517 lines
16 KiB
C
517 lines
16 KiB
C
#include "clapack.h"
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/* Table of constant values */
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static integer c__1 = 1;
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static doublereal c_b6 = 0.;
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static integer c__0 = 0;
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static doublereal c_b11 = 1.;
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/* Subroutine */ int dlalsd_(char *uplo, integer *smlsiz, integer *n, integer
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*nrhs, doublereal *d__, doublereal *e, doublereal *b, integer *ldb,
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doublereal *rcond, integer *rank, doublereal *work, integer *iwork,
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integer *info)
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{
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/* System generated locals */
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integer b_dim1, b_offset, i__1, i__2;
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doublereal d__1;
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/* Builtin functions */
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double log(doublereal), d_sign(doublereal *, doublereal *);
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/* Local variables */
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integer c__, i__, j, k;
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doublereal r__;
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integer s, u, z__;
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doublereal cs;
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integer bx;
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doublereal sn;
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integer st, vt, nm1, st1;
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doublereal eps;
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integer iwk;
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doublereal tol;
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integer difl, difr;
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doublereal rcnd;
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integer perm, nsub;
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extern /* Subroutine */ int drot_(integer *, doublereal *, integer *,
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doublereal *, integer *, doublereal *, doublereal *);
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integer nlvl, sqre, bxst;
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extern /* Subroutine */ int dgemm_(char *, char *, integer *, integer *,
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integer *, doublereal *, doublereal *, integer *, doublereal *,
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integer *, doublereal *, doublereal *, integer *),
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dcopy_(integer *, doublereal *, integer *, doublereal *, integer
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*);
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integer poles, sizei, nsize, nwork, icmpq1, icmpq2;
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extern doublereal dlamch_(char *);
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extern /* Subroutine */ int dlasda_(integer *, integer *, integer *,
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integer *, doublereal *, doublereal *, doublereal *, integer *,
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doublereal *, integer *, doublereal *, doublereal *, doublereal *,
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doublereal *, integer *, integer *, integer *, integer *,
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doublereal *, doublereal *, doublereal *, doublereal *, integer *,
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integer *), dlalsa_(integer *, integer *, integer *, integer *,
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doublereal *, integer *, doublereal *, integer *, doublereal *,
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integer *, doublereal *, integer *, doublereal *, doublereal *,
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doublereal *, doublereal *, integer *, integer *, integer *,
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integer *, doublereal *, doublereal *, doublereal *, doublereal *,
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integer *, integer *), dlascl_(char *, integer *, integer *,
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doublereal *, doublereal *, integer *, integer *, doublereal *,
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integer *, integer *);
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extern integer idamax_(integer *, doublereal *, integer *);
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extern /* Subroutine */ int dlasdq_(char *, integer *, integer *, integer
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*, integer *, integer *, doublereal *, doublereal *, doublereal *,
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integer *, doublereal *, integer *, doublereal *, integer *,
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doublereal *, integer *), dlacpy_(char *, integer *,
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integer *, doublereal *, integer *, doublereal *, integer *), dlartg_(doublereal *, doublereal *, doublereal *,
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doublereal *, doublereal *), dlaset_(char *, integer *, integer *,
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doublereal *, doublereal *, doublereal *, integer *),
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xerbla_(char *, integer *);
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integer givcol;
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extern doublereal dlanst_(char *, integer *, doublereal *, doublereal *);
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extern /* Subroutine */ int dlasrt_(char *, integer *, doublereal *,
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integer *);
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doublereal orgnrm;
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integer givnum, givptr, smlszp;
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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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/* DLALSD uses the singular value decomposition of A to solve the least */
