673 lines
22 KiB
C
673 lines
22 KiB
C
#include "clapack.h"
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/* Table of constant values */
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static integer c__6 = 6;
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static integer c_n1 = -1;
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static integer c__9 = 9;
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static integer c__0 = 0;
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static integer c__1 = 1;
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static doublereal c_b82 = 0.;
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/* Subroutine */ int dgelsd_(integer *m, integer *n, integer *nrhs,
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doublereal *a, integer *lda, doublereal *b, integer *ldb, doublereal *
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s, doublereal *rcond, integer *rank, doublereal *work, integer *lwork,
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integer *iwork, integer *info)
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{
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/* System generated locals */
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integer a_dim1, a_offset, b_dim1, b_offset, i__1, i__2, i__3, i__4;
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/* Builtin functions */
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double log(doublereal);
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/* Local variables */
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integer ie, il, mm;
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doublereal eps, anrm, bnrm;
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integer itau, nlvl, iascl, ibscl;
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doublereal sfmin;
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integer minmn, maxmn, itaup, itauq, mnthr, nwork;
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extern /* Subroutine */ int dlabad_(doublereal *, doublereal *), dgebrd_(
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integer *, integer *, doublereal *, integer *, doublereal *,
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doublereal *, doublereal *, doublereal *, doublereal *, integer *,
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integer *);
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extern doublereal dlamch_(char *), dlange_(char *, integer *,
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integer *, doublereal *, integer *, doublereal *);
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extern /* Subroutine */ int dgelqf_(integer *, integer *, doublereal *,
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integer *, doublereal *, doublereal *, integer *, integer *),
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dlalsd_(char *, integer *, integer *, integer *, doublereal *,
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doublereal *, doublereal *, integer *, doublereal *, integer *,
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doublereal *, integer *, integer *), dlascl_(char *,
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integer *, integer *, doublereal *, doublereal *, integer *,
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integer *, doublereal *, integer *, integer *), dgeqrf_(
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integer *, integer *, doublereal *, integer *, doublereal *,
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doublereal *, integer *, integer *), dlacpy_(char *, integer *,
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integer *, doublereal *, integer *, doublereal *, integer *), dlaset_(char *, integer *, integer *, doublereal *,
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doublereal *, doublereal *, integer *), xerbla_(char *,
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integer *);
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extern integer ilaenv_(integer *, char *, char *, integer *, integer *,
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integer *, integer *);
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doublereal bignum;
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extern /* Subroutine */ int dormbr_(char *, char *, char *, integer *,
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integer *, integer *, doublereal *, integer *, doublereal *,
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doublereal *, integer *, doublereal *, integer *, integer *);
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integer wlalsd;
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extern /* Subroutine */ int dormlq_(char *, char *, integer *, integer *,
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integer *, doublereal *, integer *, doublereal *, doublereal *,
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integer *, doublereal *, integer *, integer *);
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integer ldwork;
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extern /* Subroutine */ int dormqr_(char *, char *, integer *, integer *,
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integer *, doublereal *, integer *, doublereal *, doublereal *,
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integer *, doublereal *, integer *, integer *);
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integer minwrk, maxwrk;
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doublereal smlnum;
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logical lquery;
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integer smlsiz;
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/* -- LAPACK driver 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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/* DGELSD computes the minimum-norm solution to a real linear least */
