version 1.2, 2010/04/21 13:45:33
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version 1.19, 2018/05/29 07:18:24
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*> \brief \b ZLAED7 used by sstedc. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is dense. |
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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 ZLAED7 + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zlaed7.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/zlaed7.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/zlaed7.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 ZLAED7( N, CUTPNT, QSIZ, TLVLS, CURLVL, CURPBM, D, Q, |
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* LDQ, RHO, INDXQ, QSTORE, QPTR, PRMPTR, PERM, |
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* GIVPTR, GIVCOL, GIVNUM, WORK, RWORK, IWORK, |
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* INFO ) |
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* |
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* .. Scalar Arguments .. |
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* INTEGER CURLVL, CURPBM, CUTPNT, INFO, LDQ, N, QSIZ, |
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* $ TLVLS |
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* DOUBLE PRECISION RHO |
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* .. |
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* .. Array Arguments .. |
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* INTEGER GIVCOL( 2, * ), GIVPTR( * ), INDXQ( * ), |
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* $ IWORK( * ), PERM( * ), PRMPTR( * ), QPTR( * ) |
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* DOUBLE PRECISION D( * ), GIVNUM( 2, * ), QSTORE( * ), RWORK( * ) |
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* COMPLEX*16 Q( LDQ, * ), 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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*> ZLAED7 computes the updated eigensystem of a diagonal |
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*> matrix after modification by a rank-one symmetric matrix. This |
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*> routine is used only for the eigenproblem which requires all |
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*> eigenvalues and optionally eigenvectors of a dense or banded |
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*> Hermitian matrix that has been reduced to tridiagonal form. |
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*> |
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*> T = Q(in) ( D(in) + RHO * Z*Z**H ) Q**H(in) = Q(out) * D(out) * Q**H(out) |
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*> |
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*> where Z = Q**Hu, u is a vector of length N with ones in the |
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*> CUTPNT and CUTPNT + 1 th elements and zeros elsewhere. |
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*> |
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*> The eigenvectors of the original matrix are stored in Q, and the |
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*> eigenvalues are in D. The algorithm consists of three stages: |
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*> |
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*> The first stage consists of deflating the size of the problem |
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*> when there are multiple eigenvalues or if there is a zero in |
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*> the Z vector. For each such occurrence the dimension of the |
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*> secular equation problem is reduced by one. This stage is |
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*> performed by the routine DLAED2. |
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*> |
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*> The second stage consists of calculating the updated |
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*> eigenvalues. This is done by finding the roots of the secular |
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*> equation via the routine DLAED4 (as called by SLAED3). |
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*> This routine also calculates the eigenvectors of the current |
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*> problem. |
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*> |
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*> The final stage consists of computing the updated eigenvectors |
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*> directly using the updated eigenvalues. The eigenvectors for |
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*> the current problem are multiplied with the eigenvectors from |
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*> the overall problem. |
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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] N |
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*> \verbatim |
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*> N is INTEGER |
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*> The dimension of the symmetric tridiagonal matrix. N >= 0. |
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*> \endverbatim |
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*> |
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*> \param[in] CUTPNT |
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*> \verbatim |
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*> CUTPNT is INTEGER |
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*> Contains the location of the last eigenvalue in the leading |
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*> sub-matrix. min(1,N) <= CUTPNT <= N. |
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*> \endverbatim |
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*> |
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*> \param[in] QSIZ |
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*> \verbatim |
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*> QSIZ is INTEGER |
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*> The dimension of the unitary matrix used to reduce |
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*> the full matrix to tridiagonal form. QSIZ >= N. |
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*> \endverbatim |
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*> |
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*> \param[in] TLVLS |
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*> \verbatim |
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*> TLVLS is INTEGER |
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*> The total number of merging levels in the overall divide and |
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*> conquer tree. |
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*> \endverbatim |
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*> |
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*> \param[in] CURLVL |
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*> \verbatim |
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*> CURLVL is INTEGER |
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*> The current level in the overall merge routine, |
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*> 0 <= curlvl <= tlvls. |
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*> \endverbatim |
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*> |
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*> \param[in] CURPBM |
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*> \verbatim |
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*> CURPBM is INTEGER |
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*> The current problem in the current level in the overall |
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*> merge routine (counting from upper left to lower right). |
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*> \endverbatim |
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*> |
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*> \param[in,out] D |
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*> \verbatim |
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*> D is DOUBLE PRECISION array, dimension (N) |
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*> On entry, the eigenvalues of the rank-1-perturbed matrix. |
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*> On exit, the eigenvalues of the repaired matrix. |
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*> \endverbatim |
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*> |
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*> \param[in,out] Q |
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*> \verbatim |
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*> Q is COMPLEX*16 array, dimension (LDQ,N) |
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*> On entry, the eigenvectors of the rank-1-perturbed matrix. |