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/* squares problem of finding X to minimize the Euclidean norm of each */
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/* column of A*X-B, where A is N-by-N upper bidiagonal, and X and B */
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/* are N-by-NRHS. The solution X overwrites B. */
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/* The singular values of A smaller than RCOND times the largest */
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/* singular value are treated as zero in solving the least squares */
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/* problem; in this case a minimum norm solution is returned. */
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/* The actual singular values are returned in D in ascending order. */
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/* This code makes very mild assumptions about floating point */
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/* arithmetic. It will work on machines with a guard digit in */
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/* add/subtract, or on those binary machines without guard digits */
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/* which subtract like the Cray XMP, Cray YMP, Cray C 90, or Cray 2. */
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/* It could conceivably fail on hexadecimal or decimal machines */
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/* without guard digits, but we know of none. */
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/* Arguments */
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/* ========= */
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/* UPLO (input) CHARACTER*1 */
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/* = 'U': D and E define an upper bidiagonal matrix. */
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/* = 'L': D and E define a lower bidiagonal matrix. */
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/* SMLSIZ (input) INTEGER */
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/* The maximum size of the subproblems at the bottom of the */
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/* computation tree. */
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/* N (input) INTEGER */
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/* The dimension of the bidiagonal matrix. N >= 0. */
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/* NRHS (input) INTEGER */
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/* The number of columns of B. NRHS must be at least 1. */
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/* D (input/output) DOUBLE PRECISION array, dimension (N) */
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/* On entry D contains the main diagonal of the bidiagonal */
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/* matrix. On exit, if INFO = 0, D contains its singular values. */
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/* E (input/output) DOUBLE PRECISION array, dimension (N-1) */
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/* Contains the super-diagonal entries of the bidiagonal matrix. */
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/* On exit, E has been destroyed. */
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/* B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) */
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/* On input, B contains the right hand sides of the least */
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/* squares problem. On output, B contains the solution X. */
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/* LDB (input) INTEGER */
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/* The leading dimension of B in the calling subprogram. */
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/* LDB must be at least max(1,N). */
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/* RCOND (input) DOUBLE PRECISION */
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/* The singular values of A less than or equal to RCOND times */
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/* the largest singular value are treated as zero in solving */
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/* the least squares problem. If RCOND is negative, */
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/* machine precision is used instead. */
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/* For example, if diag(S)*X=B were the least squares problem, */
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/* where diag(S) is a diagonal matrix of singular values, the */
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/* solution would be X(i) = B(i) / S(i) if S(i) is greater than */
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/* RCOND*max(S), and X(i) = 0 if S(i) is less than or equal to */
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/* RCOND*max(S). */
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/* RANK (output) INTEGER */
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/* The number of singular values of A greater than RCOND times */
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/* the largest singular value. */
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/* WORK (workspace) DOUBLE PRECISION array, dimension at least */
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/* (9*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2), */
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/* where NLVL = max(0, INT(log_2 (N/(SMLSIZ+1))) + 1). */
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/* IWORK (workspace) INTEGER array, dimension at least */
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/* (3*N*NLVL + 11*N) */
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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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/* > 0: The algorithm failed to compute an singular value while */
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/* working on the submatrix lying in rows and columns */
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/* INFO/(N+1) through MOD(INFO,N+1). */
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/* Further Details */
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/* =============== */
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/* Based on contributions by */
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/* Ming Gu and Ren-Cang Li, Computer Science Division, University of */
