583 lines
19 KiB
Fortran
583 lines
19 KiB
Fortran
*> \brief <b> SSTEVR 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 SSTEVR + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sstevr.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/sstevr.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/sstevr.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 SSTEVR( 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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* REAL ABSTOL, VL, VU
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* ..
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* .. Array Arguments ..
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* INTEGER ISUPPZ( * ), IWORK( * )
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* REAL 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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*> SSTEVR 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
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*> or a range of indices for the desired eigenvalues.
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*>
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*> Whenever possible, SSTEVR calls SSTEMR to compute the
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*> eigenspectrum using Relatively Robust Representations. SSTEMR
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*> computes eigenvalues by the dqds algorithm, while orthogonal
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*> eigenvectors are computed from various "good" L D L^T representations
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*> (also known as Relatively Robust Representations). Gram-Schmidt
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*> orthogonalization is avoided as far as possible. More specifically,
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*> 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 : SSTEVR calls SSTEMR 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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*> SSTEVR calls SSTEBZ and SSTEIN 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 SSTEMR 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, SSTEBZ and
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*> SSTEIN 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 REAL 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 REAL 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 REAL
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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 REAL
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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 REAL
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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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*> SLAMCH( 'Safe minimum' ). Doing so will guarantee that
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*> eigenvalues are computed to high relative accuracy when
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*> possible in future releases. The current code does not
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*> make any guarantees about high relative accuracy, but
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*> future releases will. See J. Barlow and J. Demmel,
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*> "Computing Accurate Eigensystems of Scaled Diagonally
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*> Dominant Matrices", LAPACK Working Note #7, for a discussion
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*> 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 REAL 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 REAL 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 REAL 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 >= 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 >= 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 realOTHEReigen
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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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*> Jason Riedy, Computer Science Division, University of
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*> California at Berkeley, USA \n
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*>
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* =====================================================================
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SUBROUTINE SSTEVR( 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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* -- LAPACK driver routine (version 3.7.0) --
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* -- LAPACK is a software package provided by Univ. of Tennessee, --
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* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
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* June 2016
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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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REAL ABSTOL, VL, VU
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* ..
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* .. Array Arguments ..
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INTEGER ISUPPZ( * ), IWORK( * )
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REAL D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * )
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* ..
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*
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* =====================================================================
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*
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* .. Parameters ..
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REAL ZERO, ONE, TWO
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PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0, TWO = 2.0E+0 )
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* ..
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* .. Local Scalars ..
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LOGICAL ALLEIG, INDEIG, TEST, LQUERY, VALEIG, WANTZ,
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$ TRYRAC
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CHARACTER ORDER
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INTEGER I, IEEEOK, IMAX, INDIBL, INDIFL, INDISP,
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$ INDIWO, ISCALE, J, JJ, LIWMIN, LWMIN, NSPLIT
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REAL BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA, SMLNUM,
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$ TMP1, TNRM, VLL, VUU
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* ..
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* .. External Functions ..
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LOGICAL LSAME
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INTEGER ILAENV
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REAL SLAMCH, SLANST
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EXTERNAL LSAME, ILAENV, SLAMCH, SLANST
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* ..
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* .. External Subroutines ..
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EXTERNAL SCOPY, SSCAL, SSTEBZ, SSTEMR, SSTEIN, SSTERF,
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$ SSWAP, XERBLA
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC MAX, MIN, SQRT
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* ..
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* .. Executable Statements ..
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*
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*
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* Test the input parameters.
