769 lines
26 KiB
Fortran
769 lines
26 KiB
Fortran
*> \brief \b DSTEMR
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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 DSTEMR + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dstemr.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/dstemr.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/dstemr.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 DSTEMR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU,
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* M, W, Z, LDZ, NZC, ISUPPZ, TRYRAC, WORK, LWORK,
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* IWORK, LIWORK, INFO )
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*
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* .. Scalar Arguments ..
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* CHARACTER JOBZ, RANGE
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* LOGICAL TRYRAC
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* INTEGER IL, INFO, IU, LDZ, NZC, LIWORK, LWORK, M, N
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* DOUBLE PRECISION VL, VU
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* ..
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* .. Array Arguments ..
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* INTEGER ISUPPZ( * ), IWORK( * )
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* DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * )
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* DOUBLE PRECISION 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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*> DSTEMR computes selected eigenvalues and, optionally, eigenvectors
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*> of a real symmetric tridiagonal matrix T. Any such unreduced matrix has
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*> a well defined set of pairwise different real eigenvalues, the corresponding
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*> real eigenvectors are pairwise orthogonal.
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*>
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*> The spectrum may be computed either completely or partially by specifying
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*> either an interval (VL,VU] or a range of indices IL:IU for the desired
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*> eigenvalues.
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*>
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*> Depending on the number of desired eigenvalues, these are computed either
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*> by bisection or the dqds algorithm. Numerically orthogonal eigenvectors are
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*> computed by the use of various suitable L D L^T factorizations near clusters
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*> of close eigenvalues (referred to as RRRs, Relatively Robust
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*> Representations). An informal sketch of the algorithm 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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*> For more details, see:
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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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*> Further Details
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*> 1.DSTEMR works only on machines which follow IEEE-754
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*> floating-point standard in their handling of infinities and NaNs.
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*> This permits the use of efficient inner loops avoiding a check for
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*> zero divisors.
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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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*> \endverbatim
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*>
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*> \param[in] N
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*> \verbatim
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*> N is INTEGER
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*> The order of the matrix. N >= 0.
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*> \endverbatim
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*>
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*> \param[in,out] D
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*> \verbatim
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*> D is DOUBLE PRECISION array, dimension (N)
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*> On entry, the N diagonal elements of the tridiagonal matrix
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*> T. On exit, D is overwritten.
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*> \endverbatim
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*>
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*> \param[in,out] E
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*> \verbatim
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*> E is DOUBLE PRECISION array, dimension (N)
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*> On entry, the (N-1) subdiagonal elements of the tridiagonal
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*> matrix T in elements 1 to N-1 of E. E(N) need not be set on
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*> input, but is used internally as workspace.
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*> On exit, E is overwritten.
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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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*> \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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*>
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*> If RANGE='V', the lower and upper bounds 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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*> \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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*>
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*> If RANGE='I', the indices (in ascending order) of the
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*> smallest and largest eigenvalues to be returned.
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*> 1 <= IL <= IU <= N, 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[out] M
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*> \verbatim
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*> M is INTEGER
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*> The total number of eigenvalues found. 0 <= M <= N.
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*> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
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*> \endverbatim
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*>
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*> \param[out] W
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*> \verbatim
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*> W is DOUBLE PRECISION array, dimension (N)
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*> The first M elements contain the selected eigenvalues in
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*> ascending order.
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*> \endverbatim
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*>
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*> \param[out] Z
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*> \verbatim
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*> Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M) )
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*> If JOBZ = 'V', and if INFO = 0, then the first M columns of Z
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*> contain the orthonormal eigenvectors of the matrix T
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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 can be computed with a workspace
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*> query by setting NZC = -1, see below.
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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', then LDZ >= max(1,N).
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*> \endverbatim
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*>
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*> \param[in] NZC
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*> \verbatim
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*> NZC is INTEGER
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*> The number of eigenvectors to be held in the array Z.
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*> If RANGE = 'A', then NZC >= max(1,N).
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*> If RANGE = 'V', then NZC >= the number of eigenvalues in (VL,VU].
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*> If RANGE = 'I', then NZC >= IU-IL+1.
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*> If NZC = -1, then a workspace query is assumed; the
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*> routine calculates the number of columns of the array Z that
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*> are needed to hold the eigenvectors.
