version 1.1.1.1, 2010/01/26 15:22:45
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version 1.12, 2012/12/14 14:22:30
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*> \brief <b> DGGEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices</b> |
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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 DGGEV + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dggev.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/dggev.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/dggev.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 DGGEV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI, |
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* BETA, VL, LDVL, VR, LDVR, WORK, LWORK, INFO ) |
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* |
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* .. Scalar Arguments .. |
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* CHARACTER JOBVL, JOBVR |
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* INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N |
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* .. |
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* .. Array Arguments .. |
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* DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ), |
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* $ B( LDB, * ), BETA( * ), VL( LDVL, * ), |
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* $ VR( LDVR, * ), WORK( * ) |
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* .. |
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* |
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* |
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*> \par Purpose: |
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* ============= |
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*> |
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*> \verbatim |
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*> |
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*> DGGEV computes for a pair of N-by-N real nonsymmetric matrices (A,B) |
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*> the generalized eigenvalues, and optionally, the left and/or right |
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*> generalized eigenvectors. |
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*> |
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*> A generalized eigenvalue for a pair of matrices (A,B) is a scalar |
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*> lambda or a ratio alpha/beta = lambda, such that A - lambda*B is |
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*> singular. It is usually represented as the pair (alpha,beta), as |
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*> there is a reasonable interpretation for beta=0, and even for both |
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*> being zero. |
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*> |
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*> The right eigenvector v(j) corresponding to the eigenvalue lambda(j) |
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*> of (A,B) satisfies |
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*> |
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*> A * v(j) = lambda(j) * B * v(j). |
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*> |
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*> The left eigenvector u(j) corresponding to the eigenvalue lambda(j) |
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*> of (A,B) satisfies |
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*> |
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*> u(j)**H * A = lambda(j) * u(j)**H * B . |
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*> |
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*> where u(j)**H is the conjugate-transpose of u(j). |
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*> |
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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] JOBVL |
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*> \verbatim |
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*> JOBVL is CHARACTER*1 |
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*> = 'N': do not compute the left generalized eigenvectors; |
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*> = 'V': compute the left generalized eigenvectors. |
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*> \endverbatim |
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*> |
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*> \param[in] JOBVR |
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*> \verbatim |
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*> JOBVR is CHARACTER*1 |
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*> = 'N': do not compute the right generalized eigenvectors; |
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*> = 'V': compute the right generalized eigenvectors. |
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*> \endverbatim |
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*> |
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*> \param[in] N |
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*> \verbatim |
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*> N is INTEGER |
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*> The order of the matrices A, B, VL, and VR. 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 matrix A in the pair (A,B). |
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*> On exit, A has been overwritten. |
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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 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 matrix B in the pair (A,B). |
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*> On exit, B has been overwritten. |
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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 B. LDB >= max(1,N). |
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*> \endverbatim |
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*> |
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*> \param[out] ALPHAR |
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*> \verbatim |
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*> ALPHAR is DOUBLE PRECISION array, dimension (N) |
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*> \endverbatim |
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*> |
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*> \param[out] ALPHAI |
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*> \verbatim |
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*> ALPHAI is DOUBLE PRECISION array, dimension (N) |
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*> \endverbatim |
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*> |
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*> \param[out] BETA |
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*> \verbatim |
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*> BETA is DOUBLE PRECISION array, dimension (N) |
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*> On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will |
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*> be the generalized eigenvalues. If ALPHAI(j) is zero, then |
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*> the j-th eigenvalue is real; if positive, then the j-th and |
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*> (j+1)-st eigenvalues are a complex conjugate pair, with |
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*> ALPHAI(j+1) negative. |
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*> |
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*> Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j) |
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*> may easily over- or underflow, and BETA(j) may even be zero. |
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*> Thus, the user should avoid naively computing the ratio |
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*> alpha/beta. However, ALPHAR and ALPHAI will be always less |
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*> than and usually comparable with norm(A) in magnitude, and |
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*> BETA always less than and usually comparable with norm(B). |
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*> \endverbatim |
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*> |
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*> \param[out] VL |
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*> \verbatim |
