julie
be91086043
Those are just cosmetic changes to update version number and various other minor change.
665 lines
24 KiB
FortranFixed
665 lines
24 KiB
FortranFixed
SUBROUTINE CSTEMR( 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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IMPLICIT NONE
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*
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* -- LAPACK computational routine (version 3.2.1) --
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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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* -- April 2009 --
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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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REAL VL, VU
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* ..
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* .. Array Arguments ..
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INTEGER ISUPPZ( * ), IWORK( * )
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REAL D( * ), E( * ), W( * ), WORK( * )
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COMPLEX Z( LDZ, * )
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* ..
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*
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* Purpose
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* =======
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*
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* CSTEMR 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.CSTEMR 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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*
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* 2. LAPACK routines can be used to reduce a complex Hermitean matrix to
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* real symmetric tridiagonal form.
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*
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* (Any complex Hermitean tridiagonal matrix has real values on its diagonal
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* and potentially complex numbers on its off-diagonals. By applying a
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* similarity transform with an appropriate diagonal matrix
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* diag(1,e^{i \phy_1}, ... , e^{i \phy_{n-1}}), the complex Hermitean
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* matrix can be transformed into a real symmetric matrix and complex
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* arithmetic can be entirely avoided.)
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*
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* While the eigenvectors of the real symmetric tridiagonal matrix are real,
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* the eigenvectors of original complex Hermitean matrix have complex entries
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* in general.
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* Since LAPACK drivers overwrite the matrix data with the eigenvectors,
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* CSTEMR accepts complex workspace to facilitate interoperability
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* with CUNMTR or CUPMTR.
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*
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* Arguments
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* =========
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*
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* JOBZ (input) CHARACTER*1
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* = 'N': Compute eigenvalues only;
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* = 'V': Compute eigenvalues and eigenvectors.
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*
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* RANGE (input) CHARACTER*1
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* = 'A': all eigenvalues will be found.
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* = 'V': all eigenvalues in the half-open interval (VL,VU]
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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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*
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* N (input) INTEGER
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* The order of the matrix. N >= 0.
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*
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* D (input/output) REAL 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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*
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* E (input/output) REAL 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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*
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* VL (input) REAL
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* VU (input) REAL
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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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*
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* IL (input) INTEGER
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* IU (input) INTEGER
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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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*
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* M (output) 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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*
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* W (output) REAL array, dimension (N)
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* The first M elements contain the selected eigenvalues in
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* ascending order.
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*
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* Z (output) COMPLEX 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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*
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* LDZ (input) 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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*
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* NZC (input) 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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*
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* ISUPPZ (output) 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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*
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* TRYRAC (input/output) 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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*
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* WORK (workspace/output) REAL 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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*
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* LWORK (input) 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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*
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* IWORK (workspace/output) INTEGER array, dimension (LIWORK)
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* On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
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*
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* LIWORK (input) 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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*
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* INFO (output) 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 SLARRE,
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* if INFO = 2X, internal error in CLARRV.
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* Here, the digit X = ABS( IINFO ) < 10, where IINFO is
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* the nonzero error code returned by SLARRE or
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* CLARRV, respectively.
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*
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*
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* Further Details
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* ===============
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*
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* Based on contributions by
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* Beresford Parlett, University of California, Berkeley, USA
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* Jim Demmel, University of California, Berkeley, USA
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* Inderjit Dhillon, University of Texas, Austin, USA
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* Osni Marques, LBNL/NERSC, USA
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* Christof Voemel, University of California, Berkeley, USA
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*
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* =====================================================================
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*
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* .. Parameters ..
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REAL ZERO, ONE, FOUR, MINRGP
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PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0,
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$ FOUR = 4.0E0,
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$ MINRGP = 3.0E-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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REAL 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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REAL SLAMCH, SLANST
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EXTERNAL LSAME, SLAMCH, SLANST
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* ..
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* .. External Subroutines ..
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EXTERNAL CLARRV, CSWAP, SCOPY, SLAE2, SLAEV2, SLARRC,
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$ SLARRE, SLARRJ, SLARRR, SLASRT, SSCAL, 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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* SSTEMR needs WORK of size 6*N, IWORK of size 3*N.
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* In addition, SLARRE needs WORK of size 6*N, IWORK of size 5*N.
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* Furthermore, CLARRV 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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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 SLARRE.
