version 1.5, 2010/08/07 13:22:26
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version 1.16, 2017/06/17 10:54:03
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*> \brief <b> DSTEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER 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 DSTEVR + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dstevr.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/dstevr.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/dstevr.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 DSTEVR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, |
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* M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, |
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* LIWORK, INFO ) |
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* |
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* .. Scalar Arguments .. |
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* CHARACTER JOBZ, RANGE |
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* INTEGER IL, INFO, IU, LDZ, LIWORK, LWORK, M, N |
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* DOUBLE PRECISION ABSTOL, VL, VU |
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* .. |
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* .. Array Arguments .. |
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* INTEGER ISUPPZ( * ), IWORK( * ) |
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* DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * ) |
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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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*> DSTEVR computes selected eigenvalues and, optionally, eigenvectors |
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*> of a real symmetric tridiagonal matrix T. Eigenvalues and |
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*> eigenvectors can be selected by specifying either a range of values |
|
*> or a range of indices for the desired eigenvalues. |
|
*> |
|
*> Whenever possible, DSTEVR calls DSTEMR to compute the |
|
*> eigenspectrum using Relatively Robust Representations. DSTEMR |
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*> computes eigenvalues by the dqds algorithm, while orthogonal |
|
*> eigenvectors are computed from various "good" L D L^T representations |
|
*> (also known as Relatively Robust Representations). Gram-Schmidt |
|
*> orthogonalization is avoided as far as possible. More specifically, |
|
*> the various steps of the algorithm are as follows. For the i-th |
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*> unreduced block of T, |
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*> (a) Compute T - sigma_i = L_i D_i L_i^T, such that L_i D_i L_i^T |
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*> is a relatively robust representation, |
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*> (b) Compute the eigenvalues, lambda_j, of L_i D_i L_i^T to high |
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*> relative accuracy by the dqds algorithm, |
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*> (c) If there is a cluster of close eigenvalues, "choose" sigma_i |
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*> close to the cluster, and go to step (a), |
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*> (d) Given the approximate eigenvalue lambda_j of L_i D_i L_i^T, |
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*> compute the corresponding eigenvector by forming a |
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*> rank-revealing twisted factorization. |
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*> The desired accuracy of the output can be specified by the input |
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*> parameter ABSTOL. |
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*> |
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*> For more details, see "A new O(n^2) algorithm for the symmetric |
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*> tridiagonal eigenvalue/eigenvector problem", by Inderjit Dhillon, |
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*> Computer Science Division Technical Report No. UCB//CSD-97-971, |
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*> UC Berkeley, May 1997. |
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*> |
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*> |
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*> Note 1 : DSTEVR calls DSTEMR when the full spectrum is requested |
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*> on machines which conform to the ieee-754 floating point standard. |
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*> DSTEVR calls DSTEBZ and DSTEIN on non-ieee machines and |
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*> when partial spectrum requests are made. |
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*> |
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*> Normal execution of DSTEMR may create NaNs and infinities and |
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*> hence may abort due to a floating point exception in environments |
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*> which do not handle NaNs and infinities in the ieee standard default |
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*> manner. |
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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] JOBZ |
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*> \verbatim |
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*> JOBZ is CHARACTER*1 |
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*> = 'N': Compute eigenvalues only; |
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*> = 'V': Compute eigenvalues and eigenvectors. |
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*> \endverbatim |
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*> |
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*> \param[in] RANGE |
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*> \verbatim |
