, ILST, WORK, LWORK,
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$ STGEXC_INFO )
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K2 = K2+2
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END IF
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K = K+2
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ELSE
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* Try to deflate real eigenvalue
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TEMP = ABS( A( KWBOT, KWBOT ) )
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IF( TEMP .EQ. ZERO ) THEN
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TEMP = ABS( S )
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END IF
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IF ( ( ABS( S*QC( 1, KWBOT-KWTOP+1 ) ) ) .LE. MAX( ULP*
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$ TEMP, SMLNUM ) ) THEN
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* Deflatable
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KWBOT = KWBOT-1
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ELSE
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* Not deflatable, move out of the way
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IFST = KWBOT-KWTOP+1
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ILST = K2
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CALL STGEXC( .TRUE., .TRUE., JW, A( KWTOP, KWTOP ),
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$ LDA, B( KWTOP, KWTOP ), LDB, QC, LDQC,
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$ ZC, LDZC, IFST, ILST, WORK, LWORK,
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$ STGEXC_INFO )
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K2 = K2+1
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END IF
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K = K+1
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END IF
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END DO
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END IF
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* Store eigenvalues
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ND = IHI-KWBOT
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NS = JW-ND
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K = KWTOP
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DO WHILE ( K .LE. IHI )
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BULGE = .FALSE.
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IF ( K .LT. IHI ) THEN
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IF ( A( K+1, K ) .NE. ZERO ) THEN
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BULGE = .TRUE.
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END IF
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END IF
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IF ( BULGE ) THEN
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* 2x2 eigenvalue block
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CALL SLAG2( A( K, K ), LDA, B( K, K ), LDB, SAFMIN,
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$ BETA( K ), BETA( K+1 ), ALPHAR( K ),
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$ ALPHAR( K+1 ), ALPHAI( K ) )
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ALPHAI( K+1 ) = -ALPHAI( K )
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K = K+2
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ELSE
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* 1x1 eigenvalue block
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ALPHAR( K ) = A( K, K )
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ALPHAI( K ) = ZERO
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BETA( K ) = B( K, K )
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K = K+1
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END IF
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END DO
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IF ( KWTOP .NE. ILO .AND. S .NE. ZERO ) THEN
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* Reflect spike back, this will create optimally packed bulges
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A( KWTOP:KWBOT, KWTOP-1 ) = A( KWTOP, KWTOP-1 )*QC( 1,
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$ 1:JW-ND )
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DO K = KWBOT-1, KWTOP, -1
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CALL SLARTG( A( K, KWTOP-1 ), A( K+1, KWTOP-1 ), C1, S1,
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$ TEMP )