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/* California at Berkeley, USA */
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/* Osni Marques, LBNL/NERSC, USA */
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/* ===================================================================== */
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/* .. Parameters .. */
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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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--e;
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b_dim1 = *ldb;
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b_offset = 1 + b_dim1;
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b -= b_offset;
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--work;
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--iwork;
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/* Function Body */
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*info = 0;
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if (*n < 0) {
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*info = -3;
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} else if (*nrhs < 1) {
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*info = -4;
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} else if (*ldb < 1 || *ldb < *n) {
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*info = -8;
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}
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if (*info != 0) {
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i__1 = -(*info);
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xerbla_("DLALSD", &i__1);
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return 0;
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}
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eps = dlamch_("Epsilon");
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/* Set up the tolerance. */
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if (*rcond <= 0. || *rcond >= 1.) {
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rcnd = eps;
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} else {
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rcnd = *rcond;
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}
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*rank = 0;
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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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} else if (*n == 1) {
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if (d__[1] == 0.) {
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dlaset_("A", &c__1, nrhs, &c_b6, &c_b6, &b[b_offset], ldb);
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} else {
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*rank = 1;
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dlascl_("G", &c__0, &c__0, &d__[1], &c_b11, &c__1, nrhs, &b[
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b_offset], ldb, info);
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d__[1] = abs(d__[1]);
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}
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return 0;
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}
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/* Rotate the matrix if it is lower bidiagonal. */
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if (*(unsigned char *)uplo == 'L') {
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i__1 = *n - 1;
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for (i__ = 1; i__ <= i__1; ++i__) {
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dlartg_(&d__[i__], &e[i__], &cs, &sn, &r__);
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d__[i__] = r__;
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e[i__] = sn * d__[i__ + 1];
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d__[i__ + 1] = cs * d__[i__ + 1];
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if (*nrhs == 1) {
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drot_(&c__1, &b[i__ + b_dim1], &c__1, &b[i__ + 1 + b_dim1], &
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c__1, &cs, &sn);
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} else {
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work[(i__ << 1) - 1] = cs;
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work[i__ * 2] = sn;
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}
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/* L10: */
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}
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if (*nrhs > 1) {
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i__1 = *nrhs;
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for (i__ = 1; i__ <= i__1; ++i__) {
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i__2 = *n - 1;
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for (j = 1; j <= i__2; ++j) {
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cs = work[(j << 1) - 1];
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sn = work[j * 2];
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drot_(&c__1, &b[j + i__ * b_dim1], &c__1, &b[j + 1 + i__ *
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b_dim1], &c__1, &cs, &sn);
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/* L20: */
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}
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/* L30: */
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}
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}
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}
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/* Scale. */
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nm1 = *n - 1;
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orgnrm = dlanst_("M", n, &d__[1], &e[1]);
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if (orgnrm == 0.) {
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dlaset_("A", n, nrhs, &c_b6, &c_b6, &b[b_offset], ldb);
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return 0;
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}
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dlascl_("G", &c__0, &c__0, &orgnrm, &c_b11, n, &c__1, &d__[1], n, info);
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dlascl_("G", &c__0, &c__0, &orgnrm, &c_b11, &nm1, &c__1, &e[1], &nm1,
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info);