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*> RANGE is CHARACTER*1 |
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*> = 'A': all eigenvalues will be found. |
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*> = 'V': all eigenvalues in the half-open interval (VL,VU] |
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*> will be found. |
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*> = 'I': the IL-th through IU-th eigenvalues will be found. |
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*> For RANGE = 'V' or 'I' and IU - IL < N - 1, DSTEBZ and |
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*> DSTEIN are called |
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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. 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 n diagonal elements of the tridiagonal matrix |
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*> A. |
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*> On exit, D may be multiplied by a constant factor chosen |
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*> to avoid over/underflow in computing the eigenvalues. |
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*> \endverbatim |
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*> |
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*> \param[in,out] E |
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*> \verbatim |
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*> E is DOUBLE PRECISION array, dimension (max(1,N-1)) |
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*> On entry, the (n-1) subdiagonal elements of the tridiagonal |
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*> matrix A in elements 1 to N-1 of E. |
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*> On exit, E may be multiplied by a constant factor chosen |
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*> to avoid over/underflow in computing the eigenvalues. |
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*> \endverbatim |
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*> |
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*> \param[in] VL |
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*> \verbatim |
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*> VL is DOUBLE PRECISION |
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*> If RANGE='V', the lower bound of the interval to |
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*> be searched for eigenvalues. VL < VU. |
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*> Not referenced if RANGE = 'A' or 'I'. |
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*> \endverbatim |
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*> |
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*> \param[in] VU |
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*> \verbatim |
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*> VU is DOUBLE PRECISION |
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*> If RANGE='V', the upper bound of the interval to |
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*> be searched for eigenvalues. VL < VU. |
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*> Not referenced if RANGE = 'A' or 'I'. |
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*> \endverbatim |
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*> |
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*> \param[in] IL |
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*> \verbatim |
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*> IL is INTEGER |
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*> If RANGE='I', the index of the |
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*> smallest eigenvalue to be returned. |
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*> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. |
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*> Not referenced if RANGE = 'A' or 'V'. |
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*> \endverbatim |
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*> |
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*> \param[in] IU |
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*> \verbatim |
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*> IU is INTEGER |
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*> If RANGE='I', the index of the |
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*> largest eigenvalue to be returned. |
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*> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. |
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*> Not referenced if RANGE = 'A' or 'V'. |
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*> \endverbatim |
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*> |
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*> \param[in] ABSTOL |
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*> \verbatim |
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*> ABSTOL is DOUBLE PRECISION |
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*> The absolute error tolerance for the eigenvalues. |
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*> An approximate eigenvalue is accepted as converged |
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*> when it is determined to lie in an interval [a,b] |
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*> of width less than or equal to |
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*> |
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*> ABSTOL + EPS * max( |a|,|b| ) , |
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*> |
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*> where EPS is the machine precision. If ABSTOL is less than |
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*> or equal to zero, then EPS*|T| will be used in its place, |
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*> where |T| is the 1-norm of the tridiagonal matrix obtained |
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*> by reducing A to tridiagonal form. |
