749 lines
25 KiB
Fortran
749 lines
25 KiB
Fortran
*> \brief \b DLAQZ0
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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 DLAQZ0 + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaqz0.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/dlaqz0.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/dlaqz0.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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* RECURSIVE SUBROUTINE DLAQZ0( WANTS, WANTQ, WANTZ, N, ILO, IHI, A,
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* $ LDA, B, LDB, ALPHAR, ALPHAI, BETA,
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* $ Q, LDQ, Z, LDZ, WORK, LWORK, REC,
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* $ INFO )
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* IMPLICIT NONE
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*
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* Arguments
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* CHARACTER, INTENT( IN ) :: WANTS, WANTQ, WANTZ
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* INTEGER, INTENT( IN ) :: N, ILO, IHI, LDA, LDB, LDQ, LDZ, LWORK,
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* $ REC
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*
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* INTEGER, INTENT( OUT ) :: INFO
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*
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* DOUBLE PRECISION, INTENT( INOUT ) :: A( LDA, * ), B( LDB, * ),
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* $ Q( LDQ, * ), Z( LDZ, * ), ALPHAR( * ),
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* $ ALPHAI( * ), BETA( * ), 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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*> DLAQZ0 computes the eigenvalues of a real matrix pair (H,T),
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*> where H is an upper Hessenberg matrix and T is upper triangular,
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*> using the double-shift QZ method.
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*> Matrix pairs of this type are produced by the reduction to
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*> generalized upper Hessenberg form of a real matrix pair (A,B):
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*>
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*> A = Q1*H*Z1**T, B = Q1*T*Z1**T,
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*>
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*> as computed by DGGHRD.
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*>
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*> If JOB='S', then the Hessenberg-triangular pair (H,T) is
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*> also reduced to generalized Schur form,
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*>
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*> H = Q*S*Z**T, T = Q*P*Z**T,
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*>
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*> where Q and Z are orthogonal matrices, P is an upper triangular
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*> matrix, and S is a quasi-triangular matrix with 1-by-1 and 2-by-2
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*> diagonal blocks.
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*>
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*> The 1-by-1 blocks correspond to real eigenvalues of the matrix pair
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*> (H,T) and the 2-by-2 blocks correspond to complex conjugate pairs of
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*> eigenvalues.
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*>
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*> Additionally, the 2-by-2 upper triangular diagonal blocks of P
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*> corresponding to 2-by-2 blocks of S are reduced to positive diagonal
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*> form, i.e., if S(j+1,j) is non-zero, then P(j+1,j) = P(j,j+1) = 0,
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*> P(j,j) > 0, and P(j+1,j+1) > 0.
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*>
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*> Optionally, the orthogonal matrix Q from the generalized Schur
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*> factorization may be postmultiplied into an input matrix Q1, and the
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*> orthogonal matrix Z may be postmultiplied into an input matrix Z1.
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*> If Q1 and Z1 are the orthogonal matrices from DGGHRD that reduced
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*> the matrix pair (A,B) to generalized upper Hessenberg form, then the
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*> output matrices Q1*Q and Z1*Z are the orthogonal factors from the
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*> generalized Schur factorization of (A,B):
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*>
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*> A = (Q1*Q)*S*(Z1*Z)**T, B = (Q1*Q)*P*(Z1*Z)**T.
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*>
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*> To avoid overflow, eigenvalues of the matrix pair (H,T) (equivalently,
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*> of (A,B)) are computed as a pair of values (alpha,beta), where alpha is
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*> complex and beta real.
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*> If beta is nonzero, lambda = alpha / beta is an eigenvalue of the
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*> generalized nonsymmetric eigenvalue problem (GNEP)
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*> A*x = lambda*B*x
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*> and if alpha is nonzero, mu = beta / alpha is an eigenvalue of the
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*> alternate form of the GNEP
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*> mu*A*y = B*y.
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*> Real eigenvalues can be read directly from the generalized Schur
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*> form:
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*> alpha = S(i,i), beta = P(i,i).
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*>
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*> Ref: C.B. Moler & G.W. Stewart, "An Algorithm for Generalized Matrix
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*> Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973),
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*> pp. 241--256.
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*>
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*> Ref: B. Kagstrom, D. Kressner, "Multishift Variants of the QZ
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*> Algorithm with Aggressive Early Deflation", SIAM J. Numer.
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*> Anal., 29(2006), pp. 199--227.
