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776 lines
28 KiB
776 lines
28 KiB
*> \brief <b> ZHEEVR_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices</b>
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*
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* @precisions fortran z -> s d c
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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 ZHEEVR_2STAGE + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zheevr_2stage.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/zheevr_2stage.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/zheevr_2stage.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 ZHEEVR_2STAGE( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU,
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* IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ,
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* WORK, LWORK, RWORK, LRWORK, IWORK,
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* LIWORK, INFO )
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*
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* IMPLICIT NONE
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*
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* .. Scalar Arguments ..
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* CHARACTER JOBZ, RANGE, UPLO
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* INTEGER IL, INFO, IU, LDA, LDZ, LIWORK, LRWORK, LWORK,
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* $ 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 RWORK( * ), W( * )
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* COMPLEX*16 A( LDA, * ), 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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*> ZHEEVR_2STAGE computes selected eigenvalues and, optionally, eigenvectors
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*> of a complex Hermitian matrix A using the 2stage technique for
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*> the reduction to tridiagonal. Eigenvalues and eigenvectors can
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*> be selected by specifying either a range of values or a range of
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*> indices for the desired eigenvalues.
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*>
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*> ZHEEVR_2STAGE first reduces the matrix A to tridiagonal form T with a call
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*> to ZHETRD. Then, whenever possible, ZHEEVR_2STAGE calls ZSTEMR to compute
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*> eigenspectrum using Relatively Robust Representations. ZSTEMR
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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.
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*>
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*> For each unreduced block (submatrix) of T,
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*> (a) Compute T - sigma I = L D L^T, so that L and D
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*> define all the wanted eigenvalues to high relative accuracy.
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*> This means that small relative changes in the entries of D and L
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*> cause only small relative changes in the eigenvalues and
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*> eigenvectors. The standard (unfactored) representation of the
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*> tridiagonal matrix T does not have this property in general.
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*> (b) Compute the eigenvalues to suitable accuracy.
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*> If the eigenvectors are desired, the algorithm attains full
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*> accuracy of the computed eigenvalues only right before
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*> the corresponding vectors have to be computed, see steps c) and d).
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*> (c) For each cluster of close eigenvalues, select a new
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*> shift close to the cluster, find a new factorization, and refine
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*> the shifted eigenvalues to suitable accuracy.
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*> (d) For each eigenvalue with a large enough relative separation compute
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*> the corresponding eigenvector by forming a rank revealing twisted
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*> factorization. Go back to (c) for any clusters that remain.
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*>
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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 ZSTEMR's documentation and:
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*> - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations
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*> to compute orthogonal eigenvectors of symmetric tridiagonal matrices,"
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*> Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
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*> - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and
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*> Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25,
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*> 2004. Also LAPACK Working Note 154.
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*> - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric
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*> tridiagonal eigenvalue/eigenvector problem",
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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 : ZHEEVR_2STAGE calls ZSTEMR 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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*> ZHEEVR_2STAGE calls DSTEBZ and ZSTEIN 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 ZSTEMR 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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*> Not available in this release.
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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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*> ZSTEIN are called
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*> \endverbatim
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*>
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*> \param[in] UPLO
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*> \verbatim
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*> UPLO is CHARACTER*1
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*> = 'U': Upper triangle of A is stored;
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*> = 'L': Lower triangle of A is stored.
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*> \endverbatim
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*>
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*> \param[in] N
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*> \verbatim
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*> N is INTEGER
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*> The order of the matrix A. N >= 0.
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*> \endverbatim
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*>
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*> \param[in,out] A
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*> \verbatim
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*> A is COMPLEX*16 array, dimension (LDA, N)
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*> On entry, the Hermitian matrix A. If UPLO = 'U', the
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*> leading N-by-N upper triangular part of A contains the
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*> upper triangular part of the matrix A. If UPLO = 'L',
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*> the leading N-by-N lower triangular part of A contains
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*> the lower triangular part of the matrix A.
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*> On exit, the lower triangle (if UPLO='L') or the upper
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*> triangle (if UPLO='U') of A, including the diagonal, is
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*> destroyed.
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*> \endverbatim
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*>
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*> \param[in] LDA
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*> \verbatim
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*> LDA is INTEGER
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*> The leading dimension of the array A. LDA >= max(1,N).
