version 1.5, 2010/08/07 13:22:34
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version 1.14, 2015/11/26 11:44:24
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*> \brief \b ZHGEQZ |
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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 ZHGEQZ + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zhgeqz.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/zhgeqz.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/zhgeqz.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 ZHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT, |
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* ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, |
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* RWORK, INFO ) |
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* |
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* .. Scalar Arguments .. |
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* CHARACTER COMPQ, COMPZ, JOB |
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* INTEGER IHI, ILO, INFO, LDH, LDQ, LDT, LDZ, LWORK, N |
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* .. |
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* .. Array Arguments .. |
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* DOUBLE PRECISION RWORK( * ) |
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* COMPLEX*16 ALPHA( * ), BETA( * ), H( LDH, * ), |
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* $ Q( LDQ, * ), T( LDT, * ), WORK( * ), |
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* $ Z( LDZ, * ) |
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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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*> ZHGEQZ computes the eigenvalues of a complex 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 single-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 complex matrix pair (A,B): |
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*> |
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*> A = Q1*H*Z1**H, B = Q1*T*Z1**H, |
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*> |
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*> as computed by ZGGHRD. |
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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**H, T = Q*P*Z**H, |
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*> |
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*> where Q and Z are unitary matrices and S and P are upper triangular. |
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*> |
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*> Optionally, the unitary 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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*> unitary matrix Z may be postmultiplied into an input matrix Z1. |
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*> If Q1 and Z1 are the unitary matrices from ZGGHRD that reduced |
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*> the matrix pair (A,B) to generalized Hessenberg form, then the output |
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*> matrices Q1*Q and Z1*Z are the unitary factors from the generalized |
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*> Schur factorization of (A,B): |
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*> |
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*> A = (Q1*Q)*S*(Z1*Z)**H, B = (Q1*Q)*P*(Z1*Z)**H. |
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*> |
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*> To avoid overflow, eigenvalues of the matrix pair (H,T) |
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*> (equivalently, of (A,B)) are computed as a pair of complex values |
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*> (alpha,beta). If beta is nonzero, lambda = alpha / beta is an |
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*> eigenvalue of the 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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*> The values of alpha and beta for the i-th eigenvalue can be read |
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*> directly from the generalized Schur form: alpha = S(i,i), |
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*> 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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*> \endverbatim |
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* |
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* Arguments: |
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* ========== |
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* |
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*> \param[in] JOB |
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*> \verbatim |
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*> JOB is CHARACTER*1 |
