version 1.5, 2010/08/07 13:22:32
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version 1.13, 2015/11/26 11:44:23
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*> \brief \b ZGGHRD |
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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 ZGGHRD + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgghrd.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/zgghrd.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/zgghrd.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 ZGGHRD( COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, |
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* LDQ, Z, LDZ, INFO ) |
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* |
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* .. Scalar Arguments .. |
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* CHARACTER COMPQ, COMPZ |
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* INTEGER IHI, ILO, INFO, LDA, LDB, LDQ, LDZ, N |
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* .. |
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* .. Array Arguments .. |
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* COMPLEX*16 A( LDA, * ), B( LDB, * ), Q( LDQ, * ), |
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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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*> ZGGHRD reduces a pair of complex matrices (A,B) to generalized upper |
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*> Hessenberg form using unitary transformations, where A is a |
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*> general matrix and B is upper triangular. The form of the |
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*> generalized eigenvalue problem is |
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*> A*x = lambda*B*x, |
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*> and B is typically made upper triangular by computing its QR |
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*> factorization and moving the unitary matrix Q to the left side |
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*> of the equation. |
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*> |
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*> This subroutine simultaneously reduces A to a Hessenberg matrix H: |
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*> Q**H*A*Z = H |
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*> and transforms B to another upper triangular matrix T: |
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*> Q**H*B*Z = T |
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*> in order to reduce the problem to its standard form |
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*> H*y = lambda*T*y |
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*> where y = Z**H*x. |
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*> |
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*> The unitary matrices Q and Z are determined as products of Givens |
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*> rotations. They may either be formed explicitly, or they may be |
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*> postmultiplied into input matrices Q1 and Z1, so that |
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*> Q1 * A * Z1**H = (Q1*Q) * H * (Z1*Z)**H |
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*> Q1 * B * Z1**H = (Q1*Q) * T * (Z1*Z)**H |
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*> If Q1 is the unitary matrix from the QR factorization of B in the |
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*> original equation A*x = lambda*B*x, then ZGGHRD reduces the original |
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*> problem to generalized Hessenberg form. |
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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] COMPQ |
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*> \verbatim |
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*> COMPQ is CHARACTER*1 |
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*> = 'N': do not compute Q; |
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*> = 'I': Q is initialized to the unit matrix, and the |
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*> unitary matrix Q is returned; |
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*> = 'V': Q must contain a unitary matrix Q1 on entry, |
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*> and 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': do not compute Z; |
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*> = 'I': Z is initialized to the unit matrix, and the |
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*> unitary matrix Z is returned; |
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*> = 'V': Z must contain a unitary matrix Z1 on entry, |
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*> and 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 and B. 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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*> |
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*> ILO and IHI mark the rows and columns of A which are to be |
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*> reduced. It is assumed that A is already upper triangular |
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*> in rows and columns 1:ILO-1 and IHI+1:N. ILO and IHI are |
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*> normally set by a previous call to ZGGBAL; otherwise they |
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*> should be set to 1 and N respectively. |
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*> 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=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 COMPLEX*16 array, dimension (LDA, N) |
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*> On entry, the N-by-N general matrix to be reduced. |
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*> On exit, the upper triangle and the first subdiagonal of A |
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*> are overwritten with the upper Hessenberg matrix H, and the |
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*> rest is set to zero. |
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*> \endverbatim |
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*> |
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*> \param[in] LDA |
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*> \verbatim |
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*> LDA is INTEGER |
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*> The leading dimension of the array A. LDA >= max(1,N). |
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*> \endverbatim |
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*> |
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*> \param[in,out] B |
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*> \verbatim |
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*> B is COMPLEX*16 array, dimension (LDB, N) |
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*> On entry, the N-by-N upper triangular matrix B. |
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*> On exit, the upper triangular matrix T = Q**H B Z. The |
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*> elements below the diagonal are set to zero. |
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*> \endverbatim |
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*> |
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*> \param[in] LDB |
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*> \verbatim |
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*> LDB is INTEGER |
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*> The leading dimension of the array B. LDB >= max(1,N). |
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*> \endverbatim |
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*> |
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*> \param[in,out] Q |
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*> \verbatim |
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*> Q is COMPLEX*16 array, dimension (LDQ, N) |
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*> On entry, if COMPQ = 'V', the unitary matrix Q1, typically |
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*> from the QR factorization of B. |
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*> On exit, if COMPQ='I', the unitary matrix Q, and if |
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*> COMPQ = 'V', the product Q1*Q. |
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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. |
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*> LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise. |
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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. |
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*> On exit, if COMPZ='I', the unitary matrix Z, and if |
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*> COMPZ = 'V', the product Z1*Z. |
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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. |
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*> LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise. |
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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 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 November 2015 |
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* |
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*> \ingroup complex16OTHERcomputational |
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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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*> This routine reduces A to Hessenberg and B to triangular form by |
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*> an unblocked reduction, as described in _Matrix_Computations_, |