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/* squares problem: */
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/* minimize 2-norm(| b - A*x |) */
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/* using the singular value decomposition (SVD) of A. A is an M-by-N */
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/* matrix which may be rank-deficient. */
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/* Several right hand side vectors b and solution vectors x can be */
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/* handled in a single call; they are stored as the columns of the */
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/* M-by-NRHS right hand side matrix B and the N-by-NRHS solution */
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/* matrix X. */
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/* The problem is solved in three steps: */
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/* (1) Reduce the coefficient matrix A to bidiagonal form with */
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/* Householder transformations, reducing the original problem */
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/* into a "bidiagonal least squares problem" (BLS) */
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/* (2) Solve the BLS using a divide and conquer approach. */
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/* (3) Apply back all the Householder tranformations to solve */
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/* the original least squares problem. */
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/* The effective rank of A is determined by treating as zero those */
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/* singular values which are less than RCOND times the largest singular */
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/* value. */
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/* The divide and conquer algorithm makes very mild assumptions about */
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/* floating point arithmetic. It will work on machines with a guard */
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/* digit in add/subtract, or on those binary machines without guard */
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/* digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or */
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/* Cray-2. 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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/* M (input) INTEGER */
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/* The number of rows of A. M >= 0. */
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/* N (input) INTEGER */
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/* The number of columns of A. N >= 0. */
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/* NRHS (input) INTEGER */
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/* The number of right hand sides, i.e., the number of columns */
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/* of the matrices B and X. NRHS >= 0. */
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/* A (input) DOUBLE PRECISION array, dimension (LDA,N) */
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/* On entry, the M-by-N matrix A. */
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/* On exit, A has been destroyed. */
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/* LDA (input) INTEGER */
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/* The leading dimension of the array A. LDA >= max(1,M). */
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/* B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) */
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/* On entry, the M-by-NRHS right hand side matrix B. */
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/* On exit, B is overwritten by the N-by-NRHS solution */
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/* matrix X. If m >= n and RANK = n, the residual */
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/* sum-of-squares for the solution in the i-th column is given */
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/* by the sum of squares of elements n+1:m in that column. */
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/* LDB (input) INTEGER */
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/* The leading dimension of the array B. LDB >= max(1,max(M,N)). */
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/* S (output) DOUBLE PRECISION array, dimension (min(M,N)) */
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/* The singular values of A in decreasing order. */
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/* The condition number of A in the 2-norm = S(1)/S(min(m,n)). */
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/* RCOND (input) DOUBLE PRECISION */
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/* RCOND is used to determine the effective rank of A. */
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/* Singular values S(i) <= RCOND*S(1) are treated as zero. */
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/* If RCOND < 0, machine precision is used instead. */
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/* RANK (output) INTEGER */
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/* The effective rank of A, i.e., the number of singular values */
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/* which are greater than RCOND*S(1). */
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/* WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK)) */
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/* On exit, if INFO = 0, WORK(1) returns the optimal LWORK. */
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/* LWORK (input) INTEGER */
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/* The dimension of the array WORK. LWORK must be at least 1. */
