| 1 | SUBROUTINE CSTEMR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU,
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| 2 | $ M, W, Z, LDZ, NZC, ISUPPZ, TRYRAC, WORK, LWORK,
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| 3 | $ IWORK, LIWORK, INFO )
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| 4 | IMPLICIT NONE
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| 5 | *
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| 6 | * -- LAPACK computational routine (version 3.1) --
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| 7 | * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
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| 8 | * November 2006
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| 9 | *
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| 10 | * .. Scalar Arguments ..
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| 11 | CHARACTER JOBZ, RANGE
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| 12 | LOGICAL TRYRAC
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| 13 | INTEGER IL, INFO, IU, LDZ, NZC, LIWORK, LWORK, M, N
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| 14 | REAL VL, VU
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| 15 | * ..
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| 16 | * .. Array Arguments ..
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| 17 | INTEGER ISUPPZ( * ), IWORK( * )
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| 18 | REAL D( * ), E( * ), W( * ), WORK( * )
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| 19 | COMPLEX Z( LDZ, * )
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| 20 | * ..
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| 21 | *
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| 22 | * Purpose
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| 23 | * =======
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| 24 | *
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| 25 | * CSTEMR computes selected eigenvalues and, optionally, eigenvectors
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| 26 | * of a real symmetric tridiagonal matrix T. Any such unreduced matrix has
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| 27 | * a well defined set of pairwise different real eigenvalues, the corresponding
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| 28 | * real eigenvectors are pairwise orthogonal.
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| 29 | *
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| 30 | * The spectrum may be computed either completely or partially by specifying
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| 31 | * either an interval (VL,VU] or a range of indices IL:IU for the desired
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| 32 | * eigenvalues.
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| 33 | *
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| 34 | * Depending on the number of desired eigenvalues, these are computed either
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| 35 | * by bisection or the dqds algorithm. Numerically orthogonal eigenvectors are
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| 36 | * computed by the use of various suitable L D L^T factorizations near clusters
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| 37 | * of close eigenvalues (referred to as RRRs, Relatively Robust
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| 38 | * Representations). An informal sketch of the algorithm follows.
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| 39 | *
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| 40 | * For each unreduced block (submatrix) of T,
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| 41 | * (a) Compute T - sigma I = L D L^T, so that L and D
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| 42 | * define all the wanted eigenvalues to high relative accuracy.
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| 43 | * This means that small relative changes in the entries of D and L
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| 44 | * cause only small relative changes in the eigenvalues and
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| 45 | * eigenvectors. The standard (unfactored) representation of the
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| 46 | * tridiagonal matrix T does not have this property in general.
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| 47 | * (b) Compute the eigenvalues to suitable accuracy.
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| 48 | * If the eigenvectors are desired, the algorithm attains full
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| 49 | * accuracy of the computed eigenvalues only right before
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| 50 | * the corresponding vectors have to be computed, see steps c) and d).
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| 51 | * (c) For each cluster of close eigenvalues, select a new
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| 52 | * shift close to the cluster, find a new factorization, and refine
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| 53 | * the shifted eigenvalues to suitable accuracy.
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| 54 | * (d) For each eigenvalue with a large enough relative separation compute
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| 55 | * the corresponding eigenvector by forming a rank revealing twisted
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| 56 | * factorization. Go back to (c) for any clusters that remain.
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| 57 | *
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| 58 | * For more details, see:
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| 59 | * - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations
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| 60 | * to compute orthogonal eigenvectors of symmetric tridiagonal matrices,"
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| 61 | * Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
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| 62 | * - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and
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| 63 | * Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25,
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| 64 | * 2004. Also LAPACK Working Note 154.
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| 65 | * - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric
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| 66 | * tridiagonal eigenvalue/eigenvector problem",
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| 67 | * Computer Science Division Technical Report No. UCB/CSD-97-971,
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| 68 | * UC Berkeley, May 1997.
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| 69 | *
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| 70 | * Notes:
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| 71 | * 1.CSTEMR works only on machines which follow IEEE-754
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| 72 | * floating-point standard in their handling of infinities and NaNs.
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| 73 | * This permits the use of efficient inner loops avoiding a check for
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| 74 | * zero divisors.
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| 75 | *
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| 76 | * 2. LAPACK routines can be used to reduce a complex Hermitean matrix to
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| 77 | * real symmetric tridiagonal form.
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| 78 | *
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| 79 | * (Any complex Hermitean tridiagonal matrix has real values on its diagonal
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| 80 | * and potentially complex numbers on its off-diagonals. By applying a
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| 81 | * similarity transform with an appropriate diagonal matrix
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| 82 | * diag(1,e^{i \phy_1}, ... , e^{i \phy_{n-1}}), the complex Hermitean
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| 83 | * matrix can be transformed into a real symmetric matrix and complex
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| 84 | * arithmetic can be entirely avoided.)