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*> |
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*> See "Computing Small Singular Values of Bidiagonal Matrices |
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*> with Guaranteed High Relative Accuracy," by Demmel and |
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*> Kahan, LAPACK Working Note #3. |
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*> |
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*> If high relative accuracy is important, set ABSTOL to |
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*> DLAMCH( 'Safe minimum' ). Doing so will guarantee that |
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*> eigenvalues are computed to high relative accuracy when |
|
*> possible in future releases. The current code does not |
|
*> make any guarantees about high relative accuracy, but |
|
*> future releases will. See J. Barlow and J. Demmel, |
|
*> "Computing Accurate Eigensystems of Scaled Diagonally |
|
*> Dominant Matrices", LAPACK Working Note #7, for a discussion |
|
*> of which matrices define their eigenvalues to high relative |
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*> accuracy. |
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*> \endverbatim |
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*> |
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*> \param[out] M |
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*> \verbatim |
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*> M is INTEGER |
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*> The total number of eigenvalues found. 0 <= M <= N. |
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*> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. |
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*> \endverbatim |
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*> |
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*> \param[out] W |
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*> \verbatim |
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*> W is DOUBLE PRECISION array, dimension (N) |
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*> The first M elements contain the selected eigenvalues in |
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*> ascending order. |
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*> \endverbatim |
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*> |
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*> \param[out] Z |
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*> \verbatim |
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*> Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M) ) |
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*> If JOBZ = 'V', then if INFO = 0, the first M columns of Z |
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*> contain the orthonormal eigenvectors of the matrix A |
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*> corresponding to the selected eigenvalues, with the i-th |
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*> column of Z holding the eigenvector associated with W(i). |
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*> Note: the user must ensure that at least max(1,M) columns are |
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*> supplied in the array Z; if RANGE = 'V', the exact value of M |
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*> is not known in advance and an upper bound must be used. |
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*> \endverbatim |
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*> |
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*> \param[in] LDZ |
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*> \verbatim |
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*> LDZ is INTEGER |
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*> The leading dimension of the array Z. LDZ >= 1, and if |
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*> JOBZ = 'V', LDZ >= max(1,N). |
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*> \endverbatim |
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*> |
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*> \param[out] ISUPPZ |
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*> \verbatim |
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*> ISUPPZ is INTEGER array, dimension ( 2*max(1,M) ) |
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*> The support of the eigenvectors in Z, i.e., the indices |
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*> indicating the nonzero elements in Z. The i-th eigenvector |
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*> is nonzero only in elements ISUPPZ( 2*i-1 ) through |
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*> ISUPPZ( 2*i ). |
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*> Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1 |
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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 (and |
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*> minimal) 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(1,20*N). |
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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 sizes of the WORK and IWORK |
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*> arrays, returns these values as the first entries of the WORK |
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*> and IWORK arrays, and no error message related to LWORK or |
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*> LIWORK is issued by XERBLA. |
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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 (MAX(1,LIWORK)) |
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*> On exit, if INFO = 0, IWORK(1) returns the optimal (and |
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*> minimal) LIWORK. |
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*> \endverbatim |
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*> |
