518 lines
14 KiB
C
518 lines
14 KiB
C
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#include "clapack.h"
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/* Table of constant values */
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static real c_b3 = -1.f;
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static integer c__1 = 1;
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/* Subroutine */ int slaed2_(integer *k, integer *n, integer *n1, real *d__,
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real *q, integer *ldq, integer *indxq, real *rho, real *z__, real *
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dlamda, real *w, real *q2, integer *indx, integer *indxc, integer *
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indxp, integer *coltyp, integer *info)
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{
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/* System generated locals */
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integer q_dim1, q_offset, i__1, i__2;
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real r__1, r__2, r__3, r__4;
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/* Builtin functions */
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double sqrt(doublereal);
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/* Local variables */
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real c__;
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integer i__, j;
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real s, t;
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integer k2, n2, ct, nj, pj, js, iq1, iq2, n1p1;
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real eps, tau, tol;
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integer psm[4], imax, jmax, ctot[4];
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extern /* Subroutine */ int srot_(integer *, real *, integer *, real *,
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integer *, real *, real *), sscal_(integer *, real *, real *,
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integer *), scopy_(integer *, real *, integer *, real *, integer *
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);
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extern doublereal slapy2_(real *, real *), slamch_(char *);
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extern /* Subroutine */ int xerbla_(char *, integer *);
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extern integer isamax_(integer *, real *, integer *);
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extern /* Subroutine */ int slamrg_(integer *, integer *, real *, integer
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*, integer *, integer *), slacpy_(char *, integer *, integer *,
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real *, integer *, real *, integer *);
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/* -- LAPACK routine (version 3.1) -- */
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/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
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/* November 2006 */
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/* .. Scalar Arguments .. */
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/* .. */
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/* .. Array Arguments .. */
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/* .. */
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/* Purpose */
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/* ======= */
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/* SLAED2 merges the two sets of eigenvalues together into a single */
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/* sorted set. Then it tries to deflate the size of the problem. */
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/* There are two ways in which deflation can occur: when two or more */
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/* eigenvalues are close together or if there is a tiny entry in the */
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/* Z vector. For each such occurrence the order of the related secular */
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/* equation problem is reduced by one. */
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/* Arguments */
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/* ========= */
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/* K (output) INTEGER */
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/* The number of non-deflated eigenvalues, and the order of the */
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/* related secular equation. 0 <= K <=N. */
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/* N (input) INTEGER */
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/* The dimension of the symmetric tridiagonal matrix. N >= 0. */
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/* N1 (input) INTEGER */
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/* The location of the last eigenvalue in the leading sub-matrix. */
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/* min(1,N) <= N1 <= N/2. */
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/* D (input/output) REAL array, dimension (N) */
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/* On entry, D contains the eigenvalues of the two submatrices to */
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/* be combined. */
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/* On exit, D contains the trailing (N-K) updated eigenvalues */
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/* (those which were deflated) sorted into increasing order. */
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/* Q (input/output) REAL array, dimension (LDQ, N) */
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/* On entry, Q contains the eigenvectors of two submatrices in */
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/* the two square blocks with corners at (1,1), (N1,N1) */
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/* and (N1+1, N1+1), (N,N). */
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/* On exit, Q contains the trailing (N-K) updated eigenvectors */
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/* (those which were deflated) in its last N-K columns. */
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/* LDQ (input) INTEGER */
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/* The leading dimension of the array Q. LDQ >= max(1,N). */
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/* INDXQ (input/output) INTEGER array, dimension (N) */
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/* The permutation which separately sorts the two sub-problems */
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/* in D into ascending order. Note that elements in the second */
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/* half of this permutation must first have N1 added to their */
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/* values. Destroyed on exit. */
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/* RHO (input/output) REAL */
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/* On entry, the off-diagonal element associated with the rank-1 */
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/* cut which originally split the two submatrices which are now */
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/* being recombined. */
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/* On exit, RHO has been modified to the value required by */
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/* SLAED3. */
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/* Z (input) REAL array, dimension (N) */
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/* On entry, Z contains the updating vector (the last */
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/* row of the first sub-eigenvector matrix and the first row of */