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| 85 | *
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| 86 | * While the eigenvectors of the real symmetric tridiagonal matrix are real,
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| 87 | * the eigenvectors of original complex Hermitean matrix have complex entries
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| 88 | * in general.
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| 89 | * Since LAPACK drivers overwrite the matrix data with the eigenvectors,
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| 90 | * CSTEMR accepts complex workspace to facilitate interoperability
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| 91 | * with CUNMTR or CUPMTR.
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| 92 | *
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| 93 | * Arguments
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| 94 | * =========
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| 95 | *
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| 96 | * JOBZ (input) CHARACTER*1
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| 97 | * = 'N': Compute eigenvalues only;
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| 98 | * = 'V': Compute eigenvalues and eigenvectors.
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| 99 | *
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| 100 | * RANGE (input) CHARACTER*1
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| 101 | * = 'A': all eigenvalues will be found.
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| 102 | * = 'V': all eigenvalues in the half-open interval (VL,VU]
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| 103 | * will be found.
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| 104 | * = 'I': the IL-th through IU-th eigenvalues will be found.
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| 105 | *
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| 106 | * N (input) INTEGER
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| 107 | * The order of the matrix. N >= 0.
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| 108 | *
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| 109 | * D (input/output) REAL array, dimension (N)
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| 110 | * On entry, the N diagonal elements of the tridiagonal matrix
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| 111 | * T. On exit, D is overwritten.
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| 112 | *
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| 113 | * E (input/output) REAL array, dimension (N)
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| 114 | * On entry, the (N-1) subdiagonal elements of the tridiagonal
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| 115 | * matrix T in elements 1 to N-1 of E. E(N) need not be set on
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| 116 | * input, but is used internally as workspace.
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| 117 | * On exit, E is overwritten.
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| 118 | *
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| 119 | * VL (input) REAL
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| 120 | * VU (input) REAL
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| 121 | * If RANGE='V', the lower and upper bounds of the interval to
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| 122 | * be searched for eigenvalues. VL < VU.
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| 123 | * Not referenced if RANGE = 'A' or 'I'.
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| 124 | *
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| 125 | * IL (input) INTEGER
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| 126 | * IU (input) INTEGER
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| 127 | * If RANGE='I', the indices (in ascending order) of the
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| 128 | * smallest and largest eigenvalues to be returned.
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| 129 | * 1 <= IL <= IU <= N, if N > 0.
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| 130 | * Not referenced if RANGE = 'A' or 'V'.
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| 131 | *
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| 132 | * M (output) INTEGER
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| 133 | * The total number of eigenvalues found. 0 <= M <= N.
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| 134 | * If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
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| 135 | *
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| 136 | * W (output) REAL array, dimension (N)
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| 137 | * The first M elements contain the selected eigenvalues in
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| 138 | * ascending order.
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| 139 | *
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| 140 | * Z (output) COMPLEX array, dimension (LDZ, max(1,M) )
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| 141 | * If JOBZ = 'V', and if INFO = 0, then the first M columns of Z
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| 142 | * contain the orthonormal eigenvectors of the matrix T
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| 143 | * corresponding to the selected eigenvalues, with the i-th
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| 144 | * column of Z holding the eigenvector associated with W(i).
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| 145 | * If JOBZ = 'N', then Z is not referenced.
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| 146 | * Note: the user must ensure that at least max(1,M) columns are
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| 147 | * supplied in the array Z; if RANGE = 'V', the exact value of M
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| 148 | * is not known in advance and can be computed with a workspace
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| 149 | * query by setting NZC = -1, see below.
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| 150 | *
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| 151 | * LDZ (input) INTEGER
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| 152 | * The leading dimension of the array Z. LDZ >= 1, and if
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| 153 | * JOBZ = 'V', then LDZ >= max(1,N).
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| 154 | *
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| 155 | * NZC (input) INTEGER
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| 156 | * The number of eigenvectors to be held in the array Z.
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| 157 | * If RANGE = 'A', then NZC >= max(1,N).
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| 158 | * If RANGE = 'V', then NZC >= the number of eigenvalues in (VL,VU].
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| 159 | * If RANGE = 'I', then NZC >= IU-IL+1.
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| 160 | * If NZC = -1, then a workspace query is assumed; the
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| 161 | * routine calculates the number of columns of the array Z that
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| 162 | * are needed to hold the eigenvectors.
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| 163 | * This value is returned as the first entry of the Z array, and
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| 164 | * no error message related to NZC is issued by XERBLA.
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| 165 | *
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| 166 | * ISUPPZ (output) INTEGER ARRAY, dimension ( 2*max(1,M) )
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| 167 | * The support of the eigenvectors in Z, i.e., the indices
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| 168 | * indicating the nonzero elements in Z. The i-th computed eigenvector
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| 169 | * is nonzero only in elements ISUPPZ( 2*i-1 ) through
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| 170 | * ISUPPZ( 2*i ). This is relevant in the case when the matrix
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| 171 | * is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0.