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/* The exact minimum amount of workspace needed depends on M, */
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/* N and NRHS. As long as LWORK is at least */
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/* 12*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2, */
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/* if M is greater than or equal to N or */
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/* 12*M + 2*M*SMLSIZ + 8*M*NLVL + M*NRHS + (SMLSIZ+1)**2, */
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/* if M is less than N, the code will execute correctly. */
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/* SMLSIZ is returned by ILAENV and is equal to the maximum */
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/* size of the subproblems at the bottom of the computation */
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/* tree (usually about 25), and */
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/* NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 ) */
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/* For good performance, LWORK should generally be larger. */
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/* If LWORK = -1, then a workspace query is assumed; the routine */
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/* only calculates the optimal size of the WORK array, returns */
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/* this value as the first entry of the WORK array, and no error */
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/* message related to LWORK is issued by XERBLA. */
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/* IWORK (workspace) INTEGER array, dimension (MAX(1,LIWORK)) */
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/* LIWORK >= 3 * MINMN * NLVL + 11 * MINMN, */
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/* where MINMN = MIN( M,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 for computing the SVD failed to converge; */
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/* if INFO = i, i off-diagonal elements of an intermediate */
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/* bidiagonal form did not converge to zero. */
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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 Subroutines .. */
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/* .. */
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/* .. External Functions .. */
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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 arguments. */
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/* Parameter adjustments */
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a_dim1 = *lda;
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a_offset = 1 + a_dim1;
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a -= a_offset;
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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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--s;
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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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minmn = min(*m,*n);
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maxmn = max(*m,*n);
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mnthr = ilaenv_(&c__6, "DGELSD", " ", m, n, nrhs, &c_n1);
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lquery = *lwork == -1;
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if (*m < 0) {
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*info = -1;
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} else if (*n < 0) {
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*info = -2;
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} else if (*nrhs < 0) {
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*info = -3;
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} else if (*lda < max(1,*m)) {
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*info = -5;
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} else if (*ldb < max(1,maxmn)) {
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*info = -7;
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}
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smlsiz = ilaenv_(&c__9, "DGELSD", " ", &c__0, &c__0, &c__0, &c__0);
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/* Compute workspace. */
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/* (Note: Comments in the code beginning "Workspace:" describe the */
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/* minimal amount of workspace needed at that point in the code, */
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/* as well as the preferred amount for good performance. */
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/* NB refers to the optimal block size for the immediately */
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/* following subroutine, as returned by ILAENV.) */
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minwrk = 1;
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minmn = max(1,minmn);
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/* Computing MAX */
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i__1 = (integer) (log((doublereal) minmn / (doublereal) (smlsiz + 1)) /
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log(2.)) + 1;
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nlvl = max(i__1,0);
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if (*info == 0) {
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maxwrk = 0;
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mm = *m;
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if (*m >= *n && *m >= mnthr) {
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/* Path 1a - overdetermined, with many more rows than columns. */