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/* the second sub-eigenvector matrix). */
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/* On exit, the contents of Z have been destroyed by the updating */
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/* process. */
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/* DLAMDA (output) REAL array, dimension (N) */
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/* A copy of the first K eigenvalues which will be used by */
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/* SLAED3 to form the secular equation. */
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/* W (output) REAL array, dimension (N) */
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/* The first k values of the final deflation-altered z-vector */
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/* which will be passed to SLAED3. */
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/* Q2 (output) REAL array, dimension (N1**2+(N-N1)**2) */
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/* A copy of the first K eigenvectors which will be used by */
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/* SLAED3 in a matrix multiply (SGEMM) to solve for the new */
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/* eigenvectors. */
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/* INDX (workspace) INTEGER array, dimension (N) */
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/* The permutation used to sort the contents of DLAMDA into */
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/* ascending order. */
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/* INDXC (output) INTEGER array, dimension (N) */
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/* The permutation used to arrange the columns of the deflated */
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/* Q matrix into three groups: the first group contains non-zero */
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/* elements only at and above N1, the second contains */
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/* non-zero elements only below N1, and the third is dense. */
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/* INDXP (workspace) INTEGER array, dimension (N) */
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/* The permutation used to place deflated values of D at the end */
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/* of the array. INDXP(1:K) points to the nondeflated D-values */
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/* and INDXP(K+1:N) points to the deflated eigenvalues. */
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/* COLTYP (workspace/output) INTEGER array, dimension (N) */
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/* During execution, a label which will indicate which of the */
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/* following types a column in the Q2 matrix is: */
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/* 1 : non-zero in the upper half only; */
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/* 2 : dense; */
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/* 3 : non-zero in the lower half only; */
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/* 4 : deflated. */
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/* On exit, COLTYP(i) is the number of columns of type i, */
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/* for i=1 to 4 only. */
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/* INFO (output) INTEGER */
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/* = 0: successful exit. */
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/* < 0: if INFO = -i, the i-th argument had an illegal value. */
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/* Further Details */
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/* =============== */
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/* Based on contributions by */
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/* Jeff Rutter, Computer Science Division, University of California */
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/* at Berkeley, USA */
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/* Modified by Francoise Tisseur, University of Tennessee. */
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/* ===================================================================== */
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/* .. Parameters .. */
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/* .. */
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/* .. Local Arrays .. */
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/* .. */
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/* .. Local Scalars .. */
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/* .. */
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/* .. External Functions .. */
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/* .. */
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/* .. External Subroutines .. */
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/* .. */
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/* .. Intrinsic Functions .. */
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/* .. */
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/* .. Executable Statements .. */
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/* Test the input parameters. */
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/* Parameter adjustments */
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--d__;
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q_dim1 = *ldq;
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q_offset = 1 + q_dim1;
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q -= q_offset;
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--indxq;
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--z__;
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--dlamda;
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--w;
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--q2;
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--indx;
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--indxc;
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--indxp;
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--coltyp;
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/* Function Body */
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*info = 0;
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if (*n < 0) {
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*info = -2;
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} else if (*ldq < max(1,*n)) {
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*info = -6;
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} else /* if(complicated condition) */ {
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/* Computing MIN */
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i__1 = 1, i__2 = *n / 2;
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if (min(i__1,i__2) > *n1 || *n / 2 < *n1) {
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*info = -3;
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}
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}
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if (*info != 0) {
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i__1 = -(*info);
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xerbla_("SLAED2", &i__1);
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return 0;
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}
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/* Quick return if possible */
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if (*n == 0) {
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return 0;
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}