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/* If N is smaller than the minimum divide size SMLSIZ, then solve */
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/* the problem with another solver. */
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if (*n <= *smlsiz) {
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nwork = *n * *n + 1;
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dlaset_("A", n, n, &c_b6, &c_b11, &work[1], n);
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dlasdq_("U", &c__0, n, n, &c__0, nrhs, &d__[1], &e[1], &work[1], n, &
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work[1], n, &b[b_offset], ldb, &work[nwork], info);
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if (*info != 0) {
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return 0;
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}
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tol = rcnd * (d__1 = d__[idamax_(n, &d__[1], &c__1)], abs(d__1));
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i__1 = *n;
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for (i__ = 1; i__ <= i__1; ++i__) {
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if (d__[i__] <= tol) {
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dlaset_("A", &c__1, nrhs, &c_b6, &c_b6, &b[i__ + b_dim1], ldb);
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} else {
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dlascl_("G", &c__0, &c__0, &d__[i__], &c_b11, &c__1, nrhs, &b[
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i__ + b_dim1], ldb, info);
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++(*rank);
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}
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/* L40: */
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}
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dgemm_("T", "N", n, nrhs, n, &c_b11, &work[1], n, &b[b_offset], ldb, &
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c_b6, &work[nwork], n);
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dlacpy_("A", n, nrhs, &work[nwork], n, &b[b_offset], ldb);
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/* Unscale. */
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dlascl_("G", &c__0, &c__0, &c_b11, &orgnrm, n, &c__1, &d__[1], n,
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info);
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dlasrt_("D", n, &d__[1], info);
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dlascl_("G", &c__0, &c__0, &orgnrm, &c_b11, n, nrhs, &b[b_offset],
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ldb, info);
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return 0;
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}
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/* Book-keeping and setting up some constants. */
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nlvl = (integer) (log((doublereal) (*n) / (doublereal) (*smlsiz + 1)) /
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log(2.)) + 1;
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smlszp = *smlsiz + 1;
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u = 1;
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vt = *smlsiz * *n + 1;
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difl = vt + smlszp * *n;
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difr = difl + nlvl * *n;
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z__ = difr + (nlvl * *n << 1);
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c__ = z__ + nlvl * *n;
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s = c__ + *n;
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poles = s + *n;
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givnum = poles + (nlvl << 1) * *n;
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bx = givnum + (nlvl << 1) * *n;
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nwork = bx + *n * *nrhs;
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sizei = *n + 1;
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k = sizei + *n;
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givptr = k + *n;
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perm = givptr + *n;
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givcol = perm + nlvl * *n;
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iwk = givcol + (nlvl * *n << 1);
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st = 1;
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sqre = 0;
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icmpq1 = 1;
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icmpq2 = 0;
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nsub = 0;
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i__1 = *n;
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for (i__ = 1; i__ <= i__1; ++i__) {
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if ((d__1 = d__[i__], abs(d__1)) < eps) {
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d__[i__] = d_sign(&eps, &d__[i__]);
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}
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/* L50: */
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}
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i__1 = nm1;
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for (i__ = 1; i__ <= i__1; ++i__) {
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if ((d__1 = e[i__], abs(d__1)) < eps || i__ == nm1) {
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++nsub;
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iwork[nsub] = st;
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/* Subproblem found. First determine its size and then */
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/* apply divide and conquer on it. */
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if (i__ < nm1) {
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/* A subproblem with E(I) small for I < NM1. */
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nsize = i__ - st + 1;
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iwork[sizei + nsub - 1] = nsize;
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} else if ((d__1 = e[i__], abs(d__1)) >= eps) {
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/* A subproblem with E(NM1) not too small but I = NM1. */
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nsize = *n - st + 1;
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iwork[sizei + nsub - 1] = nsize;
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} else {