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*>
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*> Ref: T. Steel, D. Camps, K. Meerbergen, R. Vandebril "A multishift,
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*> multipole rational QZ method with aggressive early deflation"
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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] WANTS
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*> \verbatim
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*> WANTS is CHARACTER*1
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*> = 'E': Compute eigenvalues only;
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*> = 'S': Compute eigenvalues and the Schur form.
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*> \endverbatim
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*>
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*> \param[in] WANTQ
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*> \verbatim
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*> WANTQ is CHARACTER*1
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*> = 'N': Left Schur vectors (Q) are not computed;
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*> = 'I': Q is initialized to the unit matrix and the matrix Q
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*> of left Schur vectors of (A,B) is returned;
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*> = 'V': Q must contain an orthogonal matrix Q1 on entry and
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*> the product Q1*Q is returned.
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*> \endverbatim
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*>
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*> \param[in] WANTZ
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*> \verbatim
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*> WANTZ is CHARACTER*1
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*> = 'N': Right Schur vectors (Z) are not computed;
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*> = 'I': Z is initialized to the unit matrix and the matrix Z
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*> of right Schur vectors of (A,B) is returned;
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*> = 'V': Z must contain an orthogonal matrix Z1 on entry and
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*> the product Z1*Z is returned.
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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 which are in
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*> Hessenberg form. It is assumed that A is already upper
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*> triangular in rows and columns 1:ILO-1 and IHI+1:N.
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*> If N > 0, 1 <= ILO <= IHI <= N; if N = 0, ILO=1 and IHI=0.
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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 DOUBLE PRECISION array, dimension (LDA, N)
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*> On entry, the N-by-N upper Hessenberg matrix A.
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*> On exit, if JOB = 'S', A contains the upper quasi-triangular
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*> matrix S from the generalized Schur factorization.
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*> If JOB = 'E', the diagonal blocks of A match those of S, but
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*> the rest of A is unspecified.
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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 DOUBLE PRECISION array, dimension (LDB, N)
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*> On entry, the N-by-N upper triangular matrix B.
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*> On exit, if JOB = 'S', B contains the upper triangular
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*> matrix P from the generalized Schur factorization;
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*> 2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks of S
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*> are reduced to positive diagonal form, i.e., if A(j+1,j) is
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*> non-zero, then B(j+1,j) = B(j,j+1) = 0, B(j,j) > 0, and
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*> B(j+1,j+1) > 0.
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*> If JOB = 'E', the diagonal blocks of B match those of P, but
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*> the rest of B is unspecified.
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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[out] ALPHAR
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*> \verbatim
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*> ALPHAR is DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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] Q
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*> \verbatim
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*> Q is DOUBLE PRECISION array, dimension (LDQ, N)
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*> On entry, if COMPQ = 'V', the orthogonal matrix Q1 used in
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*> the reduction of (A,B) to generalized Hessenberg form.
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*> On exit, if COMPQ = 'I', the orthogonal matrix of left Schur
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*> vectors of (A,B), and if COMPQ = 'V', the orthogonal matrix
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*> of left Schur vectors of (A,B).
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*> Not referenced if COMPQ = '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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*> The leading dimension of the array Q. LDQ >= 1.
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*> If COMPQ='V' or 'I', then LDQ >= N.
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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 DOUBLE PRECISION array, dimension (LDZ, N)
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*> On entry, if COMPZ = 'V', the orthogonal matrix Z1 used in
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*> the reduction of (A,B) to generalized Hessenberg form.
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*> On exit, if COMPZ = 'I', the orthogonal matrix of
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*> right Schur vectors of (H,T), and if COMPZ = 'V', the
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*> orthogonal matrix of right Schur vectors of (A,B).
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*> Not referenced if COMPZ = '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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*> The leading dimension of the array Z. LDZ >= 1.
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*> If COMPZ='V' or 'I', then LDZ >= N.
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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 DOUBLE PRECISION 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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*> = 1,...,N: the QZ iteration did not converge. (A,B) is not
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*> in Schur form, but ALPHAR(i), ALPHAI(i), and
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*> BETA(i), i=INFO+1,...,N should be correct.