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*> This value is returned as the first entry of the Z array, and
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*> no error message related to NZC is issued by XERBLA.
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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 computed eigenvector
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*> is nonzero only in elements ISUPPZ( 2*i-1 ) through
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*> ISUPPZ( 2*i ). This is relevant in the case when the matrix
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*> is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0.
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*> \endverbatim
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*>
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*> \param[in,out] TRYRAC
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*> \verbatim
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*> TRYRAC is LOGICAL
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*> If TRYRAC.EQ..TRUE., indicates that the code should check whether
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*> the tridiagonal matrix defines its eigenvalues to high relative
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*> accuracy. If so, the code uses relative-accuracy preserving
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*> algorithms that might be (a bit) slower depending on the matrix.
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*> If the matrix does not define its eigenvalues to high relative
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*> accuracy, the code can uses possibly faster algorithms.
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*> If TRYRAC.EQ..FALSE., the code is not required to guarantee
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*> relatively accurate eigenvalues and can use the fastest possible
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*> techniques.
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*> On exit, a .TRUE. TRYRAC will be set to .FALSE. if the matrix
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*> does not define its eigenvalues to high relative accuracy.
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*> \endverbatim
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*>
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*> \param[out] WORK
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*> \verbatim
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*> WORK is DOUBLE PRECISION array, dimension (LWORK)
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*> On exit, if INFO = 0, WORK(1) returns the optimal
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*> (and minimal) LWORK.
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*> \endverbatim
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*>
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*> \param[in] LWORK
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*> \verbatim
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*> LWORK is INTEGER
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*> The dimension of the array WORK. LWORK >= max(1,18*N)
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*> if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'.
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*> If LWORK = -1, then a workspace query is assumed; the routine
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*> only calculates the optimal size of the WORK array, returns
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*> this value as the first entry of the WORK array, and no error
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*> message related to LWORK 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 (LIWORK)
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*> On exit, if INFO = 0, IWORK(1) returns the optimal 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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*> if the eigenvectors are desired, and LIWORK >= max(1,8*N)
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*> if only the eigenvalues are to be computed.
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*> If LIWORK = -1, then a workspace query is assumed; the
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*> routine only calculates the optimal size of the IWORK array,
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*> returns this value as the first entry of the IWORK array, and
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*> no error message related to 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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*> On exit, INFO
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*> = 0: successful exit
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*> < 0: if INFO = -i, the i-th argument had an illegal value
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*> > 0: if INFO = 1X, internal error in DLARRE,
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*> if INFO = 2X, internal error in DLARRV.
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*> Here, the digit X = ABS( IINFO ) < 10, where IINFO is
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*> the nonzero error code returned by DLARRE or
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*> DLARRV, respectively.
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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 November 2013
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*
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*> \ingroup doubleOTHERcomputational
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*
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*> \par Contributors:
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* ==================
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*>
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*> Beresford Parlett, University of California, Berkeley, USA \n
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*> Jim Demmel, University of California, Berkeley, USA \n
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*> Inderjit Dhillon, University of Texas, Austin, USA \n
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*> Osni Marques, LBNL/NERSC, USA \n
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*> Christof Voemel, University of California, Berkeley, USA
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*
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* =====================================================================
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SUBROUTINE DSTEMR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU,
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$ M, W, Z, LDZ, NZC, ISUPPZ, TRYRAC, WORK, LWORK,
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$ IWORK, LIWORK, INFO )
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*
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* -- LAPACK computational routine (version 3.5.0) --
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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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* November 2013
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*
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* .. Scalar Arguments ..
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CHARACTER JOBZ, RANGE
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LOGICAL TRYRAC
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INTEGER IL, INFO, IU, LDZ, NZC, LIWORK, LWORK, M, N
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DOUBLE PRECISION VL, VU
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* ..
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* .. Array Arguments ..
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INTEGER ISUPPZ( * ), IWORK( * )
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DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * )
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DOUBLE PRECISION 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, FOUR, MINRGP
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PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0,
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$ FOUR = 4.0D0,
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$ MINRGP = 1.0D-3 )
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* ..
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* .. Local Scalars ..