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/* A subproblem with E(NM1) small. This implies an */
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/* 1-by-1 subproblem at D(N), which is not solved */
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/* explicitly. */
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nsize = i__ - st + 1;
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iwork[sizei + nsub - 1] = nsize;
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++nsub;
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iwork[nsub] = *n;
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iwork[sizei + nsub - 1] = 1;
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dcopy_(nrhs, &b[*n + b_dim1], ldb, &work[bx + nm1], n);
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}
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st1 = st - 1;
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if (nsize == 1) {
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/* This is a 1-by-1 subproblem and is not solved */
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/* explicitly. */
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dcopy_(nrhs, &b[st + b_dim1], ldb, &work[bx + st1], n);
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} else if (nsize <= *smlsiz) {
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/* This is a small subproblem and is solved by DLASDQ. */
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dlaset_("A", &nsize, &nsize, &c_b6, &c_b11, &work[vt + st1],
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n);
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dlasdq_("U", &c__0, &nsize, &nsize, &c__0, nrhs, &d__[st], &e[
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st], &work[vt + st1], n, &work[nwork], n, &b[st +
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b_dim1], ldb, &work[nwork], info);
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if (*info != 0) {
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return 0;
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}
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dlacpy_("A", &nsize, nrhs, &b[st + b_dim1], ldb, &work[bx +
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st1], n);
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} else {
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/* A large problem. Solve it using divide and conquer. */
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dlasda_(&icmpq1, smlsiz, &nsize, &sqre, &d__[st], &e[st], &
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work[u + st1], n, &work[vt + st1], &iwork[k + st1], &
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work[difl + st1], &work[difr + st1], &work[z__ + st1],
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&work[poles + st1], &iwork[givptr + st1], &iwork[
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givcol + st1], n, &iwork[perm + st1], &work[givnum +
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st1], &work[c__ + st1], &work[s + st1], &work[nwork],
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&iwork[iwk], info);
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if (*info != 0) {
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return 0;
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}
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bxst = bx + st1;
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dlalsa_(&icmpq2, smlsiz, &nsize, nrhs, &b[st + b_dim1], ldb, &
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work[bxst], n, &work[u + st1], n, &work[vt + st1], &
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iwork[k + st1], &work[difl + st1], &work[difr + st1],
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&work[z__ + st1], &work[poles + st1], &iwork[givptr +
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st1], &iwork[givcol + st1], n, &iwork[perm + st1], &
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work[givnum + st1], &work[c__ + st1], &work[s + st1],
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&work[nwork], &iwork[iwk], info);
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if (*info != 0) {
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return 0;
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}
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}
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st = i__ + 1;
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}
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/* L60: */
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}
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/* Apply the singular values and treat the tiny ones as zero. */
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|
|
|
tol = rcnd * (d__1 = d__[idamax_(n, &d__[1], &c__1)], abs(d__1));
|
|
|
|
i__1 = *n;
|
|
for (i__ = 1; i__ <= i__1; ++i__) {
|
|
|
|
/* Some of the elements in D can be negative because 1-by-1 */
|
|
/* subproblems were not solved explicitly. */
|
|
|
|
if ((d__1 = d__[i__], abs(d__1)) <= tol) {
|
|
dlaset_("A", &c__1, nrhs, &c_b6, &c_b6, &work[bx + i__ - 1], n);
|
|
} else {
|
|
++(*rank);
|
|
dlascl_("G", &c__0, &c__0, &d__[i__], &c_b11, &c__1, nrhs, &work[
|
|
bx + i__ - 1], n, info);
|
|
}
|
|
d__[i__] = (d__1 = d__[i__], abs(d__1));
|
|
/* L70: */
|
|
}
|
|
|
|
/* Now apply back the right singular vectors. */
|
|
|
|
icmpq2 = 1;
|
|
i__1 = nsub;
|
|
for (i__ = 1; i__ <= i__1; ++i__) {
|
|
st = iwork[i__];
|
|
st1 = st - 1;
|
|
nsize = iwork[sizei + i__ - 1];
|
|
bxst = bx + st1;
|
|
if (nsize == 1) {
|
|
dcopy_(nrhs, &work[bxst], n, &b[st + b_dim1], ldb);
|
|
} else if (nsize <= *smlsiz) {
|
|
dgemm_("T", "N", &nsize, nrhs, &nsize, &c_b11, &work[vt + st1], n,
|
|
&work[bxst], n, &c_b6, &b[st + b_dim1], ldb);
|
|
} else {
|
|
dlalsa_(&icmpq2, smlsiz, &nsize, nrhs, &work[bxst], n, &b[st +
|
|
b_dim1], ldb, &work[u + st1], n, &work[vt + st1], &iwork[
|
|
k + st1], &work[difl + st1], &work[difr + st1], &work[z__
|
|
+ st1], &work[poles + st1], &iwork[givptr + st1], &iwork[
|
|
givcol + st1], n, &iwork[perm + st1], &work[givnum + st1],
|
|
&work[c__ + st1], &work[s + st1], &work[nwork], &iwork[
|
|
iwk], info);
|
|
if (*info != 0) {
|
|
return 0;
|
|
}
|
|
}
|
|
/* L80: */
|
|
}
|
|
|
|
/* Unscale and sort the singular values. */
|
|
|
|
dlascl_("G", &c__0, &c__0, &c_b11, &orgnrm, n, &c__1, &d__[1], n, info);
|
|
dlasrt_("D", n, &d__[1], info);
|
|
dlascl_("G", &c__0, &c__0, &orgnrm, &c_b11, n, nrhs, &b[b_offset], ldb,
|
|
info);
|
|
|
|
return 0;
|
|
|
|
/* End of DLALSD */
|
|
|
|
} /* dlalsd_ */
|