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n2 = *n - *n1;
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n1p1 = *n1 + 1;
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if (*rho < 0.f) {
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sscal_(&n2, &c_b3, &z__[n1p1], &c__1);
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}
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/* Normalize z so that norm(z) = 1. Since z is the concatenation of */
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/* two normalized vectors, norm2(z) = sqrt(2). */
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t = 1.f / sqrt(2.f);
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sscal_(n, &t, &z__[1], &c__1);
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/* RHO = ABS( norm(z)**2 * RHO ) */
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*rho = (r__1 = *rho * 2.f, dabs(r__1));
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/* Sort the eigenvalues into increasing order */
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i__1 = *n;
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for (i__ = n1p1; i__ <= i__1; ++i__) {
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indxq[i__] += *n1;
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/* L10: */
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}
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/* re-integrate the deflated parts from the last pass */
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i__1 = *n;
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for (i__ = 1; i__ <= i__1; ++i__) {
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dlamda[i__] = d__[indxq[i__]];
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/* L20: */
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}
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slamrg_(n1, &n2, &dlamda[1], &c__1, &c__1, &indxc[1]);
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i__1 = *n;
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for (i__ = 1; i__ <= i__1; ++i__) {
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indx[i__] = indxq[indxc[i__]];
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/* L30: */
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}
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/* Calculate the allowable deflation tolerance */
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imax = isamax_(n, &z__[1], &c__1);
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jmax = isamax_(n, &d__[1], &c__1);
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eps = slamch_("Epsilon");
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/* Computing MAX */
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r__3 = (r__1 = d__[jmax], dabs(r__1)), r__4 = (r__2 = z__[imax], dabs(
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r__2));
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tol = eps * 8.f * dmax(r__3,r__4);
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/* If the rank-1 modifier is small enough, no more needs to be done */
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/* except to reorganize Q so that its columns correspond with the */
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/* elements in D. */
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if (*rho * (r__1 = z__[imax], dabs(r__1)) <= tol) {
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*k = 0;
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iq2 = 1;
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i__1 = *n;
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for (j = 1; j <= i__1; ++j) {
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i__ = indx[j];
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scopy_(n, &q[i__ * q_dim1 + 1], &c__1, &q2[iq2], &c__1);
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dlamda[j] = d__[i__];
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iq2 += *n;
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/* L40: */
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}
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slacpy_("A", n, n, &q2[1], n, &q[q_offset], ldq);
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scopy_(n, &dlamda[1], &c__1, &d__[1], &c__1);
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goto L190;
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}
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/* If there are multiple eigenvalues then the problem deflates. Here */
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/* the number of equal eigenvalues are found. As each equal */
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/* eigenvalue is found, an elementary reflector is computed to rotate */
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/* the corresponding eigensubspace so that the corresponding */
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/* components of Z are zero in this new basis. */
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i__1 = *n1;
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for (i__ = 1; i__ <= i__1; ++i__) {
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coltyp[i__] = 1;
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/* L50: */
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}
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i__1 = *n;
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for (i__ = n1p1; i__ <= i__1; ++i__) {
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coltyp[i__] = 3;
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/* L60: */
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}
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*k = 0;
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k2 = *n + 1;
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i__1 = *n;
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for (j = 1; j <= i__1; ++j) {
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nj = indx[j];
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if (*rho * (r__1 = z__[nj], dabs(r__1)) <= tol) {
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/* Deflate due to small z component. */
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--k2;
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coltyp[nj] = 4;
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indxp[k2] = nj;
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if (j == *n) {
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goto L100;
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}
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} else {
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pj = nj;
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goto L80;
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}
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/* L70: */
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}
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L80:
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++j;
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nj = indx[j];
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if (j > *n) {
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goto L100;
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}
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if (*rho * (r__1 = z__[nj], dabs(r__1)) <= tol) {
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/* Deflate due to small z component. */
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--k2;
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coltyp[nj] = 4;
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indxp[k2] = nj;
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} else {
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/* Check if eigenvalues are close enough to allow deflation. */
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s = z__[pj];