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A( K, KWTOP-1 ) = TEMP
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A( K+1, KWTOP-1 ) = ZERO
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K2 = MAX( KWTOP, K-1 )
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CALL SROT( IHI-K2+1, A( K, K2 ), LDA, A( K+1, K2 ), LDA, C1,
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$ S1 )
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CALL SROT( IHI-( K-1 )+1, B( K, K-1 ), LDB, B( K+1, K-1 ),
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$ LDB, C1, S1 )
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CALL SROT( JW, QC( 1, K-KWTOP+1 ), 1, QC( 1, K+1-KWTOP+1 ),
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$ 1, C1, S1 )
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END DO
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* Chase bulges down
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ISTARTM = KWTOP
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ISTOPM = IHI
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K = KWBOT-1
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DO WHILE ( K .GE. KWTOP )
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IF ( ( K .GE. KWTOP+1 ) .AND. A( K+1, K-1 ) .NE. ZERO ) THEN
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* Move double pole block down and remove it
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DO K2 = K-1, KWBOT-2
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CALL SLAQZ2( .TRUE., .TRUE., K2, KWTOP, KWTOP+JW-1,
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$ KWBOT, A, LDA, B, LDB, JW, KWTOP, QC,
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$ LDQC, JW, KWTOP, ZC, LDZC )
|
|
END DO
|
|
|
|
K = K-2
|
|
ELSE
|
|
|
|
* k points to single shift
|
|
DO K2 = K, KWBOT-2
|
|
|
|
* Move shift down
|
|
CALL SLARTG( B( K2+1, K2+1 ), B( K2+1, K2 ), C1, S1,
|
|
$ TEMP )
|
|
B( K2+1, K2+1 ) = TEMP
|
|
B( K2+1, K2 ) = ZERO
|
|
CALL SROT( K2+2-ISTARTM+1, A( ISTARTM, K2+1 ), 1,
|
|
$ A( ISTARTM, K2 ), 1, C1, S1 )
|
|
CALL SROT( K2-ISTARTM+1, B( ISTARTM, K2+1 ), 1,
|
|
$ B( ISTARTM, K2 ), 1, C1, S1 )
|
|
CALL SROT( JW, ZC( 1, K2+1-KWTOP+1 ), 1, ZC( 1,
|
|
$ K2-KWTOP+1 ), 1, C1, S1 )
|
|
|
|
CALL SLARTG( A( K2+1, K2 ), A( K2+2, K2 ), C1, S1,
|
|
$ TEMP )
|
|
A( K2+1, K2 ) = TEMP
|
|
A( K2+2, K2 ) = ZERO
|
|
CALL SROT( ISTOPM-K2, A( K2+1, K2+1 ), LDA, A( K2+2,
|
|
$ K2+1 ), LDA, C1, S1 )
|
|
CALL SROT( ISTOPM-K2, B( K2+1, K2+1 ), LDB, B( K2+2,
|
|
$ K2+1 ), LDB, C1, S1 )
|
|
CALL SROT( JW, QC( 1, K2+1-KWTOP+1 ), 1, QC( 1,
|
|
$ K2+2-KWTOP+1 ), 1, C1, S1 )
|
|
|
|
END DO
|
|
|
|
* Remove the shift
|
|
CALL SLARTG( B( KWBOT, KWBOT ), B( KWBOT, KWBOT-1 ), C1,
|
|
$ S1, TEMP )
|
|
B( KWBOT, KWBOT ) = TEMP
|
|
B( KWBOT, KWBOT-1 ) = ZERO
|
|
CALL SROT( KWBOT-ISTARTM, B( ISTARTM, KWBOT ), 1,
|
|
$ B( ISTARTM, KWBOT-1 ), 1, C1, S1 )
|
|
CALL SROT( KWBOT-ISTARTM+1, A( ISTARTM, KWBOT ), 1,
|
|
$ A( ISTARTM, KWBOT-1 ), 1, C1, S1 )
|
|
CALL SROT( JW, ZC( 1, KWBOT-KWTOP+1 ), 1, ZC( 1,
|
|
$ KWBOT-1-KWTOP+1 ), 1, C1, S1 )
|
|
|
|
K = K-1
|
|
END IF
|
|
END DO
|
|
|
|
END IF
|
|
|
|
* Apply Qc and Zc to rest of the matrix
|
|
IF ( ILSCHUR ) THEN
|
|
ISTARTM = 1
|
|
ISTOPM = N
|
|
ELSE
|
|
ISTARTM = ILO
|
|
ISTOPM = IHI
|
|
END IF
|
|
|
|
IF ( ISTOPM-IHI > 0 ) THEN
|
|
CALL SGEMM( 'T', 'N', JW, ISTOPM-IHI, JW, ONE, QC, LDQC,
|
|
$ A( KWTOP, IHI+1 ), LDA, ZERO, WORK, JW )
|
|
CALL SLACPY( 'ALL', JW, ISTOPM-IHI, WORK, JW, A( KWTOP,
|
|
$ IHI+1 ), LDA )
|
|
CALL SGEMM( 'T', 'N', JW, ISTOPM-IHI, JW, ONE, QC, LDQC,
|
|
$ B( KWTOP, IHI+1 ), LDB, ZERO, WORK, JW )
|
|
CALL SLACPY( 'ALL', JW, ISTOPM-IHI, WORK, JW, B( KWTOP,
|
|
$ IHI+1 ), LDB )
|
|
END IF
|
|
IF ( ILQ ) THEN
|
|
CALL SGEMM( 'N', 'N', N, JW, JW, ONE, Q( 1, KWTOP ), LDQ, QC,
|
|
$ LDQC, ZERO, WORK, N )
|
|
CALL SLACPY( 'ALL', N, JW, WORK, N, Q( 1, KWTOP ), LDQ )
|
|
END IF
|
|
|
|
IF ( KWTOP-1-ISTARTM+1 > 0 ) THEN
|
|
CALL SGEMM( 'N', 'N', KWTOP-ISTARTM, JW, JW, ONE, A( ISTARTM,
|
|
$ KWTOP ), LDA, ZC, LDZC, ZERO, WORK,
|
|
$ KWTOP-ISTARTM )
|
|
CALL SLACPY( 'ALL', KWTOP-ISTARTM, JW, WORK, KWTOP-ISTARTM,
|
|
$ A( ISTARTM, KWTOP ), LDA )
|
|
CALL SGEMM( 'N', 'N', KWTOP-ISTARTM, JW, JW, ONE, B( ISTARTM,
|
|
$ KWTOP ), LDB, ZC, LDZC, ZERO, WORK,
|
|
$ KWTOP-ISTARTM )
|
|
CALL SLACPY( 'ALL', KWTOP-ISTARTM, JW, WORK, KWTOP-ISTARTM,
|
|
$ B( ISTARTM, KWTOP ), LDB )
|
|
END IF
|
|
IF ( ILZ ) THEN
|
|
CALL SGEMM( 'N', 'N', N, JW, JW, ONE, Z( 1, KWTOP ), LDZ, ZC,
|
|
$ LDZC, ZERO, WORK, N )
|
|
CALL SLACPY( 'ALL', N, JW, WORK, N, Z( 1, KWTOP ), LDZ )
|
|
END IF
|
|
|
|
END SUBROUTINE
|