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LOGICAL ALLEIG, INDEIG, LQUERY, VALEIG, WANTZ, ZQUERY
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INTEGER I, IBEGIN, IEND, IFIRST, IIL, IINDBL, IINDW,
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$ IINDWK, IINFO, IINSPL, IIU, ILAST, IN, INDD,
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$ INDE2, INDERR, INDGP, INDGRS, INDWRK, ITMP,
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$ ITMP2, J, JBLK, JJ, LIWMIN, LWMIN, NSPLIT,
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$ NZCMIN, OFFSET, WBEGIN, WEND
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DOUBLE PRECISION BIGNUM, CS, EPS, PIVMIN, R1, R2, RMAX, RMIN,
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$ RTOL1, RTOL2, SAFMIN, SCALE, SMLNUM, SN,
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$ THRESH, TMP, TNRM, WL, WU
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* ..
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* ..
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* .. External Functions ..
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LOGICAL LSAME
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DOUBLE PRECISION DLAMCH, DLANST
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EXTERNAL LSAME, DLAMCH, DLANST
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* ..
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* .. External Subroutines ..
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EXTERNAL DCOPY, DLAE2, DLAEV2, DLARRC, DLARRE, DLARRJ,
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$ DLARRR, DLARRV, DLASRT, DSCAL, DSWAP, 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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* Test the input parameters.
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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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ZQUERY = ( NZC.EQ.-1 )
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* DSTEMR needs WORK of size 6*N, IWORK of size 3*N.
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* In addition, DLARRE needs WORK of size 6*N, IWORK of size 5*N.
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* Furthermore, DLARRV needs WORK of size 12*N, IWORK of size 7*N.
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IF( WANTZ ) THEN
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LWMIN = 18*N
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LIWMIN = 10*N
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ELSE
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* need less workspace if only the eigenvalues are wanted
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LWMIN = 12*N
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LIWMIN = 8*N
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ENDIF
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WL = ZERO
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WU = ZERO
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IIL = 0
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IIU = 0
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NSPLIT = 0
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IF( VALEIG ) THEN
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* We do not reference VL, VU in the cases RANGE = 'I','A'
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* The interval (WL, WU] contains all the wanted eigenvalues.
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* It is either given by the user or computed in DLARRE.
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WL = VL
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WU = VU
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ELSEIF( INDEIG ) THEN
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* We do not reference IL, IU in the cases RANGE = 'V','A'
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IIL = IL
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IIU = IU
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ENDIF
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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 IF( VALEIG .AND. N.GT.0 .AND. WU.LE.WL ) THEN
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INFO = -7
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ELSE IF( INDEIG .AND. ( IIL.LT.1 .OR. IIL.GT.N ) ) THEN
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INFO = -8
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ELSE IF( INDEIG .AND. ( IIU.LT.IIL .OR. IIU.GT.N ) ) THEN
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INFO = -9
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ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
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INFO = -13
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ELSE 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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*
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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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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( WANTZ .AND. ALLEIG ) THEN
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NZCMIN = N
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ELSE IF( WANTZ .AND. VALEIG ) THEN
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CALL DLARRC( 'T', N, VL, VU, D, E, SAFMIN,
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$ NZCMIN, ITMP, ITMP2, INFO )
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ELSE IF( WANTZ .AND. INDEIG ) THEN
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NZCMIN = IIU-IIL+1
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ELSE
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* WANTZ .EQ. FALSE.
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NZCMIN = 0
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ENDIF
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IF( ZQUERY .AND. INFO.EQ.0 ) THEN
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Z( 1,1 ) = NZCMIN
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ELSE IF( NZC.LT.NZCMIN .AND. .NOT.ZQUERY ) 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.NE.0 ) THEN
|
|
*
|
|
CALL XERBLA( 'DSTEMR', -INFO )
|
|
*
|
|
RETURN
|
|
ELSE IF( LQUERY .OR. ZQUERY ) THEN
|
|
RETURN
|
|
END IF
|
|
*
|
|
* Handle N = 0, 1, and 2 cases immediately
|
|
*
|
|
M = 0
|
|
IF( N.EQ.0 )
|
|
$ RETURN
|
|
*
|
|
IF( N.EQ.1 ) THEN
|
|
IF( ALLEIG .OR. INDEIG ) THEN
|
|
M = 1
|
|
W( 1 ) = D( 1 )
|
|
ELSE
|
|
IF( WL.LT.D( 1 ) .AND. WU.GE.D( 1 ) ) THEN
|
|
M = 1
|
|
W( 1 ) = D( 1 )
|
|
END IF
|
|
END IF
|
|
IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
|
|
Z( 1, 1 ) = ONE
|
|
ISUPPZ(1) = 1
|
|
ISUPPZ(2) = 1
|
|
END IF
|
|
RETURN
|
|
END IF
|
|
*
|
|
IF( N.EQ.2 ) THEN
|
|
IF( .NOT.WANTZ ) THEN
|
|
CALL DLAE2( D(1), E(1), D(2), R1, R2 )
|
|
ELSE IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
|
|
CALL DLAEV2( D(1), E(1), D(2), R1, R2, CS, SN )
|
|
END IF
|
|
IF( ALLEIG.OR.