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*> = 'E': Compute eigenvalues only; |
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*> = 'S': Computer eigenvalues and the Schur form. |
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*> \endverbatim |
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*> |
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*> \param[in] COMPQ |
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*> \verbatim |
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*> COMPQ 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 (H,T) is returned; |
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*> = 'V': Q must contain a unitary 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] COMPZ |
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*> \verbatim |
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*> COMPZ is CHARACTER*1 |
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*> = 'N': Right Schur vectors (Z) are not computed; |
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*> = 'I': Q is initialized to the unit matrix and the matrix Z |
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*> of right Schur vectors of (H,T) is returned; |
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*> = 'V': Z must contain a unitary 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 H, T, 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 H 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] H |
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*> \verbatim |
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*> H is COMPLEX*16 array, dimension (LDH, N) |
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*> On entry, the N-by-N upper Hessenberg matrix H. |
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*> On exit, if JOB = 'S', H contains the upper triangular |
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*> matrix S from the generalized Schur factorization. |
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*> If JOB = 'E', the diagonal of H matches that of S, but |
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*> the rest of H is unspecified. |
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*> \endverbatim |
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*> |
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*> \param[in] LDH |
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*> \verbatim |
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*> LDH is INTEGER |
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*> The leading dimension of the array H. LDH >= max( 1, N ). |
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*> \endverbatim |
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*> |
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*> \param[in,out] T |
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*> \verbatim |
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*> T is COMPLEX*16 array, dimension (LDT, N) |
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*> On entry, the N-by-N upper triangular matrix T. |
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*> On exit, if JOB = 'S', T contains the upper triangular |
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*> matrix P from the generalized Schur factorization. |
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*> If JOB = 'E', the diagonal of T matches that of P, but |
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*> the rest of T is unspecified. |
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*> \endverbatim |
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*> |
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*> \param[in] LDT |
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*> \verbatim |
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*> LDT is INTEGER |
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*> The leading dimension of the array T. LDT >= max( 1, N ). |
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*> \endverbatim |
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*> |
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*> \param[out] ALPHA |
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*> \verbatim |
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*> ALPHA is COMPLEX*16 array, dimension (N) |
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*> The complex scalars alpha that define the eigenvalues of |
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*> GNEP. ALPHA(i) = S(i,i) in the generalized Schur |
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*> factorization. |
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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 COMPLEX*16 array, dimension (N) |
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*> The real non-negative scalars beta that define the |