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| 172 | *
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| 173 | * TRYRAC (input/output) LOGICAL
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| 174 | * If TRYRAC.EQ..TRUE., indicates that the code should check whether
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| 175 | * the tridiagonal matrix defines its eigenvalues to high relative
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| 176 | * accuracy. If so, the code uses relative-accuracy preserving
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| 177 | * algorithms that might be (a bit) slower depending on the matrix.
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| 178 | * If the matrix does not define its eigenvalues to high relative
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| 179 | * accuracy, the code can uses possibly faster algorithms.
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| 180 | * If TRYRAC.EQ..FALSE., the code is not required to guarantee
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| 181 | * relatively accurate eigenvalues and can use the fastest possible
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| 182 | * techniques.
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| 183 | * On exit, a .TRUE. TRYRAC will be set to .FALSE. if the matrix
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| 184 | * does not define its eigenvalues to high relative accuracy.
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| 185 | *
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| 186 | * WORK (workspace/output) REAL array, dimension (LWORK)
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| 187 | * On exit, if INFO = 0, WORK(1) returns the optimal
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| 188 | * (and minimal) LWORK.
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| 189 | *
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| 190 | * LWORK (input) INTEGER
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| 191 | * The dimension of the array WORK. LWORK >= max(1,18*N)
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| 192 | * if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'.
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| 193 | * If LWORK = -1, then a workspace query is assumed; the routine
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| 194 | * only calculates the optimal size of the WORK array, returns
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| 195 | * this value as the first entry of the WORK array, and no error
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| 196 | * message related to LWORK is issued by XERBLA.
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| 197 | *
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| 198 | * IWORK (workspace/output) INTEGER array, dimension (LIWORK)
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| 199 | * On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
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| 200 | *
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| 201 | * LIWORK (input) INTEGER
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| 202 | * The dimension of the array IWORK. LIWORK >= max(1,10*N)
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| 203 | * if the eigenvectors are desired, and LIWORK >= max(1,8*N)
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| 204 | * if only the eigenvalues are to be computed.
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| 205 | * If LIWORK = -1, then a workspace query is assumed; the
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| 206 | * routine only calculates the optimal size of the IWORK array,
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| 207 | * returns this value as the first entry of the IWORK array, and
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| 208 | * no error message related to LIWORK is issued by XERBLA.
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| 209 | *
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| 210 | * INFO (output) INTEGER
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| 211 | * On exit, INFO
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| 212 | * = 0: successful exit
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| 213 | * < 0: if INFO = -i, the i-th argument had an illegal value
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| 214 | * > 0: if INFO = 1X, internal error in SLARRE,
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| 215 | * if INFO = 2X, internal error in CLARRV.
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| 216 | * Here, the digit X = ABS( IINFO ) < 10, where IINFO is
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| 217 | * the nonzero error code returned by SLARRE or
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| 218 | * CLARRV, respectively.
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| 219 | *
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| 220 | *
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| 221 | * Further Details
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| 222 | * ===============
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| 223 | *
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| 224 | * Based on contributions by
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| 225 | * Beresford Parlett, University of California, Berkeley, USA
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| 226 | * Jim Demmel, University of California, Berkeley, USA
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| 227 | * Inderjit Dhillon, University of Texas, Austin, USA
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| 228 | * Osni Marques, LBNL/NERSC, USA
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| 229 | * Christof Voemel, University of California, Berkeley, USA
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| 230 | *
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| 231 | * =====================================================================
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| 232 | *
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| 233 | * .. Parameters ..
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| 234 | REAL ZERO, ONE, FOUR, MINRGP
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| 235 | PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0,
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| 236 | $ FOUR = 4.0E0,
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| 237 | $ MINRGP = 3.0E-3 )
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| 238 | * ..
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| 239 | * .. Local Scalars ..
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| 240 | LOGICAL ALLEIG, INDEIG, LQUERY, VALEIG, WANTZ, ZQUERY
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| 241 | INTEGER I, IBEGIN, IEND, IFIRST, IIL, IINDBL, IINDW,
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| 242 | $ IINDWK, IINFO, IINSPL, IIU, ILAST, IN, INDD,
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| 243 | $ INDE2, INDERR, INDGP, INDGRS, INDWRK, ITMP,
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| 244 | $ ITMP2, J, JBLK, JJ, LIWMIN, LWMIN, NSPLIT,
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| 245 | $ NZCMIN, OFFSET, WBEGIN, WEND
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| 246 | REAL BIGNUM, CS, EPS, PIVMIN, R1, R2, RMAX, RMIN,
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| 247 | $ RTOL1, RTOL2, SAFMIN, SCALE, SMLNUM, SN,
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| 248 | $ THRESH, TMP, TNRM, WL, WU
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| 249 | * ..
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| 250 | * ..
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| 251 | * .. External Functions ..