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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 DLAQZ0( WANTS, WANTQ, WANTZ, N, ILO, IHI, A,
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$ LDA, B, LDB, ALPHAR, ALPHAI, BETA,
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$ Q, LDQ, Z, LDZ, WORK, LWORK, REC,
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$ INFO )
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IMPLICIT NONE
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* Arguments
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CHARACTER, INTENT( IN ) :: WANTS, WANTQ, WANTZ
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INTEGER, INTENT( IN ) :: N, ILO, IHI, LDA, LDB, LDQ, LDZ, LWORK,
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$ REC
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INTEGER, INTENT( OUT ) :: INFO
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DOUBLE PRECISION, INTENT( INOUT ) :: A( LDA, * ), B( LDB, * ),
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$ Q( LDQ, * ), Z( LDZ, * ), ALPHAR( * ),
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$ ALPHAI( * ), BETA( * ), WORK( * )
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* Parameters
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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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DOUBLE PRECISION :: SMLNUM, ULP, ESHIFT, SAFMIN, SAFMAX, C1, S1,
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$ TEMP, SWAP, BNORM, BTOL
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INTEGER :: ISTART, ISTOP, IITER, MAXIT, ISTART2, K, LD, NSHIFTS,
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$ NBLOCK, NW, NMIN, NIBBLE, N_UNDEFLATED, N_DEFLATED,
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$ NS, SWEEP_INFO, SHIFTPOS, LWORKREQ, K2, ISTARTM,
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$ ISTOPM, IWANTS, IWANTQ, IWANTZ, NORM_INFO, AED_INFO,
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$ NWR, NBR, NSR, ITEMP1, ITEMP2, RCOST, I
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LOGICAL :: ILSCHUR, ILQ, ILZ
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CHARACTER :: JBCMPZ*3
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* External Functions
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EXTERNAL :: XERBLA, DHGEQZ, DLASET, DLAQZ3, DLAQZ4,
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$ DLARTG, DROT
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DOUBLE PRECISION, EXTERNAL :: DLAMCH, DLANHS
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LOGICAL, EXTERNAL :: LSAME
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INTEGER, EXTERNAL :: ILAENV
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*
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* Decode wantS,wantQ,wantZ
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*
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IF( LSAME( WANTS, 'E' ) ) THEN
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ILSCHUR = .FALSE.
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IWANTS = 1
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ELSE IF( LSAME( WANTS, 'S' ) ) THEN
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ILSCHUR = .TRUE.
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IWANTS = 2
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ELSE
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IWANTS = 0
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END IF
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IF( LSAME( WANTQ, 'N' ) ) THEN
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ILQ = .FALSE.
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IWANTQ = 1
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ELSE IF( LSAME( WANTQ, 'V' ) ) THEN
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ILQ = .TRUE.
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IWANTQ = 2