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mm = *n;
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/* Computing MAX */
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i__1 = maxwrk, i__2 = *n + *n * ilaenv_(&c__1, "DGEQRF", " ", m,
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n, &c_n1, &c_n1);
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maxwrk = max(i__1,i__2);
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/* Computing MAX */
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i__1 = maxwrk, i__2 = *n + *nrhs * ilaenv_(&c__1, "DORMQR", "LT",
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m, nrhs, n, &c_n1);
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maxwrk = max(i__1,i__2);
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}
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if (*m >= *n) {
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/* Path 1 - overdetermined or exactly determined. */
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/* Computing MAX */
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i__1 = maxwrk, i__2 = *n * 3 + (mm + *n) * ilaenv_(&c__1, "DGEBRD"
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, " ", &mm, n, &c_n1, &c_n1);
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maxwrk = max(i__1,i__2);
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/* Computing MAX */
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i__1 = maxwrk, i__2 = *n * 3 + *nrhs * ilaenv_(&c__1, "DORMBR",
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"QLT", &mm, nrhs, n, &c_n1);
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maxwrk = max(i__1,i__2);
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/* Computing MAX */
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i__1 = maxwrk, i__2 = *n * 3 + (*n - 1) * ilaenv_(&c__1, "DORMBR",
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"PLN", n, nrhs, n, &c_n1);
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maxwrk = max(i__1,i__2);
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/* Computing 2nd power */
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i__1 = smlsiz + 1;
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wlalsd = *n * 9 + (*n << 1) * smlsiz + (*n << 3) * nlvl + *n * *
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nrhs + i__1 * i__1;
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/* Computing MAX */
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i__1 = maxwrk, i__2 = *n * 3 + wlalsd;
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maxwrk = max(i__1,i__2);
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/* Computing MAX */
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i__1 = *n * 3 + mm, i__2 = *n * 3 + *nrhs, i__1 = max(i__1,i__2),
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i__2 = *n * 3 + wlalsd;
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minwrk = max(i__1,i__2);
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}
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if (*n > *m) {
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/* Computing 2nd power */
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i__1 = smlsiz + 1;
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wlalsd = *m * 9 + (*m << 1) * smlsiz + (*m << 3) * nlvl + *m * *
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nrhs + i__1 * i__1;
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if (*n >= mnthr) {
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/* Path 2a - underdetermined, with many more columns */
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/* than rows. */
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maxwrk = *m + *m * ilaenv_(&c__1, "DGELQF", " ", m, n, &c_n1,
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&c_n1);
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/* Computing MAX */
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i__1 = maxwrk, i__2 = *m * *m + (*m << 2) + (*m << 1) *
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ilaenv_(&c__1, "DGEBRD", " ", m, m, &c_n1, &c_n1);
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maxwrk = max(i__1,i__2);
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/* Computing MAX */
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i__1 = maxwrk, i__2 = *m * *m + (*m << 2) + *nrhs * ilaenv_(&
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c__1, "DORMBR", "QLT", m, nrhs, m, &c_n1);
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maxwrk = max(i__1,i__2);
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/* Computing MAX */
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i__1 = maxwrk, i__2 = *m * *m + (*m << 2) + (*m - 1) *
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ilaenv_(&c__1, "DORMBR", "PLN", m, nrhs, m, &c_n1);
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maxwrk = max(i__1,i__2);
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if (*nrhs > 1) {
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/* Computing MAX */
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i__1 = maxwrk, i__2 = *m * *m + *m + *m * *nrhs;
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maxwrk = max(i__1,i__2);
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} else {
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/* Computing MAX */
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i__1 = maxwrk, i__2 = *m * *m + (*m << 1);
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maxwrk = max(i__1,i__2);
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}
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/* Computing MAX */
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i__1 = maxwrk, i__2 = *m + *nrhs * ilaenv_(&c__1, "DORMLQ",
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"LT", n, nrhs, m, &c_n1);
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maxwrk = max(i__1,i__2);