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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 = SLAMCH( 'Safe minimum' )
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EPS = SLAMCH( 'Precision' )
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SMLNUM = SAFMIN / EPS
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BIGNUM = ONE / SMLNUM
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RMIN = SQRT( SMLNUM )
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RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
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*
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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 SLARRC( '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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IF( INFO.NE.0 ) THEN
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*
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CALL XERBLA( 'CSTEMR', -INFO )
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*
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RETURN
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ELSE IF( LQUERY .OR. ZQUERY ) THEN
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RETURN
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END IF
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*
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* Handle N = 0, 1, and 2 cases immediately
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*
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M = 0
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IF( N.EQ.0 )
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$ RETURN
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*
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IF( N.EQ.1 ) THEN
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IF( ALLEIG .OR. INDEIG ) THEN
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M = 1
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W( 1 ) = D( 1 )
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ELSE
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IF( WL.LT.D( 1 ) .AND. WU.GE.D( 1 ) ) THEN
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M = 1
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W( 1 ) = D( 1 )
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END IF
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END IF
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IF( WANTZ.AND.(.NOT.ZQUERY) ) 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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IF( N.EQ.2 ) THEN
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IF( .NOT.WANTZ ) THEN
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CALL SLAE2( D(1), E(1), D(2), R1, R2 )
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ELSE IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
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CALL SLAEV2( D(1), E(1), D(2), R1, R2, CS, SN )
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END IF
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IF( ALLEIG.OR.
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$ (VALEIG.AND.(R2.GT.WL).AND.
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$ (R2.LE.WU)).OR.
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$ (INDEIG.AND.(IIL.EQ.1)) ) THEN
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M = M+1
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W( M ) = R2
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IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
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Z( 1, M ) = -SN
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Z( 2, M ) = CS
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* Note: At most one of SN and CS can be zero.
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IF (SN.NE.ZERO) THEN
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IF (CS.NE.ZERO) THEN
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ISUPPZ(2*M-1) = 1
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ISUPPZ(2*M-1) = 2
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ELSE
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ISUPPZ(2*M-1) = 1
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ISUPPZ(2*M-1) = 1
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END IF
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ELSE
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ISUPPZ(2*M-1) = 2
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ISUPPZ(2*M) = 2
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END IF
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ENDIF
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ENDIF
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IF( ALLEIG.OR.
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$ (VALEIG.AND.(R1.GT.WL).AND.
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$ (R1.LE.WU)).OR.
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$ (INDEIG.AND.(IIU.EQ.2)) ) THEN
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M = M+1
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W( M ) = R1
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IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
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Z( 1, M ) = CS
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Z( 2, M ) = SN
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* Note: At most one of SN and CS can be zero.
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IF (SN.NE.ZERO) THEN
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IF (CS.NE.ZERO) THEN
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ISUPPZ(2*M-1) = 1
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ISUPPZ(2*M-1) = 2
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ELSE
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ISUPPZ(2*M-1) = 1
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ISUPPZ(2*M-1) = 1
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END IF
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ELSE
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ISUPPZ(2*M-1) = 2
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ISUPPZ(2*M) = 2
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END IF
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ENDIF
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ENDIF
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RETURN
|
|
END IF
|
|
|
|
* 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 SLARRD. The preference for scaling small values
|
|
* up is heuristic; we expect users' matrices not to be close to the
|
|
* RMAX threshold.
|
|
*
|
|
SCALE = ONE
|
|
TNRM = SLANST( '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 SSCAL( N, SCALE, D, 1 )
|
|
CALL SSCAL( 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 SLARRE
|
|
* 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 SLARRR( 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 SCOPY(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
|
|
* SLARRE computes the eigenvalues to full precision.
|
|
RTOL1 = FOUR * EPS
|
|
RTOL2 = FOUR * EPS
|
|
ELSE
|
|
* SLARRE computes the eigenvalues to less than full precision.
|
|
* CLARRV will refine the eigenvalue approximations, and we only
|
|
* need less accurate initial bisection in SLARRE.
|
|
* Note: these settings do only affect the subset case and SLARRE
|
|
RTOL1 = MAX( SQRT(EPS)*5.0E-2, FOUR * EPS )
|
|
RTOL2 = MAX( SQRT(EPS)*5.0E-3, FOUR * EPS )
|
|
ENDIF
|
|
CALL SLARRE( 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', SLARRE 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 CLARRV( 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
|
|
* SLARRE computes eigenvalues of the (shifted) root representation
|
|
* CLARRV 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 SLARRE 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 SLARRJ( 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 SSCAL( M, ONE / SCALE, W, 1 )
|
|
END IF
|
|
*
|
|
* If eigenvalues are not in increasing order, then sort them,
|
|
* possibly along with eigenvectors.
|
|
*
|
|
IF( NSPLIT.GT.1 ) THEN
|
|
IF( .NOT. WANTZ ) THEN
|
|
CALL SLASRT( '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 CSWAP( 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 CSTEMR
|
|
*
|
|
END
|