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| 252 | LOGICAL LSAME
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| 253 | REAL SLAMCH, SLANST
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| 254 | EXTERNAL LSAME, SLAMCH, SLANST
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| 255 | * ..
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| 256 | * .. External Subroutines ..
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| 257 | EXTERNAL CLARRV, CSWAP, SCOPY, SLAE2, SLAEV2, SLARRC,
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| 258 | $ SLARRE, SLARRJ, SLARRR, SLASRT, SSCAL, XERBLA
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| 259 | * ..
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| 260 | * .. Intrinsic Functions ..
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| 261 | INTRINSIC MAX, MIN, SQRT
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| 262 |
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| 263 |
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| 264 | * ..
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| 265 | * .. Executable Statements ..
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| 266 | *
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| 267 | * Test the input parameters.
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| 268 | *
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| 269 | WANTZ = LSAME( JOBZ, 'V' )
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| 270 | ALLEIG = LSAME( RANGE, 'A' )
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| 271 | VALEIG = LSAME( RANGE, 'V' )
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| 272 | INDEIG = LSAME( RANGE, 'I' )
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| 273 | *
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| 274 | LQUERY = ( ( LWORK.EQ.-1 ).OR.( LIWORK.EQ.-1 ) )
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| 275 | ZQUERY = ( NZC.EQ.-1 )
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| 276 | TRYRAC = ( INFO.NE.0 )
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| 277 |
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| 278 | * SSTEMR needs WORK of size 6*N, IWORK of size 3*N.
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| 279 | * In addition, SLARRE needs WORK of size 6*N, IWORK of size 5*N.
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| 280 | * Furthermore, CLARRV needs WORK of size 12*N, IWORK of size 7*N.
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| 281 | IF( WANTZ ) THEN
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| 282 | LWMIN = 18*N
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| 283 | LIWMIN = 10*N
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| 284 | ELSE
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| 285 | * need less workspace if only the eigenvalues are wanted
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| 286 | LWMIN = 12*N
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| 287 | LIWMIN = 8*N
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| 288 | ENDIF
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| 289 |
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| 290 | WL = ZERO
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| 291 | WU = ZERO
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| 292 | IIL = 0
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| 293 | IIU = 0
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| 294 |
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| 295 | IF( VALEIG ) THEN
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| 296 | * We do not reference VL, VU in the cases RANGE = 'I','A'
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| 297 | * The interval (WL, WU] contains all the wanted eigenvalues.
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| 298 | * It is either given by the user or computed in SLARRE.
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| 299 | WL = VL
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| 300 | WU = VU
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| 301 | ELSEIF( INDEIG ) THEN
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| 302 | * We do not reference IL, IU in the cases RANGE = 'V','A'
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| 303 | IIL = IL
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| 304 | IIU = IU
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| 305 | ENDIF
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| 306 | *
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| 307 | INFO = 0
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| 308 | IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
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| 309 | INFO = -1
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| 310 | ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
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| 311 | INFO = -2
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| 312 | ELSE IF( N.LT.0 ) THEN
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| 313 | INFO = -3
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| 314 | ELSE IF( VALEIG .AND. N.GT.0 .AND. WU.LE.WL ) THEN
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| 315 | INFO = -7
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| 316 | ELSE IF( INDEIG .AND. ( IIL.LT.1 .OR. IIL.GT.N ) ) THEN
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| 317 | INFO = -8
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| 318 | ELSE IF( INDEIG .AND. ( IIU.LT.IIL .OR. IIU.GT.N ) ) THEN
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| 319 | INFO = -9
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| 320 | ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
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| 321 | INFO = -13
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| 322 | ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
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| 323 | INFO = -17
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| 324 | ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
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| 325 | INFO = -19
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| 326 | END IF
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| 327 | *
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| 328 | * Get machine constants.
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| 329 | *
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| 330 | SAFMIN = SLAMCH( 'Safe minimum' )
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| 331 | EPS = SLAMCH( 'Precision' )
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| 332 | SMLNUM = SAFMIN / EPS
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| 333 | BIGNUM = ONE / SMLNUM
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| 334 | RMIN = SQRT( SMLNUM )
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| 335 | RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
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| 336 | *
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| 337 | IF( INFO.EQ.0 ) THEN
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| 338 | WORK( 1 ) = LWMIN
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| 339 | IWORK( 1 ) = LIWMIN
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| 340 | *
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| 341 | IF( WANTZ .AND. ALLEIG ) THEN
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| 342 | NZCMIN = N
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| 343 | ELSE IF( WANTZ .AND. VALEIG ) THEN
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| 344 | CALL SLARRC( 'T', N, VL, VU, D, E, SAFMIN,
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| 345 | $ NZCMIN, ITMP, ITMP2, INFO )
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| 346 | ELSE IF( WANTZ .AND. INDEIG ) THEN
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| 347 | NZCMIN = IIU-IIL+1
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| 348 | ELSE
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| 349 | * WANTZ .EQ. FALSE.