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c__ = z__[nj];
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/* Find sqrt(a**2+b**2) without overflow or */
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/* destructive underflow. */
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tau = slapy2_(&c__, &s);
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t = d__[nj] - d__[pj];
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c__ /= tau;
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s = -s / tau;
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if ((r__1 = t * c__ * s, dabs(r__1)) <= tol) {
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/* Deflation is possible. */
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z__[nj] = tau;
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z__[pj] = 0.f;
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if (coltyp[nj] != coltyp[pj]) {
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coltyp[nj] = 2;
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}
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coltyp[pj] = 4;
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srot_(n, &q[pj * q_dim1 + 1], &c__1, &q[nj * q_dim1 + 1], &c__1, &
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c__, &s);
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/* Computing 2nd power */
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r__1 = c__;
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/* Computing 2nd power */
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r__2 = s;
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t = d__[pj] * (r__1 * r__1) + d__[nj] * (r__2 * r__2);
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/* Computing 2nd power */
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r__1 = s;
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/* Computing 2nd power */
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r__2 = c__;
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d__[nj] = d__[pj] * (r__1 * r__1) + d__[nj] * (r__2 * r__2);
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d__[pj] = t;
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--k2;
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i__ = 1;
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L90:
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if (k2 + i__ <= *n) {
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if (d__[pj] < d__[indxp[k2 + i__]]) {
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indxp[k2 + i__ - 1] = indxp[k2 + i__];
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indxp[k2 + i__] = pj;
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++i__;
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goto L90;
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} else {
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indxp[k2 + i__ - 1] = pj;
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}
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} else {
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indxp[k2 + i__ - 1] = pj;
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}
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pj = nj;
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} else {
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++(*k);
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dlamda[*k] = d__[pj];
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w[*k] = z__[pj];
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indxp[*k] = pj;
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pj = nj;
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}
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}
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goto L80;
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L100:
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/* Record the last eigenvalue. */
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++(*k);
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dlamda[*k] = d__[pj];
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w[*k] = z__[pj];
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indxp[*k] = pj;
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/* Count up the total number of the various types of columns, then */
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/* form a permutation which positions the four column types into */
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/* four uniform groups (although one or more of these groups may be */
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/* empty). */
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for (j = 1; j <= 4; ++j) {
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ctot[j - 1] = 0;
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/* L110: */
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}
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i__1 = *n;
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for (j = 1; j <= i__1; ++j) {
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ct = coltyp[j];
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++ctot[ct - 1];
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/* L120: */
|
||
|
}
|
||
|
|
||
|
/* PSM(*) = Position in SubMatrix (of types 1 through 4) */
|
||
|
|
||
|
psm[0] = 1;
|
||
|
psm[1] = ctot[0] + 1;
|
||
|
psm[2] = psm[1] + ctot[1];
|
||
|
psm[3] = psm[2] + ctot[2];
|
||
|
*k = *n - ctot[3];
|
||
|
|
||
|
/* Fill out the INDXC array so that the permutation which it induces */
|
||
|
/* will place all type-1 columns first, all type-2 columns next, */
|
||
|
/* then all type-3's, and finally all type-4's. */
|
||
|
|
||
|
i__1 = *n;
|
||
|
for (j = 1; j <= i__1; ++j) {
|
||
|
js = indxp[j];
|
||
|
ct = coltyp[js];
|
||
|
indx[psm[ct - 1]] = js;
|
||
|
indxc[psm[ct - 1]] = j;
|
||
|
++psm[ct - 1];
|
||
|
/* L130: */
|
||
|
}
|
||
|
|
||
|
/* Sort the eigenvalues and corresponding eigenvectors into DLAMDA */
|
||
|
/* and Q2 respectively. The eigenvalues/vectors which were not */
|
||
|
/* deflated go into the first K slots of DLAMDA and Q2 respectively, */
|
||
|
/* while those which were deflated go into the last N - K slots. */
|
||
|
|
||
|
i__ = 1;
|
||
|
iq1 = 1;
|
||
|
iq2 = (ctot[0] + ctot[1]) * *n1 + 1;
|
||
|
i__1 = ctot[0];
|
||
|
for (j = 1; j <= i__1; ++j) {
|
||
|
js = indx[i__];
|
||
|
scopy_(n1, &q[js * q_dim1 + 1], &c__1, &q2[iq1], &c__1);
|
||
|
z__[i__] = d__[js];
|
||
|
++i__;
|
||
|
iq1 += *n1;
|
||
|
/* L140: */
|
||
|
}
|
||
|
|
||
|
i__1 = ctot[1];
|
||
|
for (j = 1; j <= i__1; ++j) {
|
||
|
js = indx[i__];
|
||
|
scopy_(n1, &q[js * q_dim1 + 1], &c__1, &q2[iq1], &c__1);
|
||
|
scopy_(&n2, &q[*n1 + 1 + js * q_dim1], &c__1, &q2[iq2], &c__1);
|
||
|
z__[i__] = d__[js];
|
||
|
++i__;
|
||
|
iq1 += *n1;
|
||
|
iq2 += n2;
|
||
|
/* L150: */
|
||
|
}
|
||
|
|
||
|
i__1 = ctot[2];
|
||
|
for (j = 1; j <= i__1; ++j) {
|
||
|
js = indx[i__];
|
||
|
scopy_(&n2, &q[*n1 + 1 + js * q_dim1], &c__1, &q2[iq2], &c__1);
|
||
|
z__[i__] = d__[js];
|
||
|
++i__;
|
||
|
iq2 += n2;
|
||
|
/* L160: */
|
||
|
}
|
||
|
|
||
|
iq1 = iq2;
|
||
|
i__1 = ctot[3];
|
||
|
for (j = 1; j <= i__1; ++j) {
|
||
|
js = indx[i__];
|
||
|
scopy_(n, &q[js * q_dim1 + 1], &c__1, &q2[iq2], &c__1);
|
||
|
iq2 += *n;
|
||
|
z__[i__] = d__[js];
|
||
|
++i__;
|
||
|
/* L170: */
|
||
|
}
|
||
|
|
||
|
/* The deflated eigenvalues and their corresponding vectors go back */
|
||
|
/* into the last N - K slots of D and Q respectively. */
|
||
|
|
||
|
slacpy_("A", n, &ctot[3], &q2[iq1], n, &q[(*k + 1) * q_dim1 + 1], ldq);
|
||
|
i__1 = *n - *k;
|
||
|
scopy_(&i__1, &z__[*k + 1], &c__1, &d__[*k + 1], &c__1);
|
||
|
|
||
|
/* Copy CTOT into COLTYP for referencing in SLAED3. */
|
||
|
|
||
|
for (j = 1; j <= 4; ++j) {
|
||
|
coltyp[j] = ctot[j - 1];
|
||
|
/* L180: */
|
||
|
}
|
||
|
|
||
|
L190:
|
||
|
return 0;
|
||
|
|
||
|
/* End of SLAED2 */
|
||
|
|
||
|
} /* slaed2_ */
|