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ELSE IF( LSAME( WANTQ, 'I' ) ) THEN
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ILQ = .TRUE.
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IWANTQ = 3
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ELSE
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IWANTQ = 0
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END IF
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IF( LSAME( WANTZ, 'N' ) ) THEN
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ILZ = .FALSE.
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IWANTZ = 1
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ELSE IF( LSAME( WANTZ, 'V' ) ) THEN
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ILZ = .TRUE.
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IWANTZ = 2
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ELSE IF( LSAME( WANTZ, 'I' ) ) THEN
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ILZ = .TRUE.
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IWANTZ = 3
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ELSE
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IWANTZ = 0
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END IF
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*
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* Check Argument Values
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*
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INFO = 0
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IF( IWANTS.EQ.0 ) THEN
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INFO = -1
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ELSE IF( IWANTQ.EQ.0 ) THEN
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INFO = -2
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ELSE IF( IWANTZ.EQ.0 ) THEN
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INFO = -3
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ELSE IF( N.LT.0 ) THEN
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INFO = -4
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ELSE IF( ILO.LT.1 ) THEN
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INFO = -5
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ELSE IF( IHI.GT.N .OR. IHI.LT.ILO-1 ) THEN
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INFO = -6
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ELSE IF( LDA.LT.N ) THEN
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INFO = -8
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ELSE IF( LDB.LT.N ) THEN
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INFO = -10
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ELSE IF( LDQ.LT.1 .OR. ( ILQ .AND. LDQ.LT.N ) ) THEN
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INFO = -15
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ELSE IF( LDZ.LT.1 .OR. ( ILZ .AND. LDZ.LT.N ) ) THEN
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INFO = -17
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END IF
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'DLAQZ0', -INFO )
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RETURN
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END IF
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*
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* Quick return if possible
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*
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IF( N.LE.0 ) THEN
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WORK( 1 ) = DBLE( 1 )
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RETURN
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END IF
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*
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* Get the parameters
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*
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JBCMPZ( 1:1 ) = WANTS
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JBCMPZ( 2:2 ) = WANTQ