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| 350 | NZCMIN = 0
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| 351 | ENDIF
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| 352 | IF( ZQUERY .AND. INFO.EQ.0 ) THEN
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| 353 | Z( 1,1 ) = NZCMIN
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| 354 | ELSE IF( NZC.LT.NZCMIN .AND. .NOT.ZQUERY ) THEN
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| 355 | INFO = -14
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| 356 | END IF
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| 357 | END IF
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| 358 |
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| 359 | IF( INFO.NE.0 ) THEN
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| 360 | *
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| 361 | CALL XERBLA( 'CSTEMR', -INFO )
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| 362 | *
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| 363 | RETURN
|
|---|
| 364 | ELSE IF( LQUERY .OR. ZQUERY ) THEN
|
|---|
| 365 | RETURN
|
|---|
| 366 | END IF
|
|---|
| 367 | *
|
|---|
| 368 | * Handle N = 0, 1, and 2 cases immediately
|
|---|
| 369 | *
|
|---|
| 370 | M = 0
|
|---|
| 371 | IF( N.EQ.0 )
|
|---|
| 372 | $ RETURN
|
|---|
| 373 | *
|
|---|
| 374 | IF( N.EQ.1 ) THEN
|
|---|
| 375 | IF( ALLEIG .OR. INDEIG ) THEN
|
|---|
| 376 | M = 1
|
|---|
| 377 | W( 1 ) = D( 1 )
|
|---|
| 378 | ELSE
|
|---|
| 379 | IF( WL.LT.D( 1 ) .AND. WU.GE.D( 1 ) ) THEN
|
|---|
| 380 | M = 1
|
|---|
| 381 | W( 1 ) = D( 1 )
|
|---|
| 382 | END IF
|
|---|
| 383 | END IF
|
|---|
| 384 | IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
|
|---|
| 385 | Z( 1, 1 ) = ONE
|
|---|
| 386 | ISUPPZ(1) = 1
|
|---|
| 387 | ISUPPZ(2) = 1
|
|---|
| 388 | END IF
|
|---|
| 389 | RETURN
|
|---|
| 390 | END IF
|
|---|
| 391 | *
|
|---|
| 392 | IF( N.EQ.2 ) THEN
|
|---|
| 393 | IF( .NOT.WANTZ ) THEN
|
|---|
| 394 | CALL SLAE2( D(1), E(1), D(2), R1, R2 )
|
|---|
| 395 | ELSE IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
|
|---|
| 396 | CALL SLAEV2( D(1), E(1), D(2), R1, R2, CS, SN )
|
|---|
| 397 | END IF
|
|---|
| 398 | IF( ALLEIG.OR.
|
|---|
| 399 | $ (VALEIG.AND.(R2.GT.WL).AND.
|
|---|
| 400 | $ (R2.LE.WU)).OR.
|
|---|
| 401 | $ (INDEIG.AND.(IIL.EQ.1)) ) THEN
|
|---|
| 402 | M = M+1
|
|---|
| 403 | W( M ) = R2
|
|---|
| 404 | IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
|
|---|
| 405 | Z( 1, M ) = -SN
|
|---|
| 406 | Z( 2, M ) = CS
|
|---|
| 407 | * Note: At most one of SN and CS can be zero.
|
|---|
| 408 | IF (SN.NE.ZERO) THEN
|
|---|
| 409 | IF (CS.NE.ZERO) THEN
|
|---|
| 410 | ISUPPZ(2*M-1) = 1
|
|---|
| 411 | ISUPPZ(2*M-1) = 2
|
|---|
| 412 | ELSE
|
|---|
| 413 | ISUPPZ(2*M-1) = 1
|
|---|
| 414 | ISUPPZ(2*M-1) = 1
|
|---|
| 415 | END IF
|
|---|
| 416 | ELSE
|
|---|
| 417 | ISUPPZ(2*M-1) = 2
|
|---|
| 418 | ISUPPZ(2*M) = 2
|
|---|
| 419 | END IF
|
|---|
| 420 | ENDIF
|
|---|
| 421 | ENDIF
|
|---|
| 422 | IF( ALLEIG.OR.
|
|---|
| 423 | $ (VALEIG.AND.(R1.GT.WL).AND.
|
|---|
| 424 | $ (R1.LE.WU)).OR.
|
|---|
| 425 | $ (INDEIG.AND.(IIU.EQ.2)) ) THEN
|
|---|
| 426 | M = M+1
|
|---|
| 427 | W( M ) = R1
|
|---|
| 428 | IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
|
|---|
| 429 | Z( 1, M ) = CS
|
|---|
| 430 | Z( 2, M ) = SN
|
|---|
| 431 | * Note: At most one of SN and CS can be zero.