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*> \endverbatim
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*>
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*> \param[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
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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 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 COMPLEX*16 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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*> If JOBZ = 'N', then Z is not referenced.
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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 ). This is an output of ZSTEMR (tridiagonal
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*> matrix). The support of the eigenvectors of A is typically
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*> 1:N because of the unitary transformations applied by ZUNMTR.
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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 COMPLEX*16 array, dimension (MAX(1,LWORK))
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*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
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*> \endverbatim
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*>
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*> \param[in] LWORK
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*> \verbatim
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*> LWORK is INTEGER
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*> The dimension of the array WORK.
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*> If JOBZ = 'N' and N > 1, LWORK must be queried.
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*> LWORK = MAX(1, 26*N, dimension) where
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*> dimension = max(stage1,stage2) + (KD+1)*N + N
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*> = N*KD + N*max(KD+1,FACTOPTNB)
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*> + max(2*KD*KD, KD*NTHREADS)
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*> + (KD+1)*N + N
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*> where KD is the blocking size of the reduction,
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*> FACTOPTNB is the blocking used by the QR or LQ
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*> algorithm, usually FACTOPTNB=128 is a good choice
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*> NTHREADS is the number of threads used when
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*> openMP compilation is enabled, otherwise =1.
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*> If JOBZ = 'V' and N > 1, LWORK must be queried. Not yet available
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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, RWORK and
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*> IWORK arrays, returns these values as the first entries of
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*> the WORK, RWORK and IWORK arrays, and no error message
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*> related to LWORK or LRWORK or LIWORK is issued by XERBLA.
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*> \endverbatim
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*>
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*> \param[out] RWORK
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*> \verbatim
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*> RWORK is DOUBLE PRECISION array, dimension (MAX(1,LRWORK))
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*> On exit, if INFO = 0, RWORK(1) returns the optimal
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*> (and minimal) LRWORK.
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*> \endverbatim
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*>
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*> \param[in] LRWORK
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*> \verbatim
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*> LRWORK is INTEGER
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*> The length of the array RWORK. LRWORK >= max(1,24*N).
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*>
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*> If LRWORK = -1, then a workspace query is assumed; the
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*> routine only calculates the optimal sizes of the WORK, RWORK
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*> and IWORK arrays, returns these values as the first entries
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*> of the WORK, RWORK and IWORK arrays, and no error message
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*> related to LWORK or LRWORK or 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
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*> (and 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, RWORK
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*> and IWORK arrays, returns these values as the first entries
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*> of the WORK, RWORK and IWORK arrays, and no error message
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*> related to LWORK or LRWORK 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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*> \ingroup complex16HEeigen
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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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*> \par Further Details:
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* =====================
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*>
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*> \verbatim
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*>
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*> All details about the 2stage techniques are available in:
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*>
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*> Azzam Haidar, Hatem Ltaief, and Jack Dongarra.
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*> Parallel reduction to condensed forms for symmetric eigenvalue problems
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*> using aggregated fine-grained and memory-aware kernels. In Proceedings
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*> of 2011 International Conference for High Performance Computing,
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*> Networking, Storage and Analysis (SC '11), New York, NY, USA,
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*> Article 8 , 11 pages.
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*> http://doi.acm.org/10.1145/2063384.2063394
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*>
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*> A. Haidar, J. Kurzak, P. Luszczek, 2013.
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*> An improved parallel singular value algorithm and its implementation
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*> for multicore hardware, In Proceedings of 2013 International Conference
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*> for High Performance Computing, Networking, Storage and Analysis (SC '13).
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*> Denver, Colorado, USA, 2013.
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*> Article 90, 12 pages.
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*> http://doi.acm.org/10.1145/2503210.2503292
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*>
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*> A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra.
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*> A novel hybrid CPU-GPU generalized eigensolver for electronic structure
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*> calculations based on fine-grained memory aware tasks.
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*> International Journal of High Performance Computing Applications.
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*> Volume 28 Issue 2, Pages 196-209, May 2014.