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*> \param[in] LIWORK |
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*> \verbatim |
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*> LIWORK is INTEGER |
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*> The dimension of the array IWORK. LIWORK >= max(1,10*N). |
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*> |
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*> If LIWORK = -1, then a workspace query is assumed; the |
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*> routine only calculates the optimal sizes of the WORK and |
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*> IWORK arrays, returns these values as the first entries of |
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*> the WORK and IWORK arrays, and no error message related to |
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*> LWORK or LIWORK 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: Internal error |
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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 doubleOTHEReigen |
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* |
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*> \par Contributors: |
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* ================== |
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*> |
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*> Inderjit Dhillon, IBM Almaden, USA \n |
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*> Osni Marques, LBNL/NERSC, USA \n |
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*> Ken Stanley, Computer Science Division, University of |
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*> California at Berkeley, USA \n |
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*> |
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* ===================================================================== |
SUBROUTINE DSTEVR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, |
SUBROUTINE DSTEVR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, |
$ M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, |
$ M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, |
$ LIWORK, INFO ) |
$ LIWORK, INFO ) |
* |
* |
* -- LAPACK driver routine (version 3.2) -- |
* -- 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..-- |
* November 2006 |
* June 2016 |
* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
CHARACTER JOBZ, RANGE |
CHARACTER JOBZ, RANGE |
Line 17
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Line 319
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DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * ) |
DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * ) |
* .. |
* .. |
* |
* |
* Purpose |
|
* ======= |
|
* |
|
* DSTEVR computes selected eigenvalues and, optionally, eigenvectors |
|
* of a real symmetric tridiagonal matrix T. Eigenvalues and |
|
* eigenvectors can be selected by specifying either a range of values |
|
* or a range of indices for the desired eigenvalues. |
|
* |
|
* Whenever possible, DSTEVR calls DSTEMR to compute the |
|
* eigenspectrum using Relatively Robust Representations. DSTEMR |
|
* computes eigenvalues by the dqds algorithm, while orthogonal |
|
* eigenvectors are computed from various "good" L D L^T representations |
|
* (also known as Relatively Robust Representations). Gram-Schmidt |
|
* orthogonalization is avoided as far as possible. More specifically, |
|
* the various steps of the algorithm are as follows. For the i-th |
|
* unreduced block of T, |
|
* (a) Compute T - sigma_i = L_i D_i L_i^T, such that L_i D_i L_i^T |
|
* is a relatively robust representation, |
|
* (b) Compute the eigenvalues, lambda_j, of L_i D_i L_i^T to high |
|
* relative accuracy by the dqds algorithm, |
|
* (c) If there is a cluster of close eigenvalues, "choose" sigma_i |
|
* close to the cluster, and go to step (a), |
|
* (d) Given the approximate eigenvalue lambda_j of L_i D_i L_i^T, |
|
* compute the corresponding eigenvector by forming a |
|
* rank-revealing twisted factorization. |
|
* The desired accuracy of the output can be specified by the input |
|
* parameter ABSTOL. |
|
* |
|
* For more details, see "A new O(n^2) algorithm for the symmetric |
|
* tridiagonal eigenvalue/eigenvector problem", by Inderjit Dhillon, |
|
* Computer Science Division Technical Report No. UCB//CSD-97-971, |
|
* UC Berkeley, May 1997. |
|
* |
|
* |
|
* Note 1 : DSTEVR calls DSTEMR when the full spectrum is requested |
|
* on machines which conform to the ieee-754 floating point standard. |
|
* DSTEVR calls DSTEBZ and DSTEIN on non-ieee machines and |
|
* when partial spectrum requests are made. |
|
* |
|
* Normal execution of DSTEMR may create NaNs and infinities and |
|
* hence may abort due to a floating point exception in environments |
|
* which do not handle NaNs and infinities in the ieee standard default |
|
* manner. |
|
* |
|
* Arguments |
|
* ========= |
|
* |
|
* JOBZ (input) CHARACTER*1 |
|
* = 'N': Compute eigenvalues only; |
|
* = 'V': Compute eigenvalues and eigenvectors. |
|
* |
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* RANGE (input) CHARACTER*1 |
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* = 'A': all eigenvalues will be found. |
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* = 'V': all eigenvalues in the half-open interval (VL,VU] |
|
* will be found. |
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* = 'I': the IL-th through IU-th eigenvalues will be found. |
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********** For RANGE = 'V' or 'I' and IU - IL < N - 1, DSTEBZ and |
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********** DSTEIN are called |
|
* |
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* N (input) INTEGER |
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* The order of the 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 n diagonal elements of the tridiagonal matrix |
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* A. |
|
* On exit, D may be multiplied by a constant factor chosen |
|
