version 1.11, 2012/07/31 11:06:35
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version 1.21, 2023/08/07 08:38:52
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* |
* |
* =========== DOCUMENTATION =========== |
* =========== DOCUMENTATION =========== |
* |
* |
* Online html documentation available at |
* Online html documentation available at |
* http://www.netlib.org/lapack/explore-html/ |
* http://www.netlib.org/lapack/explore-html/ |
* |
* |
*> \htmlonly |
*> \htmlonly |
*> Download DHGEQZ + dependencies |
*> Download DHGEQZ + dependencies |
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dhgeqz.f"> |
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dhgeqz.f"> |
*> [TGZ]</a> |
*> [TGZ]</a> |
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dhgeqz.f"> |
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dhgeqz.f"> |
*> [ZIP]</a> |
*> [ZIP]</a> |
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dhgeqz.f"> |
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dhgeqz.f"> |
*> [TXT]</a> |
*> [TXT]</a> |
*> \endhtmlonly |
*> \endhtmlonly |
* |
* |
* Definition: |
* Definition: |
* =========== |
* =========== |
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* SUBROUTINE DHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT, |
* SUBROUTINE DHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT, |
* ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK, |
* ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK, |
* LWORK, INFO ) |
* LWORK, INFO ) |
* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
* CHARACTER COMPQ, COMPZ, JOB |
* CHARACTER COMPQ, COMPZ, JOB |
* INTEGER IHI, ILO, INFO, LDH, LDQ, LDT, LDZ, LWORK, N |
* INTEGER IHI, ILO, INFO, LDH, LDQ, LDT, LDZ, LWORK, N |
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* $ H( LDH, * ), Q( LDQ, * ), T( LDT, * ), |
* $ H( LDH, * ), Q( LDQ, * ), T( LDT, * ), |
* $ WORK( * ), Z( LDZ, * ) |
* $ WORK( * ), Z( LDZ, * ) |
* .. |
* .. |
* |
* |
* |
* |
*> \par Purpose: |
*> \par Purpose: |
* ============= |
* ============= |
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*> |
*> |
*> If JOB='S', then the Hessenberg-triangular pair (H,T) is |
*> If JOB='S', then the Hessenberg-triangular pair (H,T) is |
*> also reduced to generalized Schur form, |
*> also reduced to generalized Schur form, |
*> |
*> |
*> H = Q*S*Z**T, T = Q*P*Z**T, |
*> H = Q*S*Z**T, T = Q*P*Z**T, |
*> |
*> |
*> where Q and Z are orthogonal matrices, P is an upper triangular |
*> where Q and Z are orthogonal matrices, P is an upper triangular |
*> matrix, and S is a quasi-triangular matrix with 1-by-1 and 2-by-2 |
*> matrix, and S is a quasi-triangular matrix with 1-by-1 and 2-by-2 |
*> diagonal blocks. |
*> diagonal blocks. |
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*> generalized Schur factorization of (A,B): |
*> generalized Schur factorization of (A,B): |
*> |
*> |
*> A = (Q1*Q)*S*(Z1*Z)**T, B = (Q1*Q)*P*(Z1*Z)**T. |
*> A = (Q1*Q)*S*(Z1*Z)**T, B = (Q1*Q)*P*(Z1*Z)**T. |
*> |
*> |
*> To avoid overflow, eigenvalues of the matrix pair (H,T) (equivalently, |
*> To avoid overflow, eigenvalues of the matrix pair (H,T) (equivalently, |
*> of (A,B)) are computed as a pair of values (alpha,beta), where alpha is |
*> of (A,B)) are computed as a pair of values (alpha,beta), where alpha is |
*> complex and beta real. |
*> complex and beta real. |
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*> alternate form of the GNEP |
*> alternate form of the GNEP |
*> mu*A*y = B*y. |
*> mu*A*y = B*y. |
*> Real eigenvalues can be read directly from the generalized Schur |
*> Real eigenvalues can be read directly from the generalized Schur |
*> form: |
*> form: |
*> alpha = S(i,i), beta = P(i,i). |
*> alpha = S(i,i), beta = P(i,i). |
*> |
*> |
*> Ref: C.B. Moler & G.W. Stewart, "An Algorithm for Generalized Matrix |
*> Ref: C.B. Moler & G.W. Stewart, "An Algorithm for Generalized Matrix |
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*> \verbatim |
*> \verbatim |
*> JOB is CHARACTER*1 |
*> JOB is CHARACTER*1 |
*> = 'E': Compute eigenvalues only; |
*> = 'E': Compute eigenvalues only; |
*> = 'S': Compute eigenvalues and the Schur form. |
*> = 'S': Compute eigenvalues and the Schur form. |
*> \endverbatim |
*> \endverbatim |
*> |
*> |
*> \param[in] COMPQ |
*> \param[in] COMPQ |
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*> \param[in,out] Q |
*> \param[in,out] Q |
*> \verbatim |
*> \verbatim |
*> Q is DOUBLE PRECISION array, dimension (LDQ, N) |
*> Q is DOUBLE PRECISION array, dimension (LDQ, N) |