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*> On exit, the eigenvectors of the repaired tridiagonal matrix. |
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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. LDQ >= max(1,N). |
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*> \endverbatim |
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*> |
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*> \param[in] RHO |
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*> \verbatim |
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*> RHO is DOUBLE PRECISION |
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*> Contains the subdiagonal element used to create the rank-1 |
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*> modification. |
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*> \endverbatim |
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*> |
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*> \param[out] INDXQ |
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*> \verbatim |
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*> INDXQ is INTEGER array, dimension (N) |
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*> This contains the permutation which will reintegrate the |
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*> subproblem just solved back into sorted order, |
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*> ie. D( INDXQ( I = 1, N ) ) will be in ascending order. |
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*> \endverbatim |
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*> |
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*> \param[out] IWORK |
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*> \verbatim |
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*> IWORK is INTEGER array, dimension (4*N) |
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*> \endverbatim |
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*> |
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*> \param[out] RWORK |
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*> \verbatim |
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*> RWORK is DOUBLE PRECISION array, |
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*> dimension (3*N+2*QSIZ*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 COMPLEX*16 array, dimension (QSIZ*N) |
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*> \endverbatim |
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*> |
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*> \param[in,out] QSTORE |
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*> \verbatim |
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*> QSTORE is DOUBLE PRECISION array, dimension (N**2+1) |
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*> Stores eigenvectors of submatrices encountered during |
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*> divide and conquer, packed together. QPTR points to |
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*> beginning of the submatrices. |
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*> \endverbatim |
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*> |
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*> \param[in,out] QPTR |
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*> \verbatim |
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*> QPTR is INTEGER array, dimension (N+2) |
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*> List of indices pointing to beginning of submatrices stored |
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*> in QSTORE. The submatrices are numbered starting at the |
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*> bottom left of the divide and conquer tree, from left to |
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*> right and bottom to top. |
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*> \endverbatim |
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*> |
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*> \param[in] PRMPTR |
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*> \verbatim |
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*> PRMPTR is INTEGER array, dimension (N lg N) |
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*> Contains a list of pointers which indicate where in PERM a |
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*> level's permutation is stored. PRMPTR(i+1) - PRMPTR(i) |
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*> indicates the size of the permutation and also the size of |
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*> the full, non-deflated problem. |
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*> \endverbatim |
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*> |
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*> \param[in] PERM |
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*> \verbatim |
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*> PERM is INTEGER array, dimension (N lg N) |
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*> Contains the permutations (from deflation and sorting) to be |
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*> applied to each eigenblock. |
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*> \endverbatim |
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*> |
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*> \param[in] GIVPTR |
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*> \verbatim |
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*> GIVPTR is INTEGER array, dimension (N lg N) |
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*> Contains a list of pointers which indicate where in GIVCOL a |
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*> level's Givens rotations are stored. GIVPTR(i+1) - GIVPTR(i) |
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*> indicates the number of Givens rotations. |
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*> \endverbatim |
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*> |
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*> \param[in] GIVCOL |
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*> \verbatim |
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*> GIVCOL is INTEGER array, dimension (2, N lg N) |
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*> Each pair of numbers indicates a pair of columns to take place |
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*> in a Givens rotation. |
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*> \endverbatim |
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*> |
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*> \param[in] GIVNUM |
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*> \verbatim |
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*> GIVNUM is DOUBLE PRECISION array, dimension (2, N lg N) |
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*> Each number indicates the S value to be used in the |
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*> corresponding Givens rotation. |
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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 = 1, an eigenvalue did not converge |
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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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*> \ingroup complex16OTHERcomputational |
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* |
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* ===================================================================== |
SUBROUTINE ZLAED7( N, CUTPNT, QSIZ, TLVLS, CURLVL, CURPBM, D, Q, |
SUBROUTINE ZLAED7( N, CUTPNT, QSIZ, TLVLS, CURLVL, CURPBM, D, Q, |
$ LDQ, RHO, INDXQ, QSTORE, QPTR, PRMPTR, PERM, |
$ LDQ, RHO, INDXQ, QSTORE, QPTR, PRMPTR, PERM, |
$ GIVPTR, GIVCOL, GIVNUM, WORK, RWORK, IWORK, |
$ GIVPTR, GIVCOL, GIVNUM, WORK, RWORK, IWORK, |
$ INFO ) |
$ INFO ) |
* |
* |
* -- LAPACK routine (version 3.2) -- |
* -- LAPACK computational 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..-- |
* November 2006 |
* June 2016 |
* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
INTEGER CURLVL, CURPBM, CUTPNT, INFO, LDQ, N, QSIZ, |
INTEGER CURLVL, CURPBM, CUTPNT, INFO, LDQ, N, QSIZ, |
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COMPLEX*16 Q( LDQ, * ), WORK( * ) |
COMPLEX*16 Q( LDQ, * ), WORK( * ) |
* .. |
* .. |
* |
* |
* Purpose |
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* ======= |
|
* |
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* ZLAED7 computes the updated eigensystem of a diagonal |
|
* matrix after modification by a rank-one symmetric matrix. This |
|
* routine is used only for the eigenproblem which requires all |
|
* eigenvalues and optionally eigenvectors of a dense or banded |
|
* Hermitian matrix that has been reduced to tridiagonal form. |
|
* |
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* T = Q(in) ( D(in) + RHO * Z*Z' ) Q'(in) = Q(out) * D(out) * Q'(out) |
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* |
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* where Z = Q'u, u is a vector of length N with ones in the |