* to avoid over/underflow in computing the eigenvalues. |
|
* |
|
* E (input/output) DOUBLE PRECISION array, dimension (max(1,N-1)) |
|
* On entry, the (n-1) subdiagonal elements of the tridiagonal |
|
* matrix A in elements 1 to N-1 of E. |
|
* On exit, E may be multiplied by a constant factor chosen |
|
* to avoid over/underflow in computing the eigenvalues. |
|
* |
|
* VL (input) DOUBLE PRECISION |
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* VU (input) DOUBLE PRECISION |
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* If RANGE='V', the lower and upper bounds of the interval to |
|
* be searched for eigenvalues. VL < VU. |
|
* Not referenced if RANGE = 'A' or 'I'. |
|
* |
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* IL (input) INTEGER |
|
* IU (input) INTEGER |
|
* If RANGE='I', the indices (in ascending order) of the |
|
* smallest and largest eigenvalues to be returned. |
|
* 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. |
|
* Not referenced if RANGE = 'A' or 'V'. |
|
* |
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* ABSTOL (input) DOUBLE PRECISION |
|
* The absolute error tolerance for the eigenvalues. |
|
* An approximate eigenvalue is accepted as converged |
|
* when it is determined to lie in an interval [a,b] |
|
* of width less than or equal to |
|
* |
|
* ABSTOL + EPS * max( |a|,|b| ) , |
|
* |
|
* where EPS is the machine precision. If ABSTOL is less than |
|
* or equal to zero, then EPS*|T| will be used in its place, |
|
* where |T| is the 1-norm of the tridiagonal matrix obtained |
|
* by reducing A to tridiagonal form. |
|
* |
|
* See "Computing Small Singular Values of Bidiagonal Matrices |
|
* with Guaranteed High Relative Accuracy," by Demmel and |
|
* Kahan, LAPACK Working Note #3. |
|
* |
|
* If high relative accuracy is important, set ABSTOL to |
|
* DLAMCH( 'Safe minimum' ). Doing so will guarantee that |
|
* eigenvalues are computed to high relative accuracy when |
|
* possible in future releases. The current code does not |
|
* make any guarantees about high relative accuracy, but |
|
* future releases will. See J. Barlow and J. Demmel, |
|
* "Computing Accurate Eigensystems of Scaled Diagonally |
|
* Dominant Matrices", LAPACK Working Note #7, for a discussion |
|
* of which matrices define their eigenvalues to high relative |
|
* accuracy. |
|
* |
|
* M (output) INTEGER |
|
* The total number of eigenvalues found. 0 <= M <= N. |
|
* If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. |
|
* |
|
* W (output) DOUBLE PRECISION array, dimension (N) |
|
* The first M elements contain the selected eigenvalues in |
|
* ascending order. |
|
* |
|
* Z (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M) ) |
|
* If JOBZ = 'V', then if INFO = 0, the first M columns of Z |
|
* contain the orthonormal eigenvectors of the matrix A |
|
* corresponding to the selected eigenvalues, with the i-th |
|
* column of Z holding the eigenvector associated with W(i). |
|
* Note: the user must ensure that at least max(1,M) columns are |
|
* supplied in the array Z; if RANGE = 'V', the exact value of M |
|
* is not known in advance and an upper bound must be used. |
|
* |
|
* LDZ (input) INTEGER |
|
* The leading dimension of the array Z. LDZ >= 1, and if |
|
* JOBZ = 'V', LDZ >= max(1,N). |
|
* |
|
* ISUPPZ (output) INTEGER array, dimension ( 2*max(1,M) ) |
|
* The support of the eigenvectors in Z, i.e., the indices |
|
* indicating the nonzero elements in Z. The i-th eigenvector |
|
* is nonzero only in elements ISUPPZ( 2*i-1 ) through |
|
* ISUPPZ( 2*i ). |
|
********** Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1 |
|
* |
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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 (and |
|
* minimal) LWORK. |
|
* |
|
* LWORK (input) INTEGER |
|
* The dimension of the array WORK. LWORK >= max(1,20*N). |
|
* |
|
* If LWORK = -1, then a workspace query is assumed; the routine |
|
* only calculates the optimal sizes of the WORK and IWORK |
|
* arrays, returns these values as the first entries of the WORK |
|
* and IWORK arrays, and no error message related to LWORK or |
|
* LIWORK is issued by XERBLA. |
|
* |
|
* IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK)) |
|
* On exit, if INFO = 0, IWORK(1) returns the optimal (and |
|
* minimal) LIWORK. |
|
* |
|
* LIWORK (input) INTEGER |
|
* The dimension of the array IWORK. LIWORK >= max(1,10*N). |
|
* |
|
* If LIWORK = -1, then a workspace query is assumed; the |
|
* routine only calculates the optimal sizes of the WORK and |
|
* IWORK arrays, returns these values as the first entries of |
|
* the WORK and IWORK arrays, and no error message related to |
|
* LWORK or LIWORK is issued by XERBLA. |
|
* |
|
* INFO (output) INTEGER |
|
* = 0: successful exit |
|
* < 0: if INFO = -i, the i-th argument had an illegal value |
|
* > 0: Internal error |
|
* |
|
* Further Details |
|
* =============== |
|
* |
|
* Based on contributions by |
|
* Inderjit Dhillon, IBM Almaden, USA |
|
* Osni Marques, LBNL/NERSC, USA |
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* Ken Stanley, Computer Science Division, University of |
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* California at Berkeley, USA |
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* |
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* ===================================================================== |
* ===================================================================== |
* |
* |
* .. Parameters .. |
* .. Parameters .. |
Line 323
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Line 442
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* Scale matrix to allowable range, if necessary. |
* Scale matrix to allowable range, if necessary. |
* |
* |
ISCALE = 0 |
ISCALE = 0 |
VLL = VL |
IF( VALEIG ) THEN |
VUU = VU |
VLL = VL |
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VUU = VU |
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END IF |
* |
* |
TNRM = DLANST( 'M', N, D, E ) |
TNRM = DLANST( 'M', N, D, E ) |
IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN |
IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN |