version 1.8, 2011/07/22 07:38:04
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version 1.19, 2018/05/29 07:17:51
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*> \brief <b> DGEEV 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 DGEEV + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgeev.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/dgeev.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/dgeev.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 DGEEV( JOBVL, JOBVR, N, A, LDA, WR, WI, VL, LDVL, VR, |
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* 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, LDVL, LDVR, LWORK, N |
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* .. |
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* .. Array Arguments .. |
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* DOUBLE PRECISION A( LDA, * ), VL( LDVL, * ), VR( LDVR, * ), |
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* $ WI( * ), WORK( * ), WR( * ) |
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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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*> DGEEV computes for an N-by-N real nonsymmetric matrix A, the |
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*> eigenvalues and, optionally, the left and/or right eigenvectors. |
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*> |
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*> The right eigenvector v(j) of A satisfies |
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*> A * v(j) = lambda(j) * v(j) |
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*> where lambda(j) is its eigenvalue. |
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*> The left eigenvector u(j) of A satisfies |
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*> u(j)**H * A = lambda(j) * u(j)**H |
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*> where u(j)**H denotes the conjugate-transpose of u(j). |
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*> |
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*> The computed eigenvectors are normalized to have Euclidean norm |
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*> equal to 1 and largest component real. |
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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': left eigenvectors of A are not computed; |
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*> = 'V': left eigenvectors of A are computed. |
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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': right eigenvectors of A are not computed; |
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*> = 'V': right eigenvectors of A are computed. |
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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 matrix A. 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 matrix A. |
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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 the array A. LDA >= max(1,N). |
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*> \endverbatim |
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*> |
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*> \param[out] WR |
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*> \verbatim |
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*> WR is DOUBLE PRECISION array, dimension (N) |
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*> \endverbatim |
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*> |
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*> \param[out] WI |
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*> \verbatim |
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*> WI is DOUBLE PRECISION array, dimension (N) |
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*> WR and WI contain the real and imaginary parts, |
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*> respectively, of the computed eigenvalues. Complex |
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*> conjugate pairs of eigenvalues appear consecutively |
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*> with the eigenvalue having the positive imaginary part |
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*> first. |
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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 |
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*> as their eigenvalues. |
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*> If JOBVL = 'N', VL is not referenced. |
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*> If the j-th eigenvalue is real, then u(j) = VL(:,j), |
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*> the j-th column of VL. |
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*> If the j-th and (j+1)-st eigenvalues form a complex |
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*> conjugate pair, then u(j) = VL(:,j) + i*VL(:,j+1) and |
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*> u(j+1) = VL(:,j) - i*VL(:,j+1). |
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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 array VL. LDVL >= 1; if |
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*> 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 |
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*> as their eigenvalues. |
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*> If JOBVR = 'N', VR is not referenced. |
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*> If the j-th eigenvalue is real, then v(j) = VR(:,j), |
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*> the j-th column of VR. |
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*> If the j-th and (j+1)-st eigenvalues form a complex |
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*> conjugate pair, then v(j) = VR(:,j) + i*VR(:,j+1) and |
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*> v(j+1) = VR(:,j) - i*VR(:,j+1). |
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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 array VR. LDVR >= 1; if |