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/* Computing MAX */
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i__1 = maxwrk, i__2 = *m * *m + (*m << 2) + wlalsd;
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maxwrk = max(i__1,i__2);
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} else {
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/* Path 2 - remaining underdetermined cases. */
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maxwrk = *m * 3 + (*n + *m) * ilaenv_(&c__1, "DGEBRD", " ", m,
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n, &c_n1, &c_n1);
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/* Computing MAX */
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i__1 = maxwrk, i__2 = *m * 3 + *nrhs * ilaenv_(&c__1, "DORMBR"
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, "QLT", m, nrhs, n, &c_n1);
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maxwrk = max(i__1,i__2);
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/* Computing MAX */
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i__1 = maxwrk, i__2 = *m * 3 + *m * ilaenv_(&c__1, "DORMBR",
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"PLN", n, nrhs, m, &c_n1);
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maxwrk = max(i__1,i__2);
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/* Computing MAX */
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i__1 = maxwrk, i__2 = *m * 3 + wlalsd;
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maxwrk = max(i__1,i__2);
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}
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/* Computing MAX */
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i__1 = *m * 3 + *nrhs, i__2 = *m * 3 + *m, i__1 = max(i__1,i__2),
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i__2 = *m * 3 + wlalsd;
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minwrk = max(i__1,i__2);
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}
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minwrk = min(minwrk,maxwrk);
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work[1] = (doublereal) maxwrk;
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if (*lwork < minwrk && ! lquery) {
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*info = -12;
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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_("DGELSD", &i__1);
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return 0;
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} else if (lquery) {
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goto L10;
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}
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/* Quick return if possible. */
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if (*m == 0 || *n == 0) {
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*rank = 0;
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return 0;
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}
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/* Get machine parameters. */
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eps = dlamch_("P");
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sfmin = dlamch_("S");
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smlnum = sfmin / eps;
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bignum = 1. / smlnum;
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dlabad_(&smlnum, &bignum);
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/* Scale A if max entry outside range [SMLNUM,BIGNUM]. */
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anrm = dlange_("M", m, n, &a[a_offset], lda, &work[1]);
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iascl = 0;
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if (anrm > 0. && anrm < smlnum) {
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/* Scale matrix norm up to SMLNUM. */
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dlascl_("G", &c__0, &c__0, &anrm, &smlnum, m, n, &a[a_offset], lda,
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info);
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iascl = 1;
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} else if (anrm > bignum) {
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/* Scale matrix norm down to BIGNUM. */
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dlascl_("G", &c__0, &c__0, &anrm, &bignum, m, n, &a[a_offset], lda,
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info);
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iascl = 2;
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} else if (anrm == 0.) {
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/* Matrix all zero. Return zero solution. */
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i__1 = max(*m,*n);
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dlaset_("F", &i__1, nrhs, &c_b82, &c_b82, &b[b_offset], ldb);
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dlaset_("F", &minmn, &c__1, &c_b82, &c_b82, &s[1], &c__1);
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*rank = 0;
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goto L10;
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}
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/* Scale B if max entry outside range [SMLNUM,BIGNUM]. */
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bnrm = dlange_("M", m, nrhs, &b[b_offset], ldb, &work[1]);
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ibscl = 0;
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if (bnrm > 0. && bnrm < smlnum) {
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/* Scale matrix norm up to SMLNUM. */
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dlascl_("G", &c__0, &c__0, &bnrm, &smlnum, m, nrhs, &b[b_offset], ldb,
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info);
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ibscl = 1;