|
|
$ (VALEIG.AND.(R2.GT.WL).AND.
|
|
$ (R2.LE.WU)).OR.
|
|
$ (INDEIG.AND.(IIL.EQ.1)) ) THEN
|
|
M = M+1
|
|
W( M ) = R2
|
|
IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
|
|
Z( 1, M ) = -SN
|
|
Z( 2, M ) = CS
|
|
* Note: At most one of SN and CS can be zero.
|
|
IF (SN.NE.ZERO) THEN
|
|
IF (CS.NE.ZERO) THEN
|
|
ISUPPZ(2*M-1) = 1
|
|
ISUPPZ(2*M) = 2
|
|
ELSE
|
|
ISUPPZ(2*M-1) = 1
|
|
ISUPPZ(2*M) = 1
|
|
END IF
|
|
ELSE
|
|
ISUPPZ(2*M-1) = 2
|
|
ISUPPZ(2*M) = 2
|
|
END IF
|
|
ENDIF
|
|
ENDIF
|
|
IF( ALLEIG.OR.
|
|
$ (VALEIG.AND.(R1.GT.WL).AND.
|
|
$ (R1.LE.WU)).OR.
|
|
$ (INDEIG.AND.(IIU.EQ.2)) ) THEN
|
|
M = M+1
|
|
W( M ) = R1
|
|
IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
|
|
Z( 1, M ) = CS
|
|
Z( 2, M ) = SN
|
|
* Note: At most one of SN and CS can be zero.
|
|
IF (SN.NE.ZERO) THEN
|
|
IF (CS.NE.ZERO) THEN
|
|
ISUPPZ(2*M-1) = 1
|
|
ISUPPZ(2*M) = 2
|
|
ELSE
|
|
ISUPPZ(2*M-1) = 1
|
|
ISUPPZ(2*M) = 1
|
|
END IF
|
|
ELSE
|
|
ISUPPZ(2*M-1) = 2
|
|
ISUPPZ(2*M) = 2
|
|
END IF
|
|
ENDIF
|
|
ENDIF
|
|
|
|
ELSE
|
|
|
|
* Continue with general N
|
|
|
|
INDGRS = 1
|
|
INDERR = 2*N + 1
|
|
INDGP = 3*N + 1
|
|
INDD = 4*N + 1
|
|
INDE2 = 5*N + 1
|
|
INDWRK = 6*N + 1
|
|
*
|
|
IINSPL = 1
|
|
IINDBL = N + 1
|
|
IINDW = 2*N + 1
|
|
IINDWK = 3*N + 1
|
|
*
|
|
* Scale matrix to allowable range, if necessary.
|
|
* The allowable range is related to the PIVMIN parameter; see the
|
|
* comments in DLARRD. The preference for scaling small values
|
|
* up is heuristic; we expect users' matrices not to be close to the
|
|
* RMAX threshold.
|
|
*
|
|
SCALE = ONE
|
|
TNRM = DLANST( 'M', N, D, E )
|
|
IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN
|
|
SCALE = RMIN / TNRM
|
|
ELSE IF( TNRM.GT.RMAX ) THEN
|
|
SCALE = RMAX / TNRM
|
|
END IF
|
|
IF( SCALE.NE.ONE ) THEN
|
|
CALL DSCAL( N, SCALE, D, 1 )
|
|
CALL DSCAL( N-1, SCALE, E, 1 )
|
|
TNRM = TNRM*SCALE
|
|
IF( VALEIG ) THEN
|
|
* If eigenvalues in interval have to be found,
|
|
* scale (WL, WU] accordingly
|
|
WL = WL*SCALE
|
|
WU = WU*SCALE
|
|
ENDIF
|
|
END IF
|
|
*
|
|
* Compute the desired eigenvalues of the tridiagonal after splitting
|
|
* into smaller subblocks if the corresponding off-diagonal elements
|
|
* are small
|
|
* THRESH is the splitting parameter for DLARRE
|
|
* A negative THRESH forces the old splitting criterion based on the
|
|
* size of the off-diagonal. A positive THRESH switches to splitting
|
|
* which preserves relative accuracy.