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*> 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(1,3*N), and |
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*> if JOBVL = 'V' or JOBVR = 'V', LWORK >= 4*N. For good |
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*> 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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*> > 0: if INFO = i, the QR algorithm failed to compute all the |
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*> eigenvalues, and no eigenvectors have been computed; |
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*> elements i+1:N of WR and WI contain eigenvalues which |
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*> have converged. |
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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 June 2016 |
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* |
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* @precisions fortran d -> s |
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* |
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*> \ingroup doubleGEeigen |
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* |
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* ===================================================================== |
SUBROUTINE DGEEV( JOBVL, JOBVR, N, A, LDA, WR, WI, VL, LDVL, VR, |
SUBROUTINE DGEEV( JOBVL, JOBVR, N, A, LDA, WR, WI, VL, LDVL, VR, |
$ LDVR, WORK, LWORK, INFO ) |
$ LDVR, WORK, LWORK, INFO ) |
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implicit none |
* |
* |
* -- LAPACK driver routine (version 3.3.1) -- |
* -- LAPACK driver routine (version 3.7.0) -- |
* -- 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..-- |
* -- April 2011 -- |
* June 2016 |
* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
CHARACTER JOBVL, JOBVR |
CHARACTER JOBVL, JOBVR |
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$ WI( * ), WORK( * ), WR( * ) |
$ WI( * ), WORK( * ), WR( * ) |
* .. |
* .. |
* |
* |
* Purpose |
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* ======= |
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* |
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* DGEEV computes for an N-by-N real nonsymmetric matrix A, the |
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* eigenvalues and, optionally, the left and/or right eigenvectors. |
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* |
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* The right eigenvector v(j) of A satisfies |
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* A * v(j) = lambda(j) * v(j) |
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* where lambda(j) is its eigenvalue. |
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* The left eigenvector u(j) of A satisfies |
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* u(j)**T * A = lambda(j) * u(j)**T |
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* where u(j)**T denotes the transpose of u(j). |
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* |
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* The computed eigenvectors are normalized to have Euclidean norm |
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* equal to 1 and largest component real. |
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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': left eigenvectors of A are not computed; |
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* = 'V': left eigenvectors of A are computed. |
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* |
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* JOBVR (input) CHARACTER*1 |
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* = 'N': right eigenvectors of A are not computed; |
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* = 'V': right eigenvectors of A are computed. |
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* |
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* N (input) INTEGER |
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* The order of the matrix A. 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 N-by-N matrix A. |
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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 the array A. LDA >= max(1,N). |
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* |
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* WR (output) DOUBLE PRECISION array, dimension (N) |
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* WI (output) DOUBLE PRECISION array, dimension (N) |
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* WR and WI contain the real and imaginary parts, |
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* respectively, of the computed eigenvalues. Complex |
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* conjugate pairs of eigenvalues appear consecutively |
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* with the eigenvalue having the positive imaginary part |
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* first. |
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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 |
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* as their eigenvalues. |
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* If JOBVL = 'N', VL is not referenced. |
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* If the j-th eigenvalue is real, then u(j) = VL(:,j), |
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* the j-th column of VL. |
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* If the j-th and (j+1)-st eigenvalues form a complex |
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* conjugate pair, then u(j) = VL(:,j) + i*VL(:,j+1) and |
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* u(j+1) = VL(:,j) - i*VL(:,j+1). |
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* |
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* LDVL (input) INTEGER |
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* The leading dimension of the array VL. LDVL >= 1; if |
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* 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 |