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*> http://hpc.sagepub.com/content/28/2/196
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*>
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*> \endverbatim
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*
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* =====================================================================
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SUBROUTINE ZHEEVR_2STAGE( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU,
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$ IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ,
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$ WORK, LWORK, RWORK, LRWORK, IWORK,
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$ LIWORK, INFO )
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*
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IMPLICIT NONE
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*
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* -- LAPACK driver routine --
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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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*
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* .. Scalar Arguments ..
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CHARACTER JOBZ, RANGE, UPLO
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INTEGER IL, INFO, IU, LDA, LDZ, LIWORK, LRWORK, LWORK,
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$ 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 RWORK( * ), W( * )
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COMPLEX*16 A( LDA, * ), 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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DOUBLE PRECISION ZERO, ONE, TWO
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PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 )
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* ..
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* .. Local Scalars ..
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LOGICAL ALLEIG, INDEIG, LOWER, LQUERY, TEST, VALEIG,
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$ WANTZ, TRYRAC
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CHARACTER ORDER
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INTEGER I, IEEEOK, IINFO, IMAX, INDIBL, INDIFL, INDISP,
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$ INDIWO, INDRD, INDRDD, INDRE, INDREE, INDRWK,
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$ INDTAU, INDWK, INDWKN, ISCALE, ITMP1, J, JJ,
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$ LIWMIN, LLWORK, LLRWORK, LLWRKN, LRWMIN,
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$ LWMIN, NSPLIT, LHTRD, LWTRD, KD, IB, INDHOUS
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DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
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$ SIGMA, SMLNUM, TMP1, VLL, VUU
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* ..
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* .. External Functions ..
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LOGICAL LSAME
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INTEGER ILAENV, ILAENV2STAGE
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DOUBLE PRECISION DLAMCH, ZLANSY
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EXTERNAL LSAME, DLAMCH, ZLANSY, ILAENV, ILAENV2STAGE
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* ..
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* .. External Subroutines ..
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EXTERNAL DCOPY, DSCAL, DSTEBZ, DSTERF, XERBLA, ZDSCAL,
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$ ZHETRD_2STAGE, ZSTEMR, ZSTEIN, ZSWAP, ZUNMTR
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC DBLE, MAX, MIN, SQRT
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* ..
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* .. Executable Statements ..
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*
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* Test the input parameters.
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*
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IEEEOK = ILAENV( 10, 'ZHEEVR', 'N', 1, 2, 3, 4 )
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*
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LOWER = LSAME( UPLO, 'L' )
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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. ( LRWORK.EQ.-1 ) .OR.
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$ ( LIWORK.EQ.-1 ) )
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*
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KD = ILAENV2STAGE( 1, 'ZHETRD_2STAGE', JOBZ, N, -1, -1, -1 )
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IB = ILAENV2STAGE( 2, 'ZHETRD_2STAGE', JOBZ, N, KD, -1, -1 )
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LHTRD = ILAENV2STAGE( 3, 'ZHETRD_2STAGE', JOBZ, N, KD, IB, -1 )
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LWTRD = ILAENV2STAGE( 4, 'ZHETRD_2STAGE', JOBZ, N, KD, IB, -1 )
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LWMIN = N + LHTRD + LWTRD
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LRWMIN = MAX( 1, 24*N )
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LIWMIN = MAX( 1, 10*N )
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*
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INFO = 0
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IF( .NOT.( 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( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
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INFO = -3
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ELSE IF( N.LT.0 ) THEN
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INFO = -4
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ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
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INFO = -6
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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 = -8
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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 = -9
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ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
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INFO = -10
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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 = -15
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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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RWORK( 1 ) = LRWMIN
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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 = -18
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ELSE IF( LRWORK.LT.LRWMIN .AND. .NOT.LQUERY ) THEN
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INFO = -20
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ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
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INFO = -22
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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( 'ZHEEVR_2STAGE', -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 ) THEN
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WORK( 1 ) = 1
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RETURN
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END IF
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*
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IF( N.EQ.1 ) THEN
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WORK( 1 ) = 2
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IF( ALLEIG .OR. INDEIG ) THEN
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M = 1
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W( 1 ) = DBLE( A( 1, 1 ) )
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ELSE
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IF( VL.LT.DBLE( A( 1, 1 ) ) .AND. VU.GE.DBLE( A( 1, 1 ) ) )
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$ THEN
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M = 1
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W( 1 ) = DBLE( A( 1, 1 ) )
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END IF
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END IF
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IF( WANTZ ) THEN
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Z( 1, 1 ) = ONE
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ISUPPZ( 1 ) = 1
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ISUPPZ( 2 ) = 1
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END IF
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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 = DLAMCH( 'Safe minimum' )
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EPS = DLAMCH( '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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* Scale matrix to allowable range, if necessary.
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*
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ISCALE = 0
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ABSTLL = ABSTOL
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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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ANRM = ZLANSY( 'M', UPLO, N, A, LDA, RWORK )
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IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
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ISCALE = 1
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SIGMA = RMIN / ANRM
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ELSE IF( ANRM.GT.RMAX ) THEN
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ISCALE = 1
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SIGMA = RMAX / ANRM
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END IF
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IF( ISCALE.EQ.1 ) THEN
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IF( LOWER ) THEN
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DO 10 J = 1, N
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CALL ZDSCAL( N-J+1, SIGMA, A( J, J ), 1 )
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10 CONTINUE
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ELSE
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DO 20 J = 1, N
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CALL ZDSCAL( J, SIGMA, A( 1, J ), 1 )
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20 CONTINUE
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END IF
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IF( ABSTOL.GT.0 )
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$ ABSTLL = ABSTOL*SIGMA
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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: The IWORK indices are
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* used only if DSTERF or ZSTEMR fail.
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* WORK(INDTAU:INDTAU+N-1) stores the complex scalar factors of the
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* elementary reflectors used in ZHETRD.
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INDTAU = 1
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* INDWK is the starting offset of the remaining complex workspace,
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* and LLWORK is the remaining complex workspace size.
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INDHOUS = INDTAU + N
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INDWK = INDHOUS + LHTRD
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LLWORK = LWORK - INDWK + 1
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* RWORK(INDRD:INDRD+N-1) stores the real tridiagonal's diagonal
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* entries.
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INDRD = 1
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* RWORK(INDRE:INDRE+N-1) stores the off-diagonal entries of the
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* tridiagonal matrix from ZHETRD.
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INDRE = INDRD + N
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* RWORK(INDRDD:INDRDD+N-1) is a copy of the diagonal entries over
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* -written by ZSTEMR (the DSTERF path copies the diagonal to W).
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INDRDD = INDRE + N
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* RWORK(INDREE:INDREE+N-1) is a copy of the off-diagonal entries over
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* -written while computing the eigenvalues in DSTERF and ZSTEMR.
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INDREE = INDRDD + N
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* INDRWK is the starting offset of the left-over real workspace, and
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* LLRWORK is the remaining workspace size.
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INDRWK = INDREE + N
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LLRWORK = LRWORK - INDRWK + 1
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* IWORK(INDIBL:INDIBL+M-1) corresponds to IBLOCK in DSTEBZ 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 DSTEBZ 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
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* ZSTEIN. This information is discarded; if any fail, the driver
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* returns INFO > 0.
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INDIFL = INDISP + N
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* INDIWO is the offset of the remaining integer workspace.
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INDIWO = INDIFL + N
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*
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* Call ZHETRD_2STAGE to reduce Hermitian matrix to tridiagonal form.
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*
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CALL ZHETRD_2STAGE( JOBZ, UPLO, N, A, LDA, RWORK( INDRD ),
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$ RWORK( INDRE ), WORK( INDTAU ),
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$ WORK( INDHOUS ), LHTRD,
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$ WORK( INDWK ), LLWORK, IINFO )
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*
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* If all eigenvalues are desired
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* then call DSTERF or ZSTEMR and ZUNMTR.
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*
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TEST = .FALSE.
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IF( INDEIG ) THEN
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IF( IL.EQ.1 .AND. IU.EQ.N ) THEN
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TEST = .TRUE.