|
|---|
| 432 | IF (SN.NE.ZERO) THEN
|
|---|
| 433 | IF (CS.NE.ZERO) THEN
|
|---|
| 434 | ISUPPZ(2*M-1) = 1
|
|---|
| 435 | ISUPPZ(2*M-1) = 2
|
|---|
| 436 | ELSE
|
|---|
| 437 | ISUPPZ(2*M-1) = 1
|
|---|
| 438 | ISUPPZ(2*M-1) = 1
|
|---|
| 439 | END IF
|
|---|
| 440 | ELSE
|
|---|
| 441 | ISUPPZ(2*M-1) = 2
|
|---|
| 442 | ISUPPZ(2*M) = 2
|
|---|
| 443 | END IF
|
|---|
| 444 | ENDIF
|
|---|
| 445 | ENDIF
|
|---|
| 446 | RETURN
|
|---|
| 447 | END IF
|
|---|
| 448 |
|
|---|
| 449 | * Continue with general N
|
|---|
| 450 |
|
|---|
| 451 | INDGRS = 1
|
|---|
| 452 | INDERR = 2*N + 1
|
|---|
| 453 | INDGP = 3*N + 1
|
|---|
| 454 | INDD = 4*N + 1
|
|---|
| 455 | INDE2 = 5*N + 1
|
|---|
| 456 | INDWRK = 6*N + 1
|
|---|
| 457 | *
|
|---|
| 458 | IINSPL = 1
|
|---|
| 459 | IINDBL = N + 1
|
|---|
| 460 | IINDW = 2*N + 1
|
|---|
| 461 | IINDWK = 3*N + 1
|
|---|
| 462 | *
|
|---|
| 463 | * Scale matrix to allowable range, if necessary.
|
|---|
| 464 | * The allowable range is related to the PIVMIN parameter; see the
|
|---|
| 465 | * comments in SLARRD. The preference for scaling small values
|
|---|
| 466 | * up is heuristic; we expect users' matrices not to be close to the
|
|---|
| 467 | * RMAX threshold.
|
|---|
| 468 | *
|
|---|
| 469 | SCALE = ONE
|
|---|
| 470 | TNRM = SLANST( 'M', N, D, E )
|
|---|
| 471 | IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN
|
|---|
| 472 | SCALE = RMIN / TNRM
|
|---|
| 473 | ELSE IF( TNRM.GT.RMAX ) THEN
|
|---|
| 474 | SCALE = RMAX / TNRM
|
|---|
| 475 | END IF
|
|---|
| 476 | IF( SCALE.NE.ONE ) THEN
|
|---|
| 477 | CALL SSCAL( N, SCALE, D, 1 )
|
|---|
| 478 | CALL SSCAL( N-1, SCALE, E, 1 )
|
|---|
| 479 | TNRM = TNRM*SCALE
|
|---|
| 480 | IF( VALEIG ) THEN
|
|---|
| 481 | * If eigenvalues in interval have to be found,
|
|---|
| 482 | * scale (WL, WU] accordingly
|
|---|
| 483 | WL = WL*SCALE
|
|---|
| 484 | WU = WU*SCALE
|
|---|
| 485 | ENDIF
|
|---|
| 486 | END IF
|
|---|
| 487 | *
|
|---|
| 488 | * Compute the desired eigenvalues of the tridiagonal after splitting
|
|---|
| 489 | * into smaller subblocks if the corresponding off-diagonal elements
|
|---|
| 490 | * are small
|
|---|
| 491 | * THRESH is the splitting parameter for SLARRE
|
|---|
| 492 | * A negative THRESH forces the old splitting criterion based on the
|
|---|
| 493 | * size of the off-diagonal. A positive THRESH switches to splitting
|
|---|
| 494 | * which preserves relative accuracy.
|
|---|
| 495 | *
|
|---|
| 496 | IF( TRYRAC ) THEN
|
|---|
| 497 | * Test whether the matrix warrants the more expensive relative approach.
|
|---|
| 498 | CALL SLARRR( N, D, E, IINFO )
|
|---|
| 499 | ELSE
|
|---|
| 500 | * The user does not care about relative accurately eigenvalues
|
|---|
| 501 | IINFO = -1
|
|---|
| 502 | ENDIF
|
|---|
| 503 | * Set the splitting criterion
|
|---|
| 504 | IF (IINFO.EQ.0) THEN
|
|---|
| 505 | THRESH = EPS
|
|---|
| 506 | ELSE
|
|---|
| 507 | THRESH = -EPS
|
|---|
| 508 | * relative accuracy is desired but T does not guarantee it
|
|---|
| 509 | TRYRAC = .FALSE.
|
|---|
| 510 | ENDIF
|
|---|
| 511 | *
|
|---|
| 512 | IF( TRYRAC ) THEN
|
|---|
| 513 | * Copy original diagonal, needed to guarantee relative accuracy
|
|---|
| 514 | CALL SCOPY(N,D,1,WORK(INDD),1)
|
|---|
| 515 | ENDIF
|
|---|
| 516 | * Store the squares of the offdiagonal values of T
|
|---|
| 517 | DO 5 J = 1, N-1
|
|---|
| 518 | WORK( INDE2+J-1 ) = E(J)**2
|
|---|
| 519 | 5 CONTINUE
|
|---|
| 520 |
|
|---|
| 521 | * Set the tolerance parameters for bisection
|
|---|
| 522 | IF( .NOT.WANTZ ) THEN
|
|---|
| 523 | * SLARRE computes the eigenvalues to full precision.