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* as their eigenvalues. |
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* If JOBVR = 'N', VR is not referenced. |
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* If the j-th eigenvalue is real, then v(j) = VR(:,j), |
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* the j-th column of VR. |
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* If the j-th and (j+1)-st eigenvalues form a complex |
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* conjugate pair, then v(j) = VR(:,j) + i*VR(:,j+1) and |
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* v(j+1) = VR(:,j) - i*VR(:,j+1). |
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* |
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* LDVR (input) INTEGER |
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* The leading dimension of the array VR. LDVR >= 1; if |
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* 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(1,3*N), and |
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* if JOBVL = 'V' or JOBVR = 'V', LWORK >= 4*N. For good |
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* 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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* > 0: if INFO = i, the QR algorithm failed to compute all the |
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* eigenvalues, and no eigenvectors have been computed; |
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* elements i+1:N of WR and WI contain eigenvalues which |
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* have converged. |
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* |
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* ===================================================================== |
* ===================================================================== |
* |
* |
* .. Parameters .. |
* .. Parameters .. |
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LOGICAL LQUERY, SCALEA, WANTVL, WANTVR |
LOGICAL LQUERY, SCALEA, WANTVL, WANTVR |
CHARACTER SIDE |
CHARACTER SIDE |
INTEGER HSWORK, I, IBAL, IERR, IHI, ILO, ITAU, IWRK, K, |
INTEGER HSWORK, I, IBAL, IERR, IHI, ILO, ITAU, IWRK, K, |
$ MAXWRK, MINWRK, NOUT |
$ LWORK_TREVC, MAXWRK, MINWRK, NOUT |
DOUBLE PRECISION ANRM, BIGNUM, CS, CSCALE, EPS, R, SCL, SMLNUM, |
DOUBLE PRECISION ANRM, BIGNUM, CS, CSCALE, EPS, R, SCL, SMLNUM, |
$ SN |
$ SN |
* .. |
* .. |
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* .. |
* .. |
* .. External Subroutines .. |
* .. External Subroutines .. |
EXTERNAL DGEBAK, DGEBAL, DGEHRD, DHSEQR, DLABAD, DLACPY, |
EXTERNAL DGEBAK, DGEBAL, DGEHRD, DHSEQR, DLABAD, DLACPY, |
$ DLARTG, DLASCL, DORGHR, DROT, DSCAL, DTREVC, |
$ DLARTG, DLASCL, DORGHR, DROT, DSCAL, DTREVC3, |
$ XERBLA |
$ XERBLA |
* .. |
* .. |
* .. External Functions .. |
* .. External Functions .. |
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MAXWRK = MAX( MAXWRK, 2*N + ( N - 1 )*ILAENV( 1, |
MAXWRK = MAX( MAXWRK, 2*N + ( N - 1 )*ILAENV( 1, |
$ 'DORGHR', ' ', N, 1, N, -1 ) ) |
$ 'DORGHR', ' ', N, 1, N, -1 ) ) |
CALL DHSEQR( 'S', 'V', N, 1, N, A, LDA, WR, WI, VL, LDVL, |
CALL DHSEQR( 'S', 'V', N, 1, N, A, LDA, WR, WI, VL, LDVL, |
$ WORK, -1, INFO ) |
$ WORK, -1, INFO ) |
HSWORK = WORK( 1 ) |
HSWORK = INT( WORK(1) ) |
MAXWRK = MAX( MAXWRK, N + 1, N + HSWORK ) |
MAXWRK = MAX( MAXWRK, N + 1, N + HSWORK ) |
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CALL DTREVC3( 'L', 'B', SELECT, N, A, LDA, |
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$ VL, LDVL, VR, LDVR, N, NOUT, |
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$ WORK, -1, IERR ) |
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LWORK_TREVC = INT( WORK(1) ) |
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MAXWRK = MAX( MAXWRK, N + LWORK_TREVC ) |
MAXWRK = MAX( MAXWRK, 4*N ) |
MAXWRK = MAX( MAXWRK, 4*N ) |
ELSE IF( WANTVR ) THEN |
ELSE IF( WANTVR ) THEN |
MINWRK = 4*N |
MINWRK = 4*N |
MAXWRK = MAX( MAXWRK, 2*N + ( N - 1 )*ILAENV( 1, |
MAXWRK = MAX( MAXWRK, 2*N + ( N - 1 )*ILAENV( 1, |
$ 'DORGHR', ' ', N, 1, N, -1 ) ) |
$ 'DORGHR', ' ', N, 1, N, -1 ) ) |
CALL DHSEQR( 'S', 'V', N, 1, N, A, LDA, WR, WI, VR, LDVR, |
CALL DHSEQR( 'S', 'V', N, 1, N, A, LDA, WR, WI, VR, LDVR, |
$ WORK, -1, INFO ) |
$ WORK, -1, INFO ) |
HSWORK = WORK( 1 ) |
HSWORK = INT( WORK(1) ) |
MAXWRK = MAX( MAXWRK, N + 1, N + HSWORK ) |
MAXWRK = MAX( MAXWRK, N + 1, N + HSWORK ) |
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CALL DTREVC3( 'R', 'B', SELECT, N, A, LDA, |
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$ VL, LDVL, VR, LDVR, N, NOUT, |
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$ WORK, -1, IERR ) |
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LWORK_TREVC = INT( WORK(1) ) |
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MAXWRK = MAX( MAXWRK, N + LWORK_TREVC ) |
MAXWRK = MAX( MAXWRK, 4*N ) |
MAXWRK = MAX( MAXWRK, 4*N ) |
ELSE |
ELSE |
MINWRK = 3*N |
MINWRK = 3*N |
CALL DHSEQR( 'E', 'N', N, 1, N, A, LDA, WR, WI, VR, LDVR, |
CALL DHSEQR( 'E', 'N', N, 1, N, A, LDA, WR, WI, VR, LDVR, |
$ WORK, -1, INFO ) |
$ WORK, -1, INFO ) |
HSWORK = WORK( 1 ) |
HSWORK = INT( WORK(1) ) |
MAXWRK = MAX( MAXWRK, N + 1, N + HSWORK ) |
MAXWRK = MAX( MAXWRK, N + 1, N + HSWORK ) |
END IF |
END IF |
MAXWRK = MAX( MAXWRK, MINWRK ) |
MAXWRK = MAX( MAXWRK, MINWRK ) |
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$ WORK( IWRK ), LWORK-IWRK+1, INFO ) |
$ WORK( IWRK ), LWORK-IWRK+1, INFO ) |
END IF |
END IF |
* |
* |
* If INFO > 0 from DHSEQR, then quit |
* If INFO .NE. 0 from DHSEQR, then quit |
* |
* |
IF( INFO.GT.0 ) |
IF( INFO.NE.0 ) |
$ GO TO 50 |
$ GO TO 50 |
* |
* |
IF( WANTVL .OR. WANTVR ) THEN |
IF( WANTVL .OR. WANTVR ) THEN |
* |
* |
* Compute left and/or right eigenvectors |
* Compute left and/or right eigenvectors |
* (Workspace: need 4*N) |
* (Workspace: need 4*N, prefer N + N + 2*N*NB) |
* |
* |
CALL DTREVC( SIDE, 'B', SELECT, N, A, LDA, VL, LDVL, VR, LDVR, |
CALL DTREVC3( SIDE, 'B', SELECT, N, A, LDA, VL, LDVL, VR, LDVR, |
$ N, NOUT, WORK( IWRK ), IERR ) |
$ N, NOUT, WORK( IWRK ), LWORK-IWRK+1, IERR ) |
END IF |
END IF |
* |
* |
IF( WANTVL ) THEN |
IF( WANTVL ) THEN |