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END IF
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END IF
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IF( ( ALLEIG.OR.TEST ) .AND. ( IEEEOK.EQ.1 ) ) THEN
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IF( .NOT.WANTZ ) THEN
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CALL DCOPY( N, RWORK( INDRD ), 1, W, 1 )
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CALL DCOPY( N-1, RWORK( INDRE ), 1, RWORK( INDREE ), 1 )
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CALL DSTERF( N, W, RWORK( INDREE ), INFO )
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ELSE
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CALL DCOPY( N-1, RWORK( INDRE ), 1, RWORK( INDREE ), 1 )
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CALL DCOPY( N, RWORK( INDRD ), 1, RWORK( INDRDD ), 1 )
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*
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IF (ABSTOL .LE. TWO*N*EPS) THEN
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TRYRAC = .TRUE.
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ELSE
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TRYRAC = .FALSE.
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END IF
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CALL ZSTEMR( JOBZ, 'A', N, RWORK( INDRDD ),
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$ RWORK( INDREE ), VL, VU, IL, IU, M, W,
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$ Z, LDZ, N, ISUPPZ, TRYRAC,
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$ RWORK( INDRWK ), LLRWORK,
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$ IWORK, LIWORK, INFO )
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*
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* Apply unitary matrix used in reduction to tridiagonal
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* form to eigenvectors returned by ZSTEMR.
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*
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IF( WANTZ .AND. INFO.EQ.0 ) THEN
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INDWKN = INDWK
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LLWRKN = LWORK - INDWKN + 1
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CALL ZUNMTR( 'L', UPLO, 'N', N, M, A, LDA,
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$ WORK( INDTAU ), Z, LDZ, WORK( INDWKN ),
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$ LLWRKN, IINFO )
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END IF
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END IF
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*
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*
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IF( INFO.EQ.0 ) THEN
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M = N
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GO TO 30
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END IF
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INFO = 0
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END IF
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*
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* Otherwise, call DSTEBZ and, if eigenvectors are desired, ZSTEIN.
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* Also call DSTEBZ and ZSTEIN if ZSTEMR fails.
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*
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IF( WANTZ ) THEN
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ORDER = 'B'
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ELSE
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ORDER = 'E'
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END IF
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CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL,
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$ RWORK( INDRD ), RWORK( INDRE ), M, NSPLIT, W,
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$ IWORK( INDIBL ), IWORK( INDISP ), RWORK( INDRWK ),
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$ IWORK( INDIWO ), INFO )
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*
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IF( WANTZ ) THEN
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CALL ZSTEIN( N, RWORK( INDRD ), RWORK( INDRE ), M, W,
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$ IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ,
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$ RWORK( INDRWK ), IWORK( INDIWO ), IWORK( INDIFL ),
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$ INFO )
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*
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* Apply unitary matrix used in reduction to tridiagonal
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* form to eigenvectors returned by ZSTEIN.
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*
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INDWKN = INDWK
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LLWRKN = LWORK - INDWKN + 1
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CALL ZUNMTR( 'L', UPLO, 'N', N, M, A, LDA, WORK( INDTAU ), Z,
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$ LDZ, WORK( INDWKN ), LLWRKN, IINFO )
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END IF
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*
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* If matrix was scaled, then rescale eigenvalues appropriately.
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*
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30 CONTINUE
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IF( ISCALE.EQ.1 ) THEN
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IF( INFO.EQ.0 ) THEN
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IMAX = M
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ELSE
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IMAX = INFO - 1
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END IF
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CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
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END IF
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*
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* If eigenvalues are not in order, then sort them, along with
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* eigenvectors.
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*
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IF( WANTZ ) THEN
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DO 50 J = 1, M - 1
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I = 0
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TMP1 = W( J )
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DO 40 JJ = J + 1, M
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IF( W( JJ ).LT.TMP1 ) THEN
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I = JJ
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TMP1 = W( JJ )
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END IF
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40 CONTINUE
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*
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IF( I.NE.0 ) THEN
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ITMP1 = IWORK( INDIBL+I-1 )
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W( I ) = W( J )
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IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 )
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W( J ) = TMP1
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IWORK( INDIBL+J-1 ) = ITMP1
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CALL ZSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
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END IF
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50 CONTINUE
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END IF
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*
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* Set WORK(1) to optimal workspace size.
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*
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WORK( 1 ) = LWMIN
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RWORK( 1 ) = LRWMIN
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IWORK( 1 ) = LIWMIN
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*
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RETURN
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*
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* End of ZHEEVR_2STAGE
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*
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END
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