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*> VL is DOUBLE PRECISION array, dimension (LDVL,N) |
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*> If JOBVL = 'V', the left eigenvectors u(j) are stored one |
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*> after another in the columns of VL, in the same order as |
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*> their eigenvalues. If the j-th eigenvalue is real, then |
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*> u(j) = VL(:,j), the j-th column of VL. If the j-th and |
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*> (j+1)-th eigenvalues form a complex conjugate pair, then |
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*> u(j) = VL(:,j)+i*VL(:,j+1) and u(j+1) = VL(:,j)-i*VL(:,j+1). |
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*> Each eigenvector is scaled so the largest component has |
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*> abs(real part)+abs(imag. part)=1. |
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*> Not referenced if JOBVL = 'N'. |
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*> \endverbatim |
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*> |
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*> \param[in] LDVL |
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*> \verbatim |
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*> LDVL is INTEGER |
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*> The leading dimension of the matrix VL. LDVL >= 1, and |
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*> if JOBVL = 'V', LDVL >= N. |
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*> \endverbatim |
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*> |
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*> \param[out] VR |
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*> \verbatim |
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*> VR is DOUBLE PRECISION array, dimension (LDVR,N) |
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*> If JOBVR = 'V', the right eigenvectors v(j) are stored one |
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*> after another in the columns of VR, in the same order as |
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*> their eigenvalues. If the j-th eigenvalue is real, then |
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*> v(j) = VR(:,j), the j-th column of VR. If the j-th and |
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*> (j+1)-th eigenvalues form a complex conjugate pair, then |
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*> v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1). |
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*> Each eigenvector is scaled so the largest component has |
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*> abs(real part)+abs(imag. part)=1. |
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*> Not referenced if JOBVR = 'N'. |
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*> \endverbatim |
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*> |
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*> \param[in] LDVR |
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*> \verbatim |
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*> LDVR is INTEGER |
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*> The leading dimension of the matrix VR. LDVR >= 1, and |
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*> if JOBVR = 'V', LDVR >= N. |
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*> \endverbatim |
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*> |
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*> \param[out] WORK |
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*> \verbatim |
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*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)) |
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*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. |
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*> \endverbatim |
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*> |
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*> \param[in] LWORK |
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*> \verbatim |
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*> LWORK is INTEGER |
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*> The dimension of the array WORK. LWORK >= max(1,8*N). |
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*> For good performance, LWORK must generally be larger. |
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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] 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: |
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*> The QZ iteration failed. No eigenvectors have been |
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*> calculated, but ALPHAR(j), ALPHAI(j), and BETA(j) |
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*> should be correct for j=INFO+1,...,N. |
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*> > N: =N+1: other than QZ iteration failed in DHGEQZ. |
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*> =N+2: error return from DTGEVC. |
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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 doubleGEeigen |
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* |
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* ===================================================================== |
SUBROUTINE DGGEV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI, |
SUBROUTINE DGGEV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI, |
$ BETA, VL, LDVL, VR, LDVR, WORK, LWORK, INFO ) |
$ BETA, VL, LDVL, VR, LDVR, WORK, LWORK, INFO ) |
* |
* |
* -- LAPACK driver routine (version 3.2) -- |
* -- LAPACK driver routine (version 3.4.1) -- |
* -- 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 JOBVL, JOBVR |
CHARACTER JOBVL, JOBVR |
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$ VR( LDVR, * ), WORK( * ) |
$ VR( LDVR, * ), WORK( * ) |
* .. |
* .. |
* |
* |
* Purpose |
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* ======= |
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* |
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* DGGEV computes for a pair of N-by-N real nonsymmetric matrices (A,B) |
|
* the generalized eigenvalues, and optionally, the left and/or right |
|
* generalized eigenvectors. |
|
* |
|
* A generalized eigenvalue for a pair of matrices (A,B) is a scalar |
|
* lambda or a ratio alpha/beta = lambda, such that A - lambda*B is |
|
* singular. It is usually represented as the pair (alpha,beta), as |
|
* there is a reasonable interpretation for beta=0, and even for both |
|
* being zero. |
|
* |
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* The right eigenvector v(j) corresponding to the eigenvalue lambda(j) |
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* of (A,B) satisfies |
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* |
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* A * v(j) = lambda(j) * B * v(j). |
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* |
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* The left eigenvector u(j) corresponding to the eigenvalue lambda(j) |
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* of (A,B) satisfies |
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* |
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* u(j)**H * A = lambda(j) * u(j)**H * B . |
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* |
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* where u(j)**H is the conjugate-transpose of u(j). |
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* |
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* |
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* Arguments |
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* ========= |
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* |
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* JOBVL (input) CHARACTER*1 |
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* = 'N': do not compute the left generalized eigenvectors; |
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* = 'V': compute the left generalized eigenvectors. |
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* |
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* JOBVR (input) CHARACTER*1 |
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* = 'N': do not compute the right generalized eigenvectors; |
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* = 'V': compute the right generalized eigenvectors. |
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* |