|
|---|
| 524 | RTOL1 = FOUR * EPS
|
|---|
| 525 | RTOL2 = FOUR * EPS
|
|---|
| 526 | ELSE
|
|---|
| 527 | * SLARRE computes the eigenvalues to less than full precision.
|
|---|
| 528 | * CLARRV will refine the eigenvalue approximations, and we only
|
|---|
| 529 | * need less accurate initial bisection in SLARRE.
|
|---|
| 530 | * Note: these settings do only affect the subset case and SLARRE
|
|---|
| 531 | RTOL1 = MAX( SQRT(EPS)*5.0E-2, FOUR * EPS )
|
|---|
| 532 | RTOL2 = MAX( SQRT(EPS)*5.0E-3, FOUR * EPS )
|
|---|
| 533 | ENDIF
|
|---|
| 534 | CALL SLARRE( RANGE, N, WL, WU, IIL, IIU, D, E,
|
|---|
| 535 | $ WORK(INDE2), RTOL1, RTOL2, THRESH, NSPLIT,
|
|---|
| 536 | $ IWORK( IINSPL ), M, W, WORK( INDERR ),
|
|---|
| 537 | $ WORK( INDGP ), IWORK( IINDBL ),
|
|---|
| 538 | $ IWORK( IINDW ), WORK( INDGRS ), PIVMIN,
|
|---|
| 539 | $ WORK( INDWRK ), IWORK( IINDWK ), IINFO )
|
|---|
| 540 | IF( IINFO.NE.0 ) THEN
|
|---|
| 541 | INFO = 10 + ABS( IINFO )
|
|---|
| 542 | RETURN
|
|---|
| 543 | END IF
|
|---|
| 544 | * Note that if RANGE .NE. 'V', SLARRE computes bounds on the desired
|
|---|
| 545 | * part of the spectrum. All desired eigenvalues are contained in
|
|---|
| 546 | * (WL,WU]
|
|---|
| 547 |
|
|---|
| 548 |
|
|---|
| 549 | IF( WANTZ ) THEN
|
|---|
| 550 | *
|
|---|
| 551 | * Compute the desired eigenvectors corresponding to the computed
|
|---|
| 552 | * eigenvalues
|
|---|
| 553 | *
|
|---|
| 554 | CALL CLARRV( N, WL, WU, D, E,
|
|---|
| 555 | $ PIVMIN, IWORK( IINSPL ), M,
|
|---|
| 556 | $ 1, M, MINRGP, RTOL1, RTOL2,
|
|---|
| 557 | $ W, WORK( INDERR ), WORK( INDGP ), IWORK( IINDBL ),
|
|---|
| 558 | $ IWORK( IINDW ), WORK( INDGRS ), Z, LDZ,
|
|---|
| 559 | $ ISUPPZ, WORK( INDWRK ), IWORK( IINDWK ), IINFO )
|
|---|
| 560 | IF( IINFO.NE.0 ) THEN
|
|---|
| 561 | INFO = 20 + ABS( IINFO )
|
|---|
| 562 | RETURN
|
|---|
| 563 | END IF
|
|---|
| 564 | ELSE
|
|---|
| 565 | * SLARRE computes eigenvalues of the (shifted) root representation
|
|---|
| 566 | * CLARRV returns the eigenvalues of the unshifted matrix.
|
|---|
| 567 | * However, if the eigenvectors are not desired by the user, we need
|
|---|
| 568 | * to apply the corresponding shifts from SLARRE to obtain the
|
|---|
| 569 | * eigenvalues of the original matrix.
|
|---|
| 570 | DO 20 J = 1, M
|
|---|
| 571 | ITMP = IWORK( IINDBL+J-1 )
|
|---|
| 572 | W( J ) = W( J ) + E( IWORK( IINSPL+ITMP-1 ) )
|
|---|
| 573 | 20 CONTINUE
|
|---|
| 574 | END IF
|
|---|
| 575 | *
|
|---|
| 576 |
|
|---|
| 577 | IF ( TRYRAC ) THEN
|
|---|
| 578 | * Refine computed eigenvalues so that they are relatively accurate
|
|---|
| 579 | * with respect to the original matrix T.