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*
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IEEEOK = ILAENV( 10, 'SSTEVR', 'N', 1, 2, 3, 4 )
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*
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WANTZ = LSAME( JOBZ, 'V' )
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ALLEIG = LSAME( RANGE, 'A' )
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VALEIG = LSAME( RANGE, 'V' )
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INDEIG = LSAME( RANGE, 'I' )
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*
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LQUERY = ( ( LWORK.EQ.-1 ) .OR. ( LIWORK.EQ.-1 ) )
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LWMIN = MAX( 1, 20*N )
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LIWMIN = MAX(1, 10*N )
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*
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*
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INFO = 0
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IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
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INFO = -1
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ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
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INFO = -2
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ELSE IF( N.LT.0 ) THEN
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INFO = -3
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ELSE
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IF( VALEIG ) THEN
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IF( N.GT.0 .AND. VU.LE.VL )
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$ INFO = -7
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ELSE IF( INDEIG ) THEN
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IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
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INFO = -8
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ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
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INFO = -9
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END IF
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END IF
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END IF
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IF( INFO.EQ.0 ) THEN
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IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
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INFO = -14
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END IF
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END IF
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*
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IF( INFO.EQ.0 ) THEN
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WORK( 1 ) = LWMIN
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IWORK( 1 ) = LIWMIN
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*
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IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
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INFO = -17
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ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
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INFO = -19
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END IF
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END IF
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*
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'SSTEVR', -INFO )
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RETURN
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ELSE IF( LQUERY ) THEN
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RETURN
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END IF
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*
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* Quick return if possible
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*
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M = 0
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IF( N.EQ.0 )
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$ RETURN
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*
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IF( N.EQ.1 ) THEN
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IF( ALLEIG .OR. INDEIG ) THEN
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M = 1
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W( 1 ) = D( 1 )
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ELSE
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IF( VL.LT.D( 1 ) .AND. VU.GE.D( 1 ) ) THEN
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M = 1
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W( 1 ) = D( 1 )
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END IF
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END IF
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IF( WANTZ )
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$ Z( 1, 1 ) = ONE
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RETURN
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END IF
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*
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* Get machine constants.
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*
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SAFMIN = SLAMCH( 'Safe minimum' )
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EPS = SLAMCH( 'Precision' )
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SMLNUM = SAFMIN / EPS
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BIGNUM = ONE / SMLNUM
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RMIN = SQRT( SMLNUM )
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RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
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*
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*
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* Scale matrix to allowable range, if necessary.
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*
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ISCALE = 0
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IF( VALEIG ) THEN
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VLL = VL
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VUU = VU
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END IF
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*
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TNRM = SLANST( 'M', N, D, E )
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IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN
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ISCALE = 1
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SIGMA = RMIN / TNRM
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ELSE IF( TNRM.GT.RMAX ) THEN
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ISCALE = 1
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SIGMA = RMAX / TNRM
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END IF
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IF( ISCALE.EQ.1 ) THEN
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CALL SSCAL( N, SIGMA, D, 1 )
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CALL SSCAL( N-1, SIGMA, E( 1 ), 1 )
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IF( VALEIG ) THEN
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VLL = VL*SIGMA
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VUU = VU*SIGMA
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END IF
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END IF
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* Initialize indices into workspaces. Note: These indices are used only
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* if SSTERF or SSTEMR fail.
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* IWORK(INDIBL:INDIBL+M-1) corresponds to IBLOCK in SSTEBZ and
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* stores the block indices of each of the M<=N eigenvalues.
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INDIBL = 1
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* IWORK(INDISP:INDISP+NSPLIT-1) corresponds to ISPLIT in SSTEBZ and
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* stores the starting and finishing indices of each block.
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INDISP = INDIBL + N
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* IWORK(INDIFL:INDIFL+N-1) stores the indices of eigenvectors
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* that corresponding to eigenvectors that fail to converge in
|
|
* SSTEIN. This information is discarded; if any fail, the driver
|
|
* returns INFO > 0.
|
|
INDIFL = INDISP + N
|
|
* INDIWO is the offset of the remaining integer workspace.
|
|
INDIWO = INDISP + N
|
|
*
|
|
* If all eigenvalues are desired, then
|
|
* call SSTERF or SSTEMR. If this fails for some eigenvalue, then
|
|
* try SSTEBZ.
|
|
*
|
|
*
|
|
TEST = .FALSE.