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*> eigenvalues of GNEP. BETA(i) = P(i,i) in the generalized |
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*> Schur factorization. |
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*> |
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*> Together, the quantities alpha = ALPHA(j) and beta = BETA(j) |
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*> represent the j-th eigenvalue of the matrix pair (A,B), in |
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*> one of the forms lambda = alpha/beta or mu = beta/alpha. |
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*> Since either lambda or mu may overflow, they should not, |
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*> 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 COMPLEX*16 array, dimension (LDQ, N) |
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*> On entry, if COMPZ = 'V', the unitary matrix Q1 used in the |
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*> reduction of (A,B) to generalized Hessenberg form. |
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*> On exit, if COMPZ = 'I', the unitary matrix of left Schur |
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*> vectors of (H,T), and if COMPZ = 'V', the unitary matrix of |
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*> left 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] 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 COMPLEX*16 array, dimension (LDZ, N) |
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*> On entry, if COMPZ = 'V', the unitary matrix Z1 used in the |
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*> reduction of (A,B) to generalized Hessenberg form. |
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*> On exit, if COMPZ = 'I', the unitary matrix of right Schur |
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*> vectors of (H,T), and if COMPZ = 'V', the unitary matrix of |
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*> 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 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] RWORK |
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*> \verbatim |
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*> RWORK is DOUBLE PRECISION array, dimension (N) |
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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. (H,T) is not |
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*> in Schur form, but ALPHA(i) and BETA(i), |
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*> i=INFO+1,...,N should be correct. |
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*> = N+1,...,2*N: the shift calculation failed. (H,T) is not |
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*> in Schur form, but ALPHA(i) and BETA(i), |
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*> i=INFO-N+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 Univ. of Tennessee |
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*> \author Univ. of California Berkeley |
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*> \author Univ. of Colorado Denver |
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*> \author NAG Ltd. |
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* |
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*> \date April 2012 |
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* |
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*> \ingroup complex16GEcomputational |
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* |
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*> \par Further Details: |
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* ===================== |
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*> |
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*> \verbatim |
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*> |
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*> We assume that complex ABS works as long as its value is less than |
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*> overflow. |
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*> \endverbatim |
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*> |
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* ===================================================================== |
SUBROUTINE ZHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT, |
SUBROUTINE ZHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT, |
$ ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, |
$ ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, |
$ RWORK, INFO ) |
$ RWORK, INFO ) |
* |
* |
* -- LAPACK routine (version 3.2) -- |
* -- LAPACK computational routine (version 3.6.0) -- |
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
* November 2006 |
* April 2012 |
* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
CHARACTER COMPQ, COMPZ, JOB |
CHARACTER COMPQ, COMPZ, JOB |
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$ Z( LDZ, * ) |
$ Z( LDZ, * ) |
* .. |
* .. |
* |
* |
* Purpose |
|
* ======= |
|
* |
|
* ZHGEQZ computes the eigenvalues of a complex matrix pair (H,T), |
|
* where H is an upper Hessenberg matrix and T is upper triangular, |
|
* using the single-shift QZ method. |
|
* Matrix pairs of this type are produced by the reduction to |
|
* generalized upper Hessenberg form of a complex matrix pair (A,B): |
|
* |
|
* A = Q1*H*Z1**H, B = Q1*T*Z1**H, |
|
* |
|
* as computed by ZGGHRD. |
|
* |
|
* If JOB='S', then the Hessenberg-triangular pair (H,T) is |
|
* also reduced to generalized Schur form, |
|
* |
|
* H = Q*S*Z**H, T = Q*P*Z**H, |
|
* |
|
* where Q and Z are unitary matrices and S and P are upper triangular. |
|
* |
|
* Optionally, the unitary matrix Q from the generalized Schur |
|
* factorization may be postmultiplied into an input matrix Q1, and the |
|
* unitary matrix Z may be postmultiplied into an input matrix Z1. |
|
* If Q1 and Z1 are the unitary matrices from ZGGHRD that reduced |
|
* the matrix pair (A,B) to generalized Hessenberg form, then the output |
|
* matrices Q1*Q and Z1*Z are the unitary factors from the generalized |
|
* Schur factorization of (A,B): |
|
* |
|
* A = (Q1*Q)*S*(Z1*Z)**H, B = (Q1*Q)*P*(Z1*Z)**H. |
|
* |
|
* To avoid overflow, eigenvalues of the matrix pair (H,T) |
|
* (equivalently, of (A,B)) are computed as a pair of complex values |
|
* (alpha,beta). If beta is nonzero, lambda = alpha / beta is an |
|
* eigenvalue of the generalized nonsymmetric eigenvalue problem (GNEP) |
|
* A*x = lambda*B*x |
|
* and if alpha is nonzero, mu = beta / alpha is an eigenvalue of the |
|
* alternate form of the GNEP |
|
* mu*A*y = B*y. |
|
* The values of alpha and beta for the i-th eigenvalue can be read |
|
* directly from the generalized Schur form: alpha = S(i,i), |
|
* beta = P(i,i). |
|
* |
|
* Ref: C.B. Moler & G.W. Stewart, "An Algorithm for Generalized Matrix |
|
* Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973), |
|
* pp. 241--256. |
|
* |
|
* Arguments |
|
* ========= |
|
* |
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* JOB (input) CHARACTER*1 |
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* = 'E': Compute eigenvalues only; |
|
* = 'S': Computer eigenvalues and the Schur form. |
|
* |
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* COMPQ (input) CHARACTER*1 |
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* = 'N': Left Schur vectors (Q) are not computed; |
|
* = 'I': Q is initialized to the unit matrix and the matrix Q |
|
* of left Schur vectors of (H,T) is returned; |
|
* = 'V': Q must contain a unitary matrix Q1 on entry and |
|
* the product Q1*Q is returned. |
|
* |
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* COMPZ (input) CHARACTER*1 |
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* = 'N': Right Schur vectors (Z) are not computed; |
|
* = 'I': Q is initialized to the unit matrix and the matrix Z |
|
* of right Schur vectors of (H,T) is returned; |
|
* = 'V': Z must contain a unitary matrix Z1 on entry and |
|
* the product Z1*Z is returned. |
|
* |
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* N (input) INTEGER |
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* The order of the matrices H, T, Q, and Z. N >= 0. |
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* |
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* ILO (input) INTEGER |
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* IHI (input) INTEGER |
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* ILO and IHI mark the rows and columns of H which are in |
|
* Hessenberg form. It is assumed that A is already upper |
|
* triangular in rows and columns 1:ILO-1 and IHI+1:N. |
|
* If N > 0, 1 <= ILO <= IHI <= N; if N = 0, ILO=1 and IHI=0. |
|
* |
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* H (input/output) COMPLEX*16 array, dimension (LDH, N) |
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* On entry, the N-by-N upper Hessenberg matrix H. |
|
* On exit, if JOB = 'S', H contains the upper triangular |
|
* matrix S from the generalized Schur factorization. |
|
* If JOB = 'E', the diagonal of H matches that of S, but |
|
* the rest of H is unspecified. |
|
* |
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* LDH (input) INTEGER |
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* The leading dimension of the array H. LDH >= max( 1, N ). |
|
* |
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* T (input/output) COMPLEX*16 array, dimension (LDT, N) |
|
* On entry, the N-by-N upper triangular matrix T. |
|
* On exit, if JOB = 'S', T contains the upper triangular |
|
* matrix P from the generalized Schur factorization. |
|
* If JOB = 'E', the diagonal of T matches that of P, but |