*> On entry, if COMPZ = 'V', the orthogonal matrix Q1 used in |
*> On entry, if COMPQ = 'V', the orthogonal matrix Q1 used in |
*> the reduction of (A,B) to generalized Hessenberg form. |
*> the reduction of (A,B) to generalized Hessenberg form. |
*> On exit, if COMPZ = 'I', the orthogonal matrix of left Schur |
*> On exit, if COMPQ = 'I', the orthogonal matrix of left Schur |
*> vectors of (H,T), and if COMPZ = 'V', the orthogonal matrix |
*> vectors of (H,T), and if COMPQ = 'V', the orthogonal matrix |
*> of left Schur vectors of (A,B). |
*> of left Schur vectors of (A,B). |
*> Not referenced if COMPZ = 'N'. |
*> Not referenced if COMPQ = 'N'. |
*> \endverbatim |
*> \endverbatim |
*> |
*> |
*> \param[in] LDQ |
*> \param[in] LDQ |
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* Authors: |
* Authors: |
* ======== |
* ======== |
* |
* |
*> \author Univ. of Tennessee |
*> \author Univ. of Tennessee |
*> \author Univ. of California Berkeley |
*> \author Univ. of California Berkeley |
*> \author Univ. of Colorado Denver |
*> \author Univ. of Colorado Denver |
*> \author NAG Ltd. |
*> \author NAG Ltd. |
* |
|
*> \date April 2012 |
|
* |
* |
*> \ingroup doubleGEcomputational |
*> \ingroup doubleGEcomputational |
* |
* |
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$ ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK, |
$ ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK, |
$ LWORK, INFO ) |
$ LWORK, INFO ) |
* |
* |
* -- LAPACK computational routine (version 3.4.1) -- |
* -- LAPACK computational routine -- |
* -- 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..-- |
* April 2012 |
|
* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
CHARACTER COMPQ, COMPZ, JOB |
CHARACTER COMPQ, COMPZ, JOB |
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* |
* |
GO TO 80 |
GO TO 80 |
ELSE |
ELSE |
IF( ABS( H( ILAST, ILAST-1 ) ).LE.ATOL ) THEN |
IF( ABS( H( ILAST, ILAST-1 ) ).LE.MAX( SAFMIN, ULP*( |
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$ ABS( H( ILAST, ILAST ) ) + ABS( H( ILAST-1, ILAST-1 ) ) |
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$ ) ) ) THEN |
H( ILAST, ILAST-1 ) = ZERO |
H( ILAST, ILAST-1 ) = ZERO |
GO TO 80 |
GO TO 80 |
END IF |
END IF |
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IF( J.EQ.ILO ) THEN |
IF( J.EQ.ILO ) THEN |
ILAZRO = .TRUE. |
ILAZRO = .TRUE. |
ELSE |
ELSE |
IF( ABS( H( J, J-1 ) ).LE.ATOL ) THEN |
IF( ABS( H( J, J-1 ) ).LE.MAX( SAFMIN, ULP*( |
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$ ABS( H( J, J ) ) + ABS( H( J-1, J-1 ) ) |
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$ ) ) ) THEN |
H( J, J-1 ) = ZERO |
H( J, J-1 ) = ZERO |
ILAZRO = .TRUE. |
ILAZRO = .TRUE. |
ELSE |
ELSE |
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* Exceptional shift. Chosen for no particularly good reason. |
* Exceptional shift. Chosen for no particularly good reason. |
* (Single shift only.) |
* (Single shift only.) |
* |
* |
IF( ( DBLE( MAXIT )*SAFMIN )*ABS( H( ILAST-1, ILAST ) ).LT. |
IF( ( DBLE( MAXIT )*SAFMIN )*ABS( H( ILAST, ILAST-1 ) ).LT. |
$ ABS( T( ILAST-1, ILAST-1 ) ) ) THEN |
$ ABS( T( ILAST-1, ILAST-1 ) ) ) THEN |
ESHIFT = ESHIFT + H( ILAST, ILAST-1 ) / |
ESHIFT = H( ILAST, ILAST-1 ) / |
$ T( ILAST-1, ILAST-1 ) |
$ T( ILAST-1, ILAST-1 ) |
ELSE |
ELSE |
ESHIFT = ESHIFT + ONE / ( SAFMIN*DBLE( MAXIT ) ) |
ESHIFT = ESHIFT + ONE / ( SAFMIN*DBLE( MAXIT ) ) |
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$ T( ILAST-1, ILAST-1 ), LDT, SAFMIN*SAFETY, S1, |
$ T( ILAST-1, ILAST-1 ), LDT, SAFMIN*SAFETY, S1, |
$ S2, WR, WR2, WI ) |
$ S2, WR, WR2, WI ) |
* |
* |
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IF ( ABS( (WR/S1)*T( ILAST, ILAST ) - H( ILAST, ILAST ) ) |
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$ .GT. ABS( (WR2/S2)*T( ILAST, ILAST ) |
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$ - H( ILAST, ILAST ) ) ) THEN |
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TEMP = WR |
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WR = WR2 |
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WR2 = TEMP |
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TEMP = S1 |
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S1 = S2 |
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S2 = TEMP |
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END IF |
TEMP = MAX( S1, SAFMIN*MAX( ONE, ABS( WR ), ABS( WI ) ) ) |
TEMP = MAX( S1, SAFMIN*MAX( ONE, ABS( WR ), ABS( WI ) ) ) |
IF( WI.NE.ZERO ) |
IF( WI.NE.ZERO ) |
$ GO TO 200 |
$ GO TO 200 |