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359 lines
10 KiB
359 lines
10 KiB
2 years ago
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*> \brief \b DGGHRD
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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 DGGHRD + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgghrd.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/dgghrd.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/dgghrd.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 DGGHRD( 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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* DOUBLE PRECISION 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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*> DGGHRD reduces a pair of real matrices (A,B) to generalized upper
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*> Hessenberg form using orthogonal 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 orthogonal 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**T*A*Z = H
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*> and transforms B to another upper triangular matrix T:
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*> Q**T*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**T*x.
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*>
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*> The orthogonal 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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*>
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*> Q1 * A * Z1**T = (Q1*Q) * H * (Z1*Z)**T
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*>
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*> Q1 * B * Z1**T = (Q1*Q) * T * (Z1*Z)**T
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*>
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*> If Q1 is the orthogonal matrix from the QR factorization of B in the
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*> original equation A*x = lambda*B*x, then DGGHRD 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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*> orthogonal matrix Q is returned;
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*> = 'V': Q must contain an orthogonal 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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*> orthogonal matrix Z is returned;
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*> = 'V': Z must contain an orthogonal 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 DGGBAL; 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 DOUBLE PRECISION 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 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, the upper triangular matrix T = Q**T 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 DOUBLE PRECISION array, dimension (LDQ, N)
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*> On entry, if COMPQ = 'V', the orthogonal matrix Q1,
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*> typically from the QR factorization of B.
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*> On exit, if COMPQ='I', the orthogonal 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 DOUBLE PRECISION array, dimension (LDZ, N)
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*> On entry, if COMPZ = 'V', the orthogonal matrix Z1.
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*> On exit, if COMPZ='I', the orthogonal 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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*> \ingroup doubleOTHERcomputational
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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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* =====================================================================
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SUBROUTINE DGGHRD( 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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* -- LAPACK computational routine --
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* -- LAPACK is a software package provided by Univ. of Tennessee, --
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* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
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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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DOUBLE PRECISION 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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*
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* .. Parameters ..
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DOUBLE PRECISION ONE, ZERO
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PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
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* ..
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* .. Local Scalars ..
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LOGICAL ILQ, ILZ
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INTEGER ICOMPQ, ICOMPZ, JCOL, JROW
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DOUBLE PRECISION C, S, TEMP
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* ..
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* .. External Functions ..
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LOGICAL LSAME
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EXTERNAL LSAME
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* ..
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* .. External Subroutines ..
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EXTERNAL DLARTG, DLASET, DROT, XERBLA
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC MAX
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* ..
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* .. Executable Statements ..
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*
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* Decode COMPQ
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*
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IF( LSAME( COMPQ, 'N' ) ) THEN
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ILQ = .FALSE.
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ICOMPQ = 1
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ELSE IF( LSAME( COMPQ, 'V' ) ) THEN
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ILQ = .TRUE.
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ICOMPQ = 2
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ELSE IF( LSAME( COMPQ, 'I' ) ) THEN
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ILQ = .TRUE.
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ICOMPQ = 3
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ELSE
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ICOMPQ = 0
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END IF
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*
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* Decode COMPZ
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*
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IF( LSAME( COMPZ, 'N' ) ) THEN
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ILZ = .FALSE.
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ICOMPZ = 1
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ELSE IF( LSAME( COMPZ, 'V' ) ) THEN
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ILZ = .TRUE.
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ICOMPZ = 2
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ELSE IF( LSAME( COMPZ, 'I' ) ) THEN
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ILZ = .TRUE.
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ICOMPZ = 3
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ELSE
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ICOMPZ = 0
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END IF
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*
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* Test the input parameters.
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*
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INFO = 0
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IF( ICOMPQ.LE.0 ) THEN
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INFO = -1
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ELSE IF( ICOMPZ.LE.0 ) THEN
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INFO = -2
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ELSE IF( N.LT.0 ) THEN
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INFO = -3
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ELSE IF( ILO.LT.1 ) THEN
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INFO = -4
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ELSE IF( IHI.GT.N .OR. IHI.LT.ILO-1 ) THEN
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INFO = -5
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ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
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INFO = -7
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ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
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INFO = -9
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ELSE IF( ( ILQ .AND. LDQ.LT.N ) .OR. LDQ.LT.1 ) THEN
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INFO = -11
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ELSE IF( ( ILZ .AND. LDZ.LT.N ) .OR. LDZ.LT.1 ) THEN
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INFO = -13
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END IF
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'DGGHRD', -INFO )
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RETURN
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END IF
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*
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* Initialize Q and Z if desired.
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*
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IF( ICOMPQ.EQ.3 )
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$ CALL DLASET( 'Full', N, N, ZERO, ONE, Q, LDQ )
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IF( ICOMPZ.EQ.3 )
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$ CALL DLASET( 'Full', N, N, ZERO, ONE, Z, LDZ )
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*
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* Quick return if possible
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*
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IF( N.LE.1 )
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$ RETURN
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*
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* Zero out lower triangle of B
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*
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DO 20 JCOL = 1, N - 1
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DO 10 JROW = JCOL + 1, N
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B( JROW, JCOL ) = ZERO
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10 CONTINUE
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20 CONTINUE
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*
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* Reduce A and B
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*
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DO 40 JCOL = ILO, IHI - 2
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*
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DO 30 JROW = IHI, JCOL + 2, -1
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*
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* Step 1: rotate rows JROW-1, JROW to kill A(JROW,JCOL)
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*
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TEMP = A( JROW-1, JCOL )
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CALL DLARTG( TEMP, A( JROW, JCOL ), C, S,
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$ A( JROW-1, JCOL ) )
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A( JROW, JCOL ) = ZERO
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CALL DROT( N-JCOL, A( JROW-1, JCOL+1 ), LDA,
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$ A( JROW, JCOL+1 ), LDA, C, S )
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CALL DROT( N+2-JROW, B( JROW-1, JROW-1 ), LDB,
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$ B( JROW, JROW-1 ), LDB, C, S )
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IF( ILQ )
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$ CALL DROT( N, Q( 1, JROW-1 ), 1, Q( 1, JROW ), 1, C, S )
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*
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* Step 2: rotate columns JROW, JROW-1 to kill B(JROW,JROW-1)
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*
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TEMP = B( JROW, JROW )
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CALL DLARTG( TEMP, B( JROW, JROW-1 ), C, S,
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$ B( JROW, JROW ) )
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B( JROW, JROW-1 ) = ZERO
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CALL DROT( IHI, A( 1, JROW ), 1, A( 1, JROW-1 ), 1, C, S )
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CALL DROT( JROW-1, B( 1, JROW ), 1, B( 1, JROW-1 ), 1, C,
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$ S )
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IF( ILZ )
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$ CALL DROT( N, Z( 1, JROW ), 1, Z( 1, JROW-1 ), 1, C, S )
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30 CONTINUE
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40 CONTINUE
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*
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RETURN
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*
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* End of DGGHRD
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*
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END
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