|
* the rest of T is unspecified. |
|
* |
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* LDT (input) INTEGER |
|
* The leading dimension of the array T. LDT >= max( 1, N ). |
|
* |
|
* ALPHA (output) COMPLEX*16 array, dimension (N) |
|
* The complex scalars alpha that define the eigenvalues of |
|
* GNEP. ALPHA(i) = S(i,i) in the generalized Schur |
|
* factorization. |
|
* |
|
* BETA (output) COMPLEX*16 array, dimension (N) |
|
* The real non-negative scalars beta that define the |
|
* eigenvalues of GNEP. BETA(i) = P(i,i) in the generalized |
|
* Schur factorization. |
|
* |
|
* Together, the quantities alpha = ALPHA(j) and beta = BETA(j) |
|
* represent the j-th eigenvalue of the matrix pair (A,B), in |
|
* one of the forms lambda = alpha/beta or mu = beta/alpha. |
|
* Since either lambda or mu may overflow, they should not, |
|
* in general, be computed. |
|
* |
|
* Q (input/output) COMPLEX*16 array, dimension (LDQ, N) |
|
* On entry, if COMPZ = 'V', the unitary matrix Q1 used in the |
|
* reduction of (A,B) to generalized Hessenberg form. |
|
* On exit, if COMPZ = 'I', the unitary matrix of left Schur |
|
* vectors of (H,T), and if COMPZ = 'V', the unitary matrix of |
|
* left Schur vectors of (A,B). |
|
* Not referenced if COMPZ = 'N'. |
|
* |
|
* LDQ (input) INTEGER |
|
* The leading dimension of the array Q. LDQ >= 1. |
|
* If COMPQ='V' or 'I', then LDQ >= N. |
|
* |
|
* Z (input/output) COMPLEX*16 array, dimension (LDZ, N) |
|
* On entry, if COMPZ = 'V', the unitary matrix Z1 used in the |
|
* reduction of (A,B) to generalized Hessenberg form. |
|
* On exit, if COMPZ = 'I', the unitary matrix of right Schur |
|
* vectors of (H,T), and if COMPZ = 'V', the unitary matrix of |
|
* right Schur vectors of (A,B). |
|
* Not referenced if COMPZ = 'N'. |
|
* |
|
* LDZ (input) INTEGER |
|
* The leading dimension of the array Z. LDZ >= 1. |
|
* If COMPZ='V' or 'I', then LDZ >= N. |
|
* |
|
* WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK)) |
|
* On exit, if INFO >= 0, WORK(1) returns the optimal LWORK. |
|
* |
|
* LWORK (input) INTEGER |
|
* The dimension of the array WORK. LWORK >= max(1,N). |
|
* |
|
* If LWORK = -1, then a workspace query is assumed; the routine |
|
* only calculates the optimal size of the WORK array, returns |
|
* this value as the first entry of the WORK array, and no error |
|
* message related to LWORK is issued by XERBLA. |
|
* |
|
* RWORK (workspace) DOUBLE PRECISION array, dimension (N) |
|
* |
|
* INFO (output) INTEGER |
|
* = 0: successful exit |
|
* < 0: if INFO = -i, the i-th argument had an illegal value |
|
* = 1,...,N: the QZ iteration did not converge. (H,T) is not |
|
* in Schur form, but ALPHA(i) and BETA(i), |
|
* i=INFO+1,...,N should be correct. |
|
* = N+1,...,2*N: the shift calculation failed. (H,T) is not |
|
* in Schur form, but ALPHA(i) and BETA(i), |
|
* i=INFO-N+1,...,N should be correct. |
|
* |
|
* Further Details |
|
* =============== |
|
* |
|
* We assume that complex ABS works as long as its value is less than |
|
* overflow. |
|
* |
|
* ===================================================================== |
* ===================================================================== |
* |
* |
* .. Parameters .. |
* .. Parameters .. |
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|
Line 454
|
CALL ZSCAL( J-1, SIGNBC, T( 1, J ), 1 ) |
CALL ZSCAL( J-1, SIGNBC, T( 1, J ), 1 ) |
CALL ZSCAL( J, SIGNBC, H( 1, J ), 1 ) |
CALL ZSCAL( J, SIGNBC, H( 1, J ), 1 ) |
ELSE |
ELSE |
H( J, J ) = H( J, J )*SIGNBC |
CALL ZSCAL( 1, SIGNBC, H( J, J ), 1 ) |
END IF |
END IF |
IF( ILZ ) |
IF( ILZ ) |
$ CALL ZSCAL( N, SIGNBC, Z( 1, J ), 1 ) |
$ CALL ZSCAL( N, SIGNBC, Z( 1, J ), 1 ) |
Line 550
|
Line 666
|
CALL ZSCAL( ILAST+1-IFRSTM, SIGNBC, H( IFRSTM, ILAST ), |
CALL ZSCAL( ILAST+1-IFRSTM, SIGNBC, H( IFRSTM, ILAST ), |
$ 1 ) |
$ 1 ) |
ELSE |
ELSE |
H( ILAST, ILAST ) = H( ILAST, ILAST )*SIGNBC |
CALL ZSCAL( 1, SIGNBC, H( ILAST, ILAST ), 1 ) |
END IF |
END IF |
IF( ILZ ) |
IF( ILZ ) |
$ CALL ZSCAL( N, SIGNBC, Z( 1, ILAST ), 1 ) |
$ CALL ZSCAL( N, SIGNBC, Z( 1, ILAST ), 1 ) |
Line 628
|
Line 744
|
* |
* |
* Exceptional shift. Chosen for no particularly good reason. |
* Exceptional shift. Chosen for no particularly good reason. |
* |
* |
ESHIFT = ESHIFT + DCONJG( ( ASCALE*H( ILAST-1, ILAST ) ) / |
ESHIFT = ESHIFT + (ASCALE*H(ILAST,ILAST-1))/ |
$ ( BSCALE*T( ILAST-1, ILAST-1 ) ) ) |
$ (BSCALE*T(ILAST-1,ILAST-1)) |
SHIFT = ESHIFT |
SHIFT = ESHIFT |
END IF |
END IF |
* |
* |
Line 734
|
Line 850
|
CALL ZSCAL( J-1, SIGNBC, T( 1, J ), 1 ) |
CALL ZSCAL( J-1, SIGNBC, T( 1, J ), 1 ) |
CALL ZSCAL( J, SIGNBC, H( 1, J ), 1 ) |
CALL ZSCAL( J, SIGNBC, H( 1, J ), 1 ) |
ELSE |
ELSE |
H( J, J ) = H( J, J )*SIGNBC |
CALL ZSCAL( 1, SIGNBC, H( J, J ), 1 ) |
END IF |
END IF |
IF( ILZ ) |
IF( ILZ ) |
$ CALL ZSCAL( N, SIGNBC, Z( 1, J ), 1 ) |
$ CALL ZSCAL( N, SIGNBC, Z( 1, J ), 1 ) |