|
|---|
| 580 | IBEGIN = 1
|
|---|
| 581 | WBEGIN = 1
|
|---|
| 582 | DO 39 JBLK = 1, IWORK( IINDBL+M-1 )
|
|---|
| 583 | IEND = IWORK( IINSPL+JBLK-1 )
|
|---|
| 584 | IN = IEND - IBEGIN + 1
|
|---|
| 585 | WEND = WBEGIN - 1
|
|---|
| 586 | * check if any eigenvalues have to be refined in this block
|
|---|
| 587 | 36 CONTINUE
|
|---|
| 588 | IF( WEND.LT.M ) THEN
|
|---|
| 589 | IF( IWORK( IINDBL+WEND ).EQ.JBLK ) THEN
|
|---|
| 590 | WEND = WEND + 1
|
|---|
| 591 | GO TO 36
|
|---|
| 592 | END IF
|
|---|
| 593 | END IF
|
|---|
| 594 | IF( WEND.LT.WBEGIN ) THEN
|
|---|
| 595 | IBEGIN = IEND + 1
|
|---|
| 596 | GO TO 39
|
|---|
| 597 | END IF
|
|---|
| 598 |
|
|---|
| 599 | OFFSET = IWORK(IINDW+WBEGIN-1)-1
|
|---|
| 600 | IFIRST = IWORK(IINDW+WBEGIN-1)
|
|---|
| 601 | ILAST = IWORK(IINDW+WEND-1)
|
|---|
| 602 | RTOL2 = FOUR * EPS
|
|---|
| 603 | CALL SLARRJ( IN,
|
|---|
| 604 | $ WORK(INDD+IBEGIN-1), WORK(INDE2+IBEGIN-1),
|
|---|
| 605 | $ IFIRST, ILAST, RTOL2, OFFSET, W(WBEGIN),
|
|---|
| 606 | $ WORK( INDERR+WBEGIN-1 ),
|
|---|
| 607 | $ WORK( INDWRK ), IWORK( IINDWK ), PIVMIN,
|
|---|
| 608 | $ TNRM, IINFO )
|
|---|
| 609 | IBEGIN = IEND + 1
|
|---|
| 610 | WBEGIN = WEND + 1
|
|---|
| 611 | 39 CONTINUE
|
|---|
| 612 | ENDIF
|
|---|
| 613 | *
|
|---|
| 614 | * If matrix was scaled, then rescale eigenvalues appropriately.
|
|---|
| 615 | *
|
|---|
| 616 | IF( SCALE.NE.ONE ) THEN
|
|---|
| 617 | CALL SSCAL( M, ONE / SCALE, W, 1 )
|
|---|
| 618 | END IF
|
|---|
| 619 | *
|
|---|
| 620 | * If eigenvalues are not in increasing order, then sort them,
|
|---|
| 621 | * possibly along with eigenvectors.
|
|---|
| 622 | *
|
|---|
| 623 | IF( NSPLIT.GT.1 ) THEN
|
|---|
| 624 | IF( .NOT. WANTZ ) THEN
|
|---|
| 625 | CALL SLASRT( 'I', M, W, IINFO )
|
|---|
| 626 | IF( IINFO.NE.0 ) THEN
|
|---|
| 627 | INFO = 3
|
|---|
| 628 | RETURN
|
|---|
| 629 | END IF
|
|---|
| 630 | ELSE
|
|---|
| 631 | DO 60 J = 1, M - 1
|
|---|
| 632 | I = 0
|
|---|
| 633 | TMP = W( J )
|
|---|
| 634 | DO 50 JJ = J + 1, M
|
|---|
| 635 | IF( W( JJ ).LT.TMP ) THEN
|
|---|
| 636 | I = JJ
|
|---|
| 637 | TMP = W( JJ )
|
|---|
| 638 | END IF
|
|---|
| 639 | 50 CONTINUE
|
|---|
| 640 | IF( I.NE.0 ) THEN
|
|---|
| 641 | W( I ) = W( J )
|
|---|
| 642 | W( J ) = TMP
|
|---|
| 643 | IF( WANTZ ) THEN
|
|---|
| 644 | CALL CSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
|
|---|
| 645 | ITMP = ISUPPZ( 2*I-1 )
|
|---|
| 646 | ISUPPZ( 2*I-1 ) = ISUPPZ( 2*J-1 )
|
|---|
| 647 | ISUPPZ( 2*J-1 ) = ITMP
|
|---|
| 648 | ITMP = ISUPPZ( 2*I )
|
|---|
| 649 | ISUPPZ( 2*I ) = ISUPPZ( 2*J )
|
|---|
| 650 | ISUPPZ( 2*J ) = ITMP
|
|---|
| 651 | END IF
|
|---|
| 652 | END IF
|
|---|
| 653 | 60 CONTINUE
|
|---|
| 654 | END IF
|
|---|
| 655 | ENDIF
|
|---|
| 656 | *
|
|---|
| 657 | *
|
|---|
| 658 | WORK( 1 ) = LWMIN
|
|---|
| 659 | IWORK( 1 ) = LIWMIN
|
|---|
| 660 | RETURN
|
|---|
| 661 | *
|
|---|
| 662 | * End of CSTEMR
|
|---|
| 663 | *
|
|---|
| 664 | END
|
|---|
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