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version 1.14, 2014/01/27 09:28:19
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*> \brief \b DLAED1 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 tridiagonal. |
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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 DLAED1 + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaed1.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/dlaed1.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/dlaed1.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 DLAED1( N, D, Q, LDQ, INDXQ, RHO, CUTPNT, WORK, IWORK, |
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* INFO ) |
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* |
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* .. Scalar Arguments .. |
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* INTEGER CUTPNT, INFO, LDQ, N |
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* DOUBLE PRECISION RHO |
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* .. |
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* .. Array Arguments .. |
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* INTEGER INDXQ( * ), IWORK( * ) |
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* DOUBLE PRECISION D( * ), 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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*> DLAED1 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 eigenvectors of a tridiagonal matrix. DLAED7 handles |
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*> the case in which eigenvalues only or eigenvalues and eigenvectors |
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*> of a full symmetric matrix (which was reduced to tridiagonal form) |
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*> are desired. |
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*> |
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*> T = Q(in) ( D(in) + RHO * Z*Z**T ) Q**T(in) = Q(out) * D(out) * Q**T(out) |
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*> |
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*> where Z = Q**T*u, 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 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 |
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*> equation via the routine DLAED4 (as called by DLAED3). |
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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,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 DOUBLE PRECISION 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,out] INDXQ |
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*> \verbatim |
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*> INDXQ is INTEGER array, dimension (N) |
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*> On entry, the permutation which separately sorts the two |
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*> subproblems in D into ascending order. |
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*> On exit, the permutation which will reintegrate the |
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*> subproblems back into sorted order, |
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*> i.e. D( INDXQ( I = 1, N ) ) will be in ascending order. |
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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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*> The subdiagonal entry used to create the rank-1 modification. |
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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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*> The location of the last eigenvalue in the leading sub-matrix. |
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*> min(1,N) <= CUTPNT <= N/2. |
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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 (4*N + N**2) |
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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] 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 September 2012 |
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* |
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*> \ingroup auxOTHERcomputational |
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* |
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*> \par Contributors: |
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* ================== |
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*> |
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*> Jeff Rutter, Computer Science Division, University of California |
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*> at Berkeley, USA \n |
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*> Modified by Francoise Tisseur, University of Tennessee |
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*> |
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* ===================================================================== |
SUBROUTINE DLAED1( N, D, Q, LDQ, INDXQ, RHO, CUTPNT, WORK, IWORK, |
SUBROUTINE DLAED1( N, D, Q, LDQ, INDXQ, RHO, CUTPNT, WORK, IWORK, |
$ INFO ) |
$ INFO ) |
* |
* |
* -- LAPACK routine (version 3.2) -- |
* -- LAPACK computational routine (version 3.4.2) -- |
* -- 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 |
* September 2012 |
* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
INTEGER CUTPNT, INFO, LDQ, N |
INTEGER CUTPNT, INFO, LDQ, N |
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DOUBLE PRECISION D( * ), Q( LDQ, * ), WORK( * ) |
DOUBLE PRECISION D( * ), Q( LDQ, * ), WORK( * ) |
* .. |
* .. |
* |
* |
* Purpose |
|
* ======= |
|
* |
|
* DLAED1 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 eigenvectors of a tridiagonal matrix. DLAED7 handles |
|
* the case in which eigenvalues only or eigenvalues and eigenvectors |
|
* of a full symmetric matrix (which was reduced to tridiagonal form) |
|
* are desired. |
|
* |
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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 |
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* CUTPNT and CUTPNT + 1 th elements and zeros elsewhere. |
|
* |
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* 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 |
|
* secular equation problem is reduced by one. This stage is |
|
* performed by the routine DLAED2. |
|
* |
|
* The second stage consists of calculating the updated |
|
* eigenvalues. This is done by finding the roots of the secular |
|
* equation via the routine DLAED4 (as called by DLAED3). |
|
* 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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* N (input) INTEGER |
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* The dimension of the symmetric tridiagonal matrix. N >= 0. |
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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) DOUBLE PRECISION 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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* INDXQ (input/output) INTEGER array, dimension (N) |
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* On entry, the permutation which separately sorts the two |
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* subproblems in D into ascending order. |
|
* On exit, the permutation which will reintegrate the |
|
* subproblems back into sorted order, |
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* i.e. D( INDXQ( I = 1, N ) ) will be in ascending order. |
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* |
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* RHO (input) DOUBLE PRECISION |
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* The subdiagonal entry used to create the rank-1 modification. |
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* |
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* CUTPNT (input) INTEGER |
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* The location of the last eigenvalue in the leading sub-matrix. |
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* min(1,N) <= CUTPNT <= N/2. |
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* |
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* WORK (workspace) DOUBLE PRECISION array, dimension (4*N + N**2) |
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* |
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* IWORK (workspace) INTEGER array, dimension (4*N) |
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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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* Further Details |
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* =============== |
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* |
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* Based on contributions by |
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* Jeff Rutter, Computer Science Division, University of California |
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* at Berkeley, USA |
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* Modified by Francoise Tisseur, University of Tennessee. |
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* |
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* ===================================================================== |
* ===================================================================== |
* |
* |
* .. Local Scalars .. |
* .. Local Scalars .. |