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*> by Golub and van Loan (Johns Hopkins Press). |
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*> \endverbatim |
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*> |
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* ===================================================================== |
SUBROUTINE ZGGHRD( COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, |
SUBROUTINE ZGGHRD( COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, |
$ LDQ, Z, LDZ, INFO ) |
$ LDQ, Z, LDZ, 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 |
* November 2015 |
* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
CHARACTER COMPQ, COMPZ |
CHARACTER COMPQ, COMPZ |
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$ Z( LDZ, * ) |
$ Z( LDZ, * ) |
* .. |
* .. |
* |
* |
* Purpose |
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* ======= |
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* |
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* ZGGHRD reduces a pair of complex matrices (A,B) to generalized upper |
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* Hessenberg form using unitary transformations, where A is a |
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* general matrix and B is upper triangular. The form of the |
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* generalized eigenvalue problem is |
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* A*x = lambda*B*x, |
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* and B is typically made upper triangular by computing its QR |
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* factorization and moving the unitary matrix Q to the left side |
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* of the equation. |
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* |
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* This subroutine simultaneously reduces A to a Hessenberg matrix H: |
|
* Q**H*A*Z = H |
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* and transforms B to another upper triangular matrix T: |
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* Q**H*B*Z = T |
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* in order to reduce the problem to its standard form |
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* H*y = lambda*T*y |
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* where y = Z**H*x. |
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* |
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* The unitary matrices Q and Z are determined as products of Givens |
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* rotations. They may either be formed explicitly, or they may be |
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* postmultiplied into input matrices Q1 and Z1, so that |
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* Q1 * A * Z1**H = (Q1*Q) * H * (Z1*Z)**H |
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* Q1 * B * Z1**H = (Q1*Q) * T * (Z1*Z)**H |
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* If Q1 is the unitary matrix from the QR factorization of B in the |
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* original equation A*x = lambda*B*x, then ZGGHRD reduces the original |
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* problem to generalized Hessenberg form. |
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* |
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* Arguments |
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* ========= |
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* |
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* COMPQ (input) CHARACTER*1 |
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* = 'N': do not compute Q; |
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* = 'I': Q is initialized to the unit matrix, and the |
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* unitary matrix Q is returned; |
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* = 'V': Q must contain a unitary matrix Q1 on entry, |
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* and the product Q1*Q is returned. |
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* |
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* COMPZ (input) CHARACTER*1 |
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* = 'N': do not compute Q; |
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* = 'I': Q is initialized to the unit matrix, and the |
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* unitary matrix Q is returned; |
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* = 'V': Q must contain a unitary matrix Q1 on entry, |
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* and the product Q1*Q is returned. |
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* |
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* N (input) INTEGER |
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* The order of the matrices A and B. 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 A which are to be |
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* reduced. It is assumed that A is already upper triangular |
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* in rows and columns 1:ILO-1 and IHI+1:N. ILO and IHI are |
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* normally set by a previous call to ZGGBAL; otherwise they |
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* should be set to 1 and N respectively. |
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* 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0. |
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* |
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* A (input/output) COMPLEX*16 array, dimension (LDA, N) |
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* On entry, the N-by-N general matrix to be reduced. |
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* On exit, the upper triangle and the first subdiagonal of A |
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* are overwritten with the upper Hessenberg matrix H, and the |
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* rest is set to zero. |
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* |
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* LDA (input) INTEGER |
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* The leading dimension of the array A. LDA >= max(1,N). |
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* |
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* B (input/output) COMPLEX*16 array, dimension (LDB, N) |
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* On entry, the N-by-N upper triangular matrix B. |
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* On exit, the upper triangular matrix T = Q**H B Z. The |
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* elements below the diagonal are set to zero. |
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* |
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* LDB (input) INTEGER |
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* The leading dimension of the array B. LDB >= max(1,N). |
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* |
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* Q (input/output) COMPLEX*16 array, dimension (LDQ, N) |
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* On entry, if COMPQ = 'V', the unitary matrix Q1, typically |
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* from the QR factorization of B. |
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* On exit, if COMPQ='I', the unitary matrix Q, and if |
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* COMPQ = 'V', the product Q1*Q. |
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* Not referenced if COMPQ='N'. |
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* |
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* LDQ (input) INTEGER |
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* The leading dimension of the array Q. |
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* LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise. |
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* |
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* Z (input/output) COMPLEX*16 array, dimension (LDZ, N) |
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* On entry, if COMPZ = 'V', the unitary matrix Z1. |
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* On exit, if COMPZ='I', the unitary matrix Z, and if |
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* COMPZ = 'V', the product Z1*Z. |
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* Not referenced if COMPZ='N'. |
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* |
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* LDZ (input) INTEGER |
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* The leading dimension of the array Z. |
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* LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise. |
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* |
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* INFO (output) 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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* |
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* Further Details |
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* =============== |
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* |
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* This routine reduces A to Hessenberg and B to triangular form by |
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* an unblocked reduction, as described in _Matrix_Computations_, |
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* by Golub and van Loan (Johns Hopkins Press). |
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* |
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* ===================================================================== |
* ===================================================================== |
* |
* |
* .. Parameters .. |
* .. Parameters .. |