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* N (input) INTEGER |
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* The order of the matrices A, B, VL, and VR. N >= 0. |
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* |
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* A (input/output) DOUBLE PRECISION array, dimension (LDA, N) |
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* On entry, the matrix A in the pair (A,B). |
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* On exit, A has been overwritten. |
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* |
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* LDA (input) INTEGER |
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* The leading dimension of A. LDA >= max(1,N). |
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* |
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* B (input/output) DOUBLE PRECISION array, dimension (LDB, N) |
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* On entry, the matrix B in the pair (A,B). |
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* On exit, B has been overwritten. |
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* |
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* LDB (input) INTEGER |
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* The leading dimension of B. LDB >= max(1,N). |
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* |
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* ALPHAR (output) DOUBLE PRECISION array, dimension (N) |
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* ALPHAI (output) DOUBLE PRECISION array, dimension (N) |
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* BETA (output) DOUBLE PRECISION array, dimension (N) |
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* On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will |
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* be the generalized eigenvalues. If ALPHAI(j) is zero, then |
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* the j-th eigenvalue is real; if positive, then the j-th and |
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* (j+1)-st eigenvalues are a complex conjugate pair, with |
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* ALPHAI(j+1) negative. |
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* |
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* Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j) |
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* may easily over- or underflow, and BETA(j) may even be zero. |
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* Thus, the user should avoid naively computing the ratio |
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* alpha/beta. However, ALPHAR and ALPHAI will be always less |
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* than and usually comparable with norm(A) in magnitude, and |
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* BETA always less than and usually comparable with norm(B). |
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* |
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* VL (output) DOUBLE PRECISION array, dimension (LDVL,N) |
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* If JOBVL = 'V', the left eigenvectors u(j) are stored one |
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* after another in the columns of VL, in the same order as |
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* their eigenvalues. If the j-th eigenvalue is real, then |
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* u(j) = VL(:,j), the j-th column of VL. If the j-th and |
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* (j+1)-th eigenvalues form a complex conjugate pair, then |
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* u(j) = VL(:,j)+i*VL(:,j+1) and u(j+1) = VL(:,j)-i*VL(:,j+1). |
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* Each eigenvector is scaled so the largest component has |
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* abs(real part)+abs(imag. part)=1. |
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* Not referenced if JOBVL = 'N'. |
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* |
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* LDVL (input) INTEGER |
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* The leading dimension of the matrix VL. LDVL >= 1, and |
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* if JOBVL = 'V', LDVL >= N. |
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* |
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* VR (output) DOUBLE PRECISION array, dimension (LDVR,N) |
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* If JOBVR = 'V', the right eigenvectors v(j) are stored one |
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* after another in the columns of VR, in the same order as |
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* their eigenvalues. If the j-th eigenvalue is real, then |
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* v(j) = VR(:,j), the j-th column of VR. If the j-th and |
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* (j+1)-th eigenvalues form a complex conjugate pair, then |
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* v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1). |
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* Each eigenvector is scaled so the largest component has |
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* abs(real part)+abs(imag. part)=1. |
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* Not referenced if JOBVR = 'N'. |
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* |
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* LDVR (input) INTEGER |
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* The leading dimension of the matrix VR. LDVR >= 1, and |
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* if JOBVR = 'V', LDVR >= N. |
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* |
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* WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK)) |
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* On exit, if INFO = 0, WORK(1) returns the optimal LWORK. |
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* |
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* LWORK (input) INTEGER |
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* The dimension of the array WORK. LWORK >= max(1,8*N). |
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* For good performance, LWORK must generally be larger. |
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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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* |
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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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* = 1,...,N: |
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* The QZ iteration failed. No eigenvectors have been |
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* calculated, but ALPHAR(j), ALPHAI(j), and BETA(j) |
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* should be correct for j=INFO+1,...,N. |
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* > N: =N+1: other than QZ iteration failed in DHGEQZ. |
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* =N+2: error return from DTGEVC. |
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* |
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* ===================================================================== |
* ===================================================================== |
* |
* |
* .. Parameters .. |
* .. Parameters .. |
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* |
* |
* Undo scaling if necessary |
* Undo scaling if necessary |
* |
* |
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110 CONTINUE |
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* |
IF( ILASCL ) THEN |
IF( ILASCL ) THEN |
CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAR, N, IERR ) |
CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAR, N, IERR ) |
CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAI, N, IERR ) |
CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAI, N, IERR ) |
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CALL DLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR ) |
CALL DLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR ) |
END IF |
END IF |
* |
* |
110 CONTINUE |
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* |
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WORK( 1 ) = MAXWRK |
WORK( 1 ) = MAXWRK |
* |
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RETURN |
RETURN |
* |
* |
* End of DGGEV |
* End of DGGEV |