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JBCMPZ( 3:3 ) = WANTZ
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NMIN = ILAENV( 12, 'DLAQZ0', JBCMPZ, N, ILO, IHI, LWORK )
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NWR = ILAENV( 13, 'DLAQZ0', JBCMPZ, N, ILO, IHI, LWORK )
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NWR = MAX( 2, NWR )
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NWR = MIN( IHI-ILO+1, ( N-1 ) / 3, NWR )
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NIBBLE = ILAENV( 14, 'DLAQZ0', JBCMPZ, N, ILO, IHI, LWORK )
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NSR = ILAENV( 15, 'DLAQZ0', JBCMPZ, N, ILO, IHI, LWORK )
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NSR = MIN( NSR, ( N+6 ) / 9, IHI-ILO )
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NSR = MAX( 2, NSR-MOD( NSR, 2 ) )
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RCOST = ILAENV( 17, 'DLAQZ0', JBCMPZ, N, ILO, IHI, LWORK )
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ITEMP1 = INT( NSR/SQRT( 1+2*NSR/( DBLE( RCOST )/100*N ) ) )
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ITEMP1 = ( ( ITEMP1-1 )/4 )*4+4
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NBR = NSR+ITEMP1
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IF( N .LT. NMIN .OR. REC .GE. 2 ) THEN
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CALL DHGEQZ( WANTS, WANTQ, WANTZ, N, ILO, IHI, A, LDA, B, LDB,
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$ ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK,
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$ LWORK, INFO )
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RETURN
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END IF
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*
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* Find out required workspace
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*
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* Workspace query to dlaqz3
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|
NW = MAX( NWR, NMIN )
|
|
CALL DLAQZ3( ILSCHUR, ILQ, ILZ, N, ILO, IHI, NW, A, LDA, B, LDB,
|
|
$ Q, LDQ, Z, LDZ, N_UNDEFLATED, N_DEFLATED, ALPHAR,
|
|
$ ALPHAI, BETA, WORK, NW, WORK, NW, WORK, -1, REC,
|
|
$ AED_INFO )
|
|
ITEMP1 = INT( WORK( 1 ) )
|
|
* Workspace query to dlaqz4
|
|
CALL DLAQZ4( ILSCHUR, ILQ, ILZ, N, ILO, IHI, NSR, NBR, ALPHAR,
|
|
$ ALPHAI, BETA, A, LDA, B, LDB, Q, LDQ, Z, LDZ, WORK,
|
|
$ NBR, WORK, NBR, WORK, -1, SWEEP_INFO )
|
|
ITEMP2 = INT( WORK( 1 ) )
|
|
|
|
LWORKREQ = MAX( ITEMP1+2*NW**2, ITEMP2+2*NBR**2 )
|
|
IF ( LWORK .EQ.-1 ) THEN
|
|
WORK( 1 ) = DBLE( LWORKREQ )
|
|
RETURN
|
|
ELSE IF ( LWORK .LT. LWORKREQ ) THEN
|
|
INFO = -19
|
|
END IF
|
|
IF( INFO.NE.0 ) THEN
|
|
CALL XERBLA( 'DLAQZ0', INFO )
|
|
RETURN
|
|
END IF
|
|
*
|
|
* Initialize Q and Z
|
|
*
|
|
IF( IWANTQ.EQ.3 ) CALL DLASET( 'FULL', N, N, ZERO, ONE, Q, LDQ )
|
|
IF( IWANTZ.EQ.3 ) CALL DLASET( 'FULL', N, N, ZERO, ONE, Z, LDZ )
|
|
|
|
* Get machine constants
|
|
SAFMIN = DLAMCH( 'SAFE MINIMUM' )
|
|
SAFMAX = ONE/SAFMIN
|
|
ULP = DLAMCH( 'PRECISION' )
|
|
SMLNUM = SAFMIN*( DBLE( N )/ULP )
|
|
|
|
BNORM = DLANHS( 'F', IHI-ILO+1, B( ILO, ILO ), LDB, WORK )
|
|
BTOL = MAX( SAFMIN, ULP*BNORM )
|
|
|
|
ISTART = ILO
|
|
ISTOP = IHI
|
|
MAXIT = 3*( IHI-ILO+1 )
|
|
LD = 0
|
|
|
|
DO IITER = 1, MAXIT
|
|
IF( IITER .GE. MAXIT ) THEN
|
|
INFO = ISTOP+1
|
|
GOTO 80
|
|
END IF
|
|
IF ( ISTART+1 .GE. ISTOP ) THEN
|
|
ISTOP = ISTART
|
|
EXIT
|
|
END IF
|
|
|
|
* Check deflations at the end
|
|
IF ( ABS( A( ISTOP-1, ISTOP-2 ) ) .LE. MAX( SMLNUM,
|
|
$ ULP*( ABS( A( ISTOP-1, ISTOP-1 ) )+ABS( A( ISTOP-2,
|
|
$ ISTOP-2 ) ) ) ) ) THEN
|
|
A( ISTOP-1, ISTOP-2 ) = ZERO
|
|
ISTOP = ISTOP-2
|
|
LD = 0
|
|
ESHIFT = ZERO
|
|
ELSE IF ( ABS( A( ISTOP, ISTOP-1 ) ) .LE. MAX( SMLNUM,
|
|
$ ULP*( ABS( A( ISTOP, ISTOP ) )+ABS( A( ISTOP-1,
|
|
$ ISTOP-1 ) ) ) ) ) THEN
|
|
A( ISTOP, ISTOP-1 ) = ZERO