|
|
IF( INDEIG ) THEN
|
|
IF( IL.EQ.1 .AND. IU.EQ.N ) THEN
|
|
TEST = .TRUE.
|
|
END IF
|
|
END IF
|
|
IF( ( ALLEIG .OR. TEST ) .AND. IEEEOK.EQ.1 ) THEN
|
|
CALL SCOPY( N-1, E( 1 ), 1, WORK( 1 ), 1 )
|
|
IF( .NOT.WANTZ ) THEN
|
|
CALL SCOPY( N, D, 1, W, 1 )
|
|
CALL SSTERF( N, W, WORK, INFO )
|
|
ELSE
|
|
CALL SCOPY( N, D, 1, WORK( N+1 ), 1 )
|
|
IF (ABSTOL .LE. TWO*N*EPS) THEN
|
|
TRYRAC = .TRUE.
|
|
ELSE
|
|
TRYRAC = .FALSE.
|
|
END IF
|
|
CALL SSTEMR( JOBZ, 'A', N, WORK( N+1 ), WORK, VL, VU, IL,
|
|
$ IU, M, W, Z, LDZ, N, ISUPPZ, TRYRAC,
|
|
$ WORK( 2*N+1 ), LWORK-2*N, IWORK, LIWORK, INFO )
|
|
*
|
|
END IF
|
|
IF( INFO.EQ.0 ) THEN
|
|
M = N
|
|
GO TO 10
|
|
END IF
|
|
INFO = 0
|
|
END IF
|
|
*
|
|
* Otherwise, call SSTEBZ and, if eigenvectors are desired, SSTEIN.
|
|
*
|
|
IF( WANTZ ) THEN
|
|
ORDER = 'B'
|
|
ELSE
|
|
ORDER = 'E'
|
|
END IF
|
|
|
|
CALL SSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTOL, D, E, M,
|
|
$ NSPLIT, W, IWORK( INDIBL ), IWORK( INDISP ), WORK,
|
|
$ IWORK( INDIWO ), INFO )
|
|
*
|
|
IF( WANTZ ) THEN
|
|
CALL SSTEIN( N, D, E, M, W, IWORK( INDIBL ), IWORK( INDISP ),
|
|
$ Z, LDZ, WORK, IWORK( INDIWO ), IWORK( INDIFL ),
|
|
$ INFO )
|
|
END IF
|
|
*
|
|
* If matrix was scaled, then rescale eigenvalues appropriately.
|
|
*
|
|
10 CONTINUE
|
|
IF( ISCALE.EQ.1 ) THEN
|
|
IF( INFO.EQ.0 ) THEN
|
|
IMAX = M
|
|
ELSE
|
|
IMAX = INFO - 1
|
|
END IF
|
|
CALL SSCAL( IMAX, ONE / SIGMA, W, 1 )
|
|
END IF
|
|
*
|
|
* If eigenvalues are not in order, then sort them, along with
|
|
* eigenvectors.
|
|
*
|
|
IF( WANTZ ) THEN
|
|
DO 30 J = 1, M - 1
|
|
I = 0
|
|
TMP1 = W( J )
|
|
DO 20 JJ = J + 1, M
|
|
IF( W( JJ ).LT.TMP1 ) THEN
|
|
I = JJ
|
|
TMP1 = W( JJ )
|
|
END IF
|
|
20 CONTINUE
|
|
*
|
|
IF( I.NE.0 ) THEN
|
|
W( I ) = W( J )
|
|
W( J ) = TMP1
|
|
CALL SSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
|
|
END IF
|
|
30 CONTINUE
|
|
END IF
|
|
*
|
|
* Causes problems with tests 19 & 20:
|
|
* IF (wantz .and. INDEIG ) Z( 1,1) = Z(1,1) / 1.002 + .002
|
|
*
|
|
*
|
|
WORK( 1 ) = LWMIN
|
|
IWORK( 1 ) = LIWMIN
|
|
RETURN
|
|
*
|
|
* End of SSTEVR
|
|
*
|
|
END
|