|
|
*
|
|
IF( TRYRAC ) THEN
|
|
* Test whether the matrix warrants the more expensive relative approach.
|
|
CALL DLARRR( N, D, E, IINFO )
|
|
ELSE
|
|
* The user does not care about relative accurately eigenvalues
|
|
IINFO = -1
|
|
ENDIF
|
|
* Set the splitting criterion
|
|
IF (IINFO.EQ.0) THEN
|
|
THRESH = EPS
|
|
ELSE
|
|
THRESH = -EPS
|
|
* relative accuracy is desired but T does not guarantee it
|
|
TRYRAC = .FALSE.
|
|
ENDIF
|
|
*
|
|
IF( TRYRAC ) THEN
|
|
* Copy original diagonal, needed to guarantee relative accuracy
|
|
CALL DCOPY(N,D,1,WORK(INDD),1)
|
|
ENDIF
|
|
* Store the squares of the offdiagonal values of T
|
|
DO 5 J = 1, N-1
|
|
WORK( INDE2+J-1 ) = E(J)**2
|
|
5 CONTINUE
|
|
|
|
* Set the tolerance parameters for bisection
|
|
IF( .NOT.WANTZ ) THEN
|
|
* DLARRE computes the eigenvalues to full precision.
|
|
RTOL1 = FOUR * EPS
|
|
RTOL2 = FOUR * EPS
|
|
ELSE
|
|
* DLARRE computes the eigenvalues to less than full precision.
|
|
* DLARRV will refine the eigenvalue approximations, and we can
|
|
* need less accurate initial bisection in DLARRE.
|
|
* Note: these settings do only affect the subset case and DLARRE
|
|
RTOL1 = SQRT(EPS)
|
|
RTOL2 = MAX( SQRT(EPS)*5.0D-3, FOUR * EPS )
|
|
ENDIF
|
|
CALL DLARRE( RANGE, N, WL, WU, IIL, IIU, D, E,
|
|
$ WORK(INDE2), RTOL1, RTOL2, THRESH, NSPLIT,
|
|
$ IWORK( IINSPL ), M, W, WORK( INDERR ),
|
|
$ WORK( INDGP ), IWORK( IINDBL ),
|
|
$ IWORK( IINDW ), WORK( INDGRS ), PIVMIN,
|
|
$ WORK( INDWRK ), IWORK( IINDWK ), IINFO )
|
|
IF( IINFO.NE.0 ) THEN
|
|
INFO = 10 + ABS( IINFO )
|
|
RETURN
|
|
END IF
|
|
* Note that if RANGE .NE. 'V', DLARRE computes bounds on the desired
|
|
* part of the spectrum. All desired eigenvalues are contained in
|
|
* (WL,WU]
|
|
|
|
|
|
IF( WANTZ ) THEN
|
|
*
|
|
* Compute the desired eigenvectors corresponding to the computed
|
|
* eigenvalues
|
|
*
|
|
CALL DLARRV( N, WL, WU, D, E,
|
|
$ PIVMIN, IWORK( IINSPL ), M,
|
|
$ 1, M, MINRGP, RTOL1, RTOL2,
|
|
$ W, WORK( INDERR ), WORK( INDGP ), IWORK( IINDBL ),
|
|
$ IWORK( IINDW ), WORK( INDGRS ), Z, LDZ,
|
|
$ ISUPPZ, WORK( INDWRK ), IWORK( IINDWK ), IINFO )
|
|
IF( IINFO.NE.0 ) THEN
|
|
INFO = 20 + ABS( IINFO )
|
|
RETURN
|
|
END IF
|
|
ELSE
|
|
* DLARRE computes eigenvalues of the (shifted) root representation
|
|
* DLARRV returns the eigenvalues of the unshifted matrix.
|
|
* However, if the eigenvectors are not desired by the user, we need
|
|
* to apply the corresponding shifts from DLARRE to obtain the
|
|
* eigenvalues of the original matrix.