|
|
ISTOP = ISTOP-1
|
|
LD = 0
|
|
ESHIFT = ZERO
|
|
END IF
|
|
* Check deflations at the start
|
|
IF ( ABS( A( ISTART+2, ISTART+1 ) ) .LE. MAX( SMLNUM,
|
|
$ ULP*( ABS( A( ISTART+1, ISTART+1 ) )+ABS( A( ISTART+2,
|
|
$ ISTART+2 ) ) ) ) ) THEN
|
|
A( ISTART+2, ISTART+1 ) = ZERO
|
|
ISTART = ISTART+2
|
|
LD = 0
|
|
ESHIFT = ZERO
|
|
ELSE IF ( ABS( A( ISTART+1, ISTART ) ) .LE. MAX( SMLNUM,
|
|
$ ULP*( ABS( A( ISTART, ISTART ) )+ABS( A( ISTART+1,
|
|
$ ISTART+1 ) ) ) ) ) THEN
|
|
A( ISTART+1, ISTART ) = ZERO
|
|
ISTART = ISTART+1
|
|
LD = 0
|
|
ESHIFT = ZERO
|
|
END IF
|
|
|
|
IF ( ISTART+1 .GE. ISTOP ) THEN
|
|
EXIT
|
|
END IF
|
|
|
|
* Check interior deflations
|
|
ISTART2 = ISTART
|
|
DO K = ISTOP, ISTART+1, -1
|
|
IF ( ABS( A( K, K-1 ) ) .LE. MAX( SMLNUM, ULP*( ABS( A( K,
|
|
$ K ) )+ABS( A( K-1, K-1 ) ) ) ) ) THEN
|
|
A( K, K-1 ) = ZERO
|
|
ISTART2 = K
|
|
EXIT
|
|
END IF
|
|
END DO
|
|
|
|
* Get range to apply rotations to
|
|
IF ( ILSCHUR ) THEN
|
|
ISTARTM = 1
|
|
ISTOPM = N
|
|
ELSE
|
|
ISTARTM = ISTART2
|
|
ISTOPM = ISTOP
|
|
END IF
|
|
|
|
* Check infinite eigenvalues, this is done without blocking so might
|
|
* slow down the method when many infinite eigenvalues are present
|
|
K = ISTOP
|
|
DO WHILE ( K.GE.ISTART2 )
|
|
|
|
IF( ABS( B( K, K ) ) .LT. BTOL ) THEN
|
|
* A diagonal element of B is negligible, move it
|
|
* to the top and deflate it
|
|
|
|
DO K2 = K, ISTART2+1, -1
|
|
CALL DLARTG( B( K2-1, K2 ), B( K2-1, K2-1 ), C1, S1,
|
|
$ TEMP )
|
|
B( K2-1, K2 ) = TEMP
|
|
B( K2-1, K2-1 ) = ZERO
|
|
|
|
CALL DROT( K2-2-ISTARTM+1, B( ISTARTM, K2 ), 1,
|
|
$ B( ISTARTM, K2-1 ), 1, C1, S1 )
|
|
CALL DROT( MIN( K2+1, ISTOP )-ISTARTM+1, A( ISTARTM,
|
|
$ K2 ), 1, A( ISTARTM, K2-1 ), 1, C1, S1 )
|
|
IF ( ILZ ) THEN
|
|
CALL DROT( N, Z( 1, K2 ), 1, Z( 1, K2-1 ), 1, C1,
|
|
$ S1 )
|
|
END IF
|
|
|
|
IF( K2.LT.ISTOP ) THEN
|
|
CALL DLARTG( A( K2, K2-1 ), A( K2+1, K2-1 ), C1,
|
|
$ S1, TEMP )
|
|
A( K2, K2-1 ) = TEMP
|
|
A( K2+1, K2-1 ) = ZERO
|
|
|
|
CALL DROT( ISTOPM-K2+1, A( K2, K2 ), LDA, A( K2+1,
|
|
$ K2 ), LDA, C1, S1 )
|
|
CALL DROT( ISTOPM-K2+1, B( K2, K2 ), LDB, B( K2+1,
|
|
$ K2 ), LDB, C1, S1 )
|
|
IF( ILQ ) THEN
|
|
CALL DROT( N, Q( 1, K2 ), 1, Q( 1, K2+1 ), 1,
|
|
$ C1, S1 )
|
|
END IF
|
|
END IF
|
|
|
|
END DO
|
|
|
|
IF( ISTART2.LT.ISTOP )THEN
|
|
CALL DLARTG( A( ISTART2, ISTART2 ), A( ISTART2+1,
|
|
$ ISTART2 ), C1, S1, TEMP )
|
|
A( ISTART2, ISTART2 ) = TEMP
|
|
A( ISTART2+1, ISTART2 ) = ZERO
|
|
|
|
CALL DROT( ISTOPM-( ISTART2+1 )+1, A( ISTART2,
|
|
$ ISTART2+1 ), LDA, A( ISTART2+1,
|
|
$ ISTART2+1 ), LDA, C1, S1 )
|
|
CALL DROT( ISTOPM-( ISTART2+1 )+1, B( ISTART2,
|
|
$ ISTART2+1 ), LDB, B( ISTART2+1,
|
|
$ ISTART2+1 ), LDB, C1, S1 )
|
|
IF( ILQ ) THEN
|
|
CALL DROT( N, Q( 1, ISTART2 ), 1, Q( 1,
|
|
$ ISTART2+1 ), 1, C1, S1 )
|
|
END IF
|
|
END IF
|
|
|
|
ISTART2 = ISTART2+1
|
|
|
|
END IF
|
|
K = K-1
|
|
END DO
|
|
|
|
* istart2 now points to the top of the bottom right
|
|
* unreduced Hessenberg block
|
|
IF ( ISTART2 .GE. ISTOP ) THEN
|
|
ISTOP = ISTART2-1
|
|
LD = 0
|
|
ESHIFT = ZERO
|
|
CYCLE
|
|
END IF
|
|
|
|
NW = NWR
|
|
NSHIFTS = NSR
|
|
NBLOCK = NBR
|
|
|
|
IF ( ISTOP-ISTART2+1 .LT. NMIN ) THEN
|
|
* Setting nw to the size of the subblock will make AED deflate
|
|
* all the eigenvalues. This is slightly more efficient than just
|
|
* using DHGEQZ because the off diagonal part gets updated via BLAS.
|
|
IF ( ISTOP-ISTART+1 .LT. NMIN ) THEN
|
|
NW = ISTOP-ISTART+1
|
|
ISTART2 = ISTART
|
|
ELSE
|
|
NW = ISTOP-ISTART2+1
|
|
END IF
|
|
END IF
|
|
|
|
*
|
|
* Time for AED
|
|
*
|
|
CALL DLAQZ3( ILSCHUR, ILQ, ILZ, N, ISTART2, ISTOP, NW, A, LDA,
|
|
$ B, LDB, Q, LDQ, Z, LDZ, N_UNDEFLATED, N_DEFLATED,
|
|
$ ALPHAR, ALPHAI, BETA, WORK, NW, WORK( NW**2+1 ),
|
|
$ NW, WORK( 2*NW**2+1 ), LWORK-2*NW**2, REC,
|
|
$ AED_INFO )
|
|
|
|
IF ( N_DEFLATED > 0 ) THEN
|
|
ISTOP = ISTOP-N_DEFLATED
|
|
LD = 0
|
|
ESHIFT = ZERO
|
|
END IF
|
|
|
|
IF ( 100*N_DEFLATED > NIBBLE*( N_DEFLATED+N_UNDEFLATED ) .OR.
|
|
$ ISTOP-ISTART2+1 .LT. NMIN ) THEN
|
|
* AED has uncovered many eigenvalues. Skip a QZ sweep and run
|
|
* AED again.
|
|
CYCLE
|
|
END IF
|
|
|
|
LD = LD+1
|
|
|
|
NS = MIN( NSHIFTS, ISTOP-ISTART2 )
|
|
NS = MIN( NS, N_UNDEFLATED )
|
|
SHIFTPOS = ISTOP-N_UNDEFLATED+1
|
|
*
|
|
* Shuffle shifts to put double shifts in front
|
|
* This ensures that we don't split up a double shift
|
|
*
|
|
DO I = SHIFTPOS, SHIFTPOS+N_UNDEFLATED-1, 2
|
|
IF( ALPHAI( I ).NE.-ALPHAI( I+1 ) ) THEN
|
|
*
|
|
SWAP = ALPHAR( I )
|
|
ALPHAR( I ) = ALPHAR( I+1 )
|
|
ALPHAR( I+1 ) = ALPHAR( I+2 )
|
|
ALPHAR( I+2 ) = SWAP
|
|
|
|
SWAP = ALPHAI( I )
|
|
ALPHAI( I ) = ALPHAI( I+1 )
|
|
ALPHAI( I+1 ) = ALPHAI( I+2 )
|
|
ALPHAI( I+2 ) = SWAP
|
|
|
|
SWAP = BETA( I )
|
|
BETA( I ) = BETA( I+1 )
|
|
BETA( I+1 ) = BETA( I+2 )
|
|
BETA( I+2 ) = SWAP
|
|
END IF
|
|
END DO
|
|
|
|
IF ( MOD( LD, 6 ) .EQ. 0 ) THEN
|
|
*
|
|
* Exceptional shift. Chosen for no particularly good reason.
|
|
*
|
|
IF( ( DBLE( MAXIT )*SAFMIN )*ABS( A( ISTOP,
|
|
$ ISTOP-1 ) ).LT.ABS( A( ISTOP-1, ISTOP-1 ) ) ) THEN
|
|
ESHIFT = A( ISTOP, ISTOP-1 )/B( ISTOP-1, ISTOP-1 )
|
|
ELSE
|
|
ESHIFT = ESHIFT+ONE/( SAFMIN*DBLE( MAXIT ) )
|
|
END IF
|
|
ALPHAR( SHIFTPOS ) = ONE
|
|
ALPHAR( SHIFTPOS+1 ) = ZERO
|
|
ALPHAI( SHIFTPOS ) = ZERO
|
|
ALPHAI( SHIFTPOS+1 ) = ZERO
|
|
BETA( SHIFTPOS ) = ESHIFT
|
|
BETA( SHIFTPOS+1 ) = ESHIFT
|
|
NS = 2
|
|
END IF
|
|
|
|
*
|
|
* Time for a QZ sweep
|
|
*
|
|
CALL DLAQZ4( ILSCHUR, ILQ, ILZ, N, ISTART2, ISTOP, NS, NBLOCK,
|
|
$ ALPHAR( SHIFTPOS ), ALPHAI( SHIFTPOS ),
|
|
$ BETA( SHIFTPOS ), A, LDA, B, LDB, Q, LDQ, Z, LDZ,
|
|
$ WORK, NBLOCK, WORK( NBLOCK**2+1 ), NBLOCK,
|
|
$ WORK( 2*NBLOCK**2+1 ), LWORK-2*NBLOCK**2,
|
|
$ SWEEP_INFO )
|
|
|
|
END DO
|
|
|
|
*
|
|
* Call DHGEQZ to normalize the eigenvalue blocks and set the eigenvalues
|
|
* If all the eigenvalues have been found, DHGEQZ will not do any iterations
|
|
* and only normalize the blocks. In case of a rare convergence failure,
|
|
* the single shift might perform better.
|
|
*
|
|
80 CALL DHGEQZ( WANTS, WANTQ, WANTZ, N, ILO, IHI, A, LDA, B, LDB,
|
|
$ ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK, LWORK,
|
|
$ NORM_INFO )
|
|
|
|
INFO = NORM_INFO
|
|
|
|
END SUBROUTINE
|