|
* CUTPNT and CUTPNT + 1 th elements and zeros elsewhere. |
|
* |
|
* The eigenvectors of the original matrix are stored in Q, and the |
|
* eigenvalues are in D. The algorithm consists of three stages: |
|
* |
|
* The first stage consists of deflating the size of the problem |
|
* when there are multiple eigenvalues or if there is a zero in |
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* the Z vector. For each such occurence the dimension of the |
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* secular equation problem is reduced by one. This stage is |
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* performed by the routine DLAED2. |
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* |
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* The second stage consists of calculating the updated |
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* eigenvalues. This is done by finding the roots of the secular |
|
* equation via the routine DLAED4 (as called by SLAED3). |
|
* This routine also calculates the eigenvectors of the current |
|
* problem. |
|
* |
|
* The final stage consists of computing the updated eigenvectors |
|
* directly using the updated eigenvalues. The eigenvectors for |
|
* the current problem are multiplied with the eigenvectors from |
|
* the overall problem. |
|
* |
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* Arguments |
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* ========= |
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* |
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* N (input) INTEGER |
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* The dimension of the symmetric tridiagonal matrix. N >= 0. |
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* |
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* CUTPNT (input) INTEGER |
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* Contains the location of the last eigenvalue in the leading |
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* sub-matrix. min(1,N) <= CUTPNT <= N. |
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* |
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* QSIZ (input) INTEGER |
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* The dimension of the unitary matrix used to reduce |
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* the full matrix to tridiagonal form. QSIZ >= N. |
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* |
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* TLVLS (input) INTEGER |
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* The total number of merging levels in the overall divide and |
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* conquer tree. |
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* |
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* CURLVL (input) INTEGER |
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* The current level in the overall merge routine, |
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* 0 <= curlvl <= tlvls. |
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* |
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* CURPBM (input) INTEGER |
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* The current problem in the current level in the overall |
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* merge routine (counting from upper left to lower right). |
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* |
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* D (input/output) DOUBLE PRECISION array, dimension (N) |
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* On entry, the eigenvalues of the rank-1-perturbed matrix. |
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* On exit, the eigenvalues of the repaired matrix. |
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* |
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* Q (input/output) COMPLEX*16 array, dimension (LDQ,N) |
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* On entry, the eigenvectors of the rank-1-perturbed matrix. |
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* On exit, the eigenvectors of the repaired tridiagonal matrix. |
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* |
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* LDQ (input) INTEGER |
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* The leading dimension of the array Q. LDQ >= max(1,N). |
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* |
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* RHO (input) DOUBLE PRECISION |
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* Contains the subdiagonal element used to create the rank-1 |
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* modification. |
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* |
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* INDXQ (output) INTEGER array, dimension (N) |
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* This contains the permutation which will reintegrate the |
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* subproblem just solved back into sorted order, |
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* ie. D( INDXQ( I = 1, N ) ) will be in ascending order. |
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* |
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* IWORK (workspace) INTEGER array, dimension (4*N) |
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* |
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* RWORK (workspace) DOUBLE PRECISION array, |
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* dimension (3*N+2*QSIZ*N) |
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* |
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* WORK (workspace) COMPLEX*16 array, dimension (QSIZ*N) |
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* |
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* QSTORE (input/output) DOUBLE PRECISION array, dimension (N**2+1) |
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* Stores eigenvectors of submatrices encountered during |
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* divide and conquer, packed together. QPTR points to |
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* beginning of the submatrices. |
|
* |
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* QPTR (input/output) INTEGER array, dimension (N+2) |
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* List of indices pointing to beginning of submatrices stored |
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* in QSTORE. The submatrices are numbered starting at the |
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* bottom left of the divide and conquer tree, from left to |
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* right and bottom to top. |
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* |
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* PRMPTR (input) INTEGER array, dimension (N lg N) |
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* Contains a list of pointers which indicate where in PERM a |
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* level's permutation is stored. PRMPTR(i+1) - PRMPTR(i) |
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* indicates the size of the permutation and also the size of |
|
* the full, non-deflated problem. |
|
* |
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* PERM (input) INTEGER array, dimension (N lg N) |
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* Contains the permutations (from deflation and sorting) to be |
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* applied to each eigenblock. |
|
* |
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* GIVPTR (input) INTEGER array, dimension (N lg N) |
|
* Contains a list of pointers which indicate where in GIVCOL a |
|
* level's Givens rotations are stored. GIVPTR(i+1) - GIVPTR(i) |
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* indicates the number of Givens rotations. |
|
* |
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* GIVCOL (input) INTEGER array, dimension (2, N lg N) |
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* Each pair of numbers indicates a pair of columns to take place |
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* in a Givens rotation. |
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* |
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* GIVNUM (input) DOUBLE PRECISION array, dimension (2, N lg N) |
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* Each number indicates the S value to be used in the |
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* corresponding Givens rotation. |
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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 = 1, an eigenvalue did not converge |
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* |
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* ===================================================================== |
* ===================================================================== |
* |
* |
* .. Local Scalars .. |
* .. Local Scalars .. |