|
|
DO 20 J = 1, M
|
|
ITMP = IWORK( IINDBL+J-1 )
|
|
W( J ) = W( J ) + E( IWORK( IINSPL+ITMP-1 ) )
|
|
20 CONTINUE
|
|
END IF
|
|
*
|
|
|
|
IF ( TRYRAC ) THEN
|
|
* Refine computed eigenvalues so that they are relatively accurate
|
|
* with respect to the original matrix T.
|
|
IBEGIN = 1
|
|
WBEGIN = 1
|
|
DO 39 JBLK = 1, IWORK( IINDBL+M-1 )
|
|
IEND = IWORK( IINSPL+JBLK-1 )
|
|
IN = IEND - IBEGIN + 1
|
|
WEND = WBEGIN - 1
|
|
* check if any eigenvalues have to be refined in this block
|
|
36 CONTINUE
|
|
IF( WEND.LT.M ) THEN
|
|
IF( IWORK( IINDBL+WEND ).EQ.JBLK ) THEN
|
|
WEND = WEND + 1
|
|
GO TO 36
|
|
END IF
|
|
END IF
|
|
IF( WEND.LT.WBEGIN ) THEN
|
|
IBEGIN = IEND + 1
|
|
GO TO 39
|
|
END IF
|
|
|
|
OFFSET = IWORK(IINDW+WBEGIN-1)-1
|
|
IFIRST = IWORK(IINDW+WBEGIN-1)
|
|
ILAST = IWORK(IINDW+WEND-1)
|
|
RTOL2 = FOUR * EPS
|
|
CALL DLARRJ( IN,
|
|
$ WORK(INDD+IBEGIN-1), WORK(INDE2+IBEGIN-1),
|
|
$ IFIRST, ILAST, RTOL2, OFFSET, W(WBEGIN),
|
|
$ WORK( INDERR+WBEGIN-1 ),
|
|
$ WORK( INDWRK ), IWORK( IINDWK ), PIVMIN,
|
|
$ TNRM, IINFO )
|
|
IBEGIN = IEND + 1
|
|
WBEGIN = WEND + 1
|
|
39 CONTINUE
|
|
ENDIF
|
|
*
|
|
* If matrix was scaled, then rescale eigenvalues appropriately.
|
|
*
|
|
IF( SCALE.NE.ONE ) THEN
|
|
CALL DSCAL( M, ONE / SCALE, W, 1 )
|
|
END IF
|
|
|
|
END IF
|
|
|
|
*
|
|
* If eigenvalues are not in increasing order, then sort them,
|
|
* possibly along with eigenvectors.
|
|
*
|
|
IF( NSPLIT.GT.1 .OR. N.EQ.2 ) THEN
|
|
IF( .NOT. WANTZ ) THEN
|
|
CALL DLASRT( 'I', M, W, IINFO )
|
|
IF( IINFO.NE.0 ) THEN
|
|
INFO = 3
|
|
RETURN
|
|
END IF
|
|
ELSE
|
|
DO 60 J = 1, M - 1
|
|
I = 0
|
|
TMP = W( J )
|
|
DO 50 JJ = J + 1, M
|
|
IF( W( JJ ).LT.TMP ) THEN
|
|
I = JJ
|
|
TMP = W( JJ )
|
|
END IF
|
|
50 CONTINUE
|
|
IF( I.NE.0 ) THEN
|
|
W( I ) = W( J )
|
|
W( J ) = TMP
|
|
IF( WANTZ ) THEN
|
|
CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
|
|
ITMP = ISUPPZ( 2*I-1 )
|
|
ISUPPZ( 2*I-1 ) = ISUPPZ( 2*J-1 )
|
|
ISUPPZ( 2*J-1 ) = ITMP
|
|
ITMP = ISUPPZ( 2*I )
|
|
ISUPPZ( 2*I ) = ISUPPZ( 2*J )
|
|
ISUPPZ( 2*J ) = ITMP
|
|
END IF
|
|
END IF
|
|
60 CONTINUE
|
|
END IF
|
|
ENDIF
|
|
*
|
|
*
|
|
WORK( 1 ) = LWMIN
|
|
IWORK( 1 ) = LIWMIN
|
|
RETURN
|
|
*
|
|
* End of DSTEMR
|
|
*
|
|
END
|