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} else if (bnrm > bignum) {
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/* Scale matrix norm down to BIGNUM. */
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dlascl_("G", &c__0, &c__0, &bnrm, &bignum, m, nrhs, &b[b_offset], ldb,
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info);
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ibscl = 2;
|
|
}
|
|
|
|
/* If M < N make sure certain entries of B are zero. */
|
|
|
|
if (*m < *n) {
|
|
i__1 = *n - *m;
|
|
dlaset_("F", &i__1, nrhs, &c_b82, &c_b82, &b[*m + 1 + b_dim1], ldb);
|
|
}
|
|
|
|
/* Overdetermined case. */
|
|
|
|
if (*m >= *n) {
|
|
|
|
/* Path 1 - overdetermined or exactly determined. */
|
|
|
|
mm = *m;
|
|
if (*m >= mnthr) {
|
|
|
|
/* Path 1a - overdetermined, with many more rows than columns. */
|
|
|
|
mm = *n;
|
|
itau = 1;
|
|
nwork = itau + *n;
|
|
|
|
/* Compute A=Q*R. */
|
|
/* (Workspace: need 2*N, prefer N+N*NB) */
|
|
|
|
i__1 = *lwork - nwork + 1;
|
|
dgeqrf_(m, n, &a[a_offset], lda, &work[itau], &work[nwork], &i__1,
|
|
info);
|
|
|
|
/* Multiply B by transpose(Q). */
|
|
/* (Workspace: need N+NRHS, prefer N+NRHS*NB) */
|
|
|
|
i__1 = *lwork - nwork + 1;
|
|
dormqr_("L", "T", m, nrhs, n, &a[a_offset], lda, &work[itau], &b[
|
|
b_offset], ldb, &work[nwork], &i__1, info);
|
|
|
|
/* Zero out below R. */
|
|
|
|
if (*n > 1) {
|
|
i__1 = *n - 1;
|
|
i__2 = *n - 1;
|
|
dlaset_("L", &i__1, &i__2, &c_b82, &c_b82, &a[a_dim1 + 2],
|
|
lda);
|
|
}
|
|
}
|
|
|
|
ie = 1;
|
|
itauq = ie + *n;
|
|
itaup = itauq + *n;
|
|
nwork = itaup + *n;
|
|
|
|
/* Bidiagonalize R in A. */
|
|
/* (Workspace: need 3*N+MM, prefer 3*N+(MM+N)*NB) */
|
|
|
|
i__1 = *lwork - nwork + 1;
|
|
dgebrd_(&mm, n, &a[a_offset], lda, &s[1], &work[ie], &work[itauq], &
|
|
work[itaup], &work[nwork], &i__1, info);
|
|
|
|
/* Multiply B by transpose of left bidiagonalizing vectors of R. */
|
|
/* (Workspace: need 3*N+NRHS, prefer 3*N+NRHS*NB) */
|
|
|
|
i__1 = *lwork - nwork + 1;
|
|
dormbr_("Q", "L", "T", &mm, nrhs, n, &a[a_offset], lda, &work[itauq],
|
|
&b[b_offset], ldb, &work[nwork], &i__1, info);
|
|
|
|
/* Solve the bidiagonal least squares problem. */
|
|
|
|
dlalsd_("U", &smlsiz, n, nrhs, &s[1], &work[ie], &b[b_offset], ldb,
|
|
rcond, rank, &work[nwork], &iwork[1], info);
|
|
if (*info != 0) {
|
|
goto L10;
|
|
}
|
|
|
|
/* Multiply B by right bidiagonalizing vectors of R. */
|
|
|
|
i__1 = *lwork - nwork + 1;
|
|
dormbr_("P", "L", "N", n, nrhs, n, &a[a_offset], lda, &work[itaup], &
|
|
b[b_offset], ldb, &work[nwork], &i__1, info);
|
|
|
|
} else /* if(complicated condition) */ {
|
|
/* Computing MAX */
|
|
i__1 = *m, i__2 = (*m << 1) - 4, i__1 = max(i__1,i__2), i__1 = max(
|
|
i__1,*nrhs), i__2 = *n - *m * 3, i__1 = max(i__1,i__2);
|
|
if (*n >= mnthr && *lwork >= (*m << 2) + *m * *m + max(i__1,wlalsd)) {
|
|
|
|
/* Path 2a - underdetermined, with many more columns than rows */
|
|
/* and sufficient workspace for an efficient algorithm. */
|
|
|
|
ldwork = *m;
|
|
/* Computing MAX */
|
|
/* Computing MAX */
|
|
i__3 = *m, i__4 = (*m << 1) - 4, i__3 = max(i__3,i__4), i__3 =
|
|
max(i__3,*nrhs), i__4 = *n - *m * 3;
|
|
i__1 = (*m << 2) + *m * *lda + max(i__3,i__4), i__2 = *m * *lda +
|
|
*m + *m * *nrhs, i__1 = max(i__1,i__2), i__2 = (*m << 2)
|
|
+ *m * *lda + wlalsd;
|
|
if (*lwork >= max(i__1,i__2)) {
|
|
ldwork = *lda;
|
|
}
|
|
itau = 1;
|
|
nwork = *m + 1;
|
|
|
|
/* Compute A=L*Q. */
|
|
/* (Workspace: need 2*M, prefer M+M*NB) */
|
|
|
|
i__1 = *lwork - nwork + 1;
|
|
dgelqf_(m, n, &a[a_offset], lda, &work[itau], &work[nwork], &i__1,
|
|
info);
|
|
il = nwork;
|
|
|
|
/* Copy L to WORK(IL), zeroing out above its diagonal. */
|
|
|
|
dlacpy_("L", m, m, &a[a_offset], lda, &work[il], &ldwork);
|
|
i__1 = *m - 1;
|
|
i__2 = *m - 1;
|
|
dlaset_("U", &i__1, &i__2, &c_b82, &c_b82, &work[il + ldwork], &
|
|
ldwork);
|
|
ie = il + ldwork * *m;
|
|
itauq = ie + *m;
|
|
itaup = itauq + *m;
|
|
nwork = itaup + *m;
|
|
|
|
/* Bidiagonalize L in WORK(IL). */
|
|
/* (Workspace: need M*M+5*M, prefer M*M+4*M+2*M*NB) */
|
|
|
|
i__1 = *lwork - nwork + 1;
|
|
dgebrd_(m, m, &work[il], &ldwork, &s[1], &work[ie], &work[itauq],
|
|
&work[itaup], &work[nwork], &i__1, info);
|
|
|
|
/* Multiply B by transpose of left bidiagonalizing vectors of L. */
|
|
/* (Workspace: need M*M+4*M+NRHS, prefer M*M+4*M+NRHS*NB) */
|
|
|
|
i__1 = *lwork - nwork + 1;
|
|
dormbr_("Q", "L", "T", m, nrhs, m, &work[il], &ldwork, &work[
|
|
itauq], &b[b_offset], ldb, &work[nwork], &i__1, info);
|
|
|
|
/* Solve the bidiagonal least squares problem. */
|
|
|
|
dlalsd_("U", &smlsiz, m, nrhs, &s[1], &work[ie], &b[b_offset],
|
|
ldb, rcond, rank, &work[nwork], &iwork[1], info);
|
|
if (*info != 0) {
|
|
goto L10;
|
|
}
|
|
|
|
/* Multiply B by right bidiagonalizing vectors of L. */
|
|
|
|
i__1 = *lwork - nwork + 1;
|
|
dormbr_("P", "L", "N", m, nrhs, m, &work[il], &ldwork, &work[
|
|
itaup], &b[b_offset], ldb, &work[nwork], &i__1, info);
|
|
|
|
/* Zero out below first M rows of B. */
|
|
|
|
i__1 = *n - *m;
|
|
dlaset_("F", &i__1, nrhs, &c_b82, &c_b82, &b[*m + 1 + b_dim1],
|
|
ldb);
|
|
nwork = itau + *m;
|
|
|
|
/* Multiply transpose(Q) by B. */
|
|
/* (Workspace: need M+NRHS, prefer M+NRHS*NB) */
|
|
|
|
i__1 = *lwork - nwork + 1;
|
|
dormlq_("L", "T", n, nrhs, m, &a[a_offset], lda, &work[itau], &b[
|
|
b_offset], ldb, &work[nwork], &i__1, info);
|
|
|
|
} else {
|
|
|
|
/* Path 2 - remaining underdetermined cases. */
|
|
|
|
ie = 1;
|
|
itauq = ie + *m;
|
|
itaup = itauq + *m;
|
|
nwork = itaup + *m;
|
|
|
|
/* Bidiagonalize A. */
|
|
/* (Workspace: need 3*M+N, prefer 3*M+(M+N)*NB) */
|
|
|
|
i__1 = *lwork - nwork + 1;
|
|
dgebrd_(m, n, &a[a_offset], lda, &s[1], &work[ie], &work[itauq], &
|
|
work[itaup], &work[nwork], &i__1, info);
|
|
|
|
/* Multiply B by transpose of left bidiagonalizing vectors. */
|
|
/* (Workspace: need 3*M+NRHS, prefer 3*M+NRHS*NB) */
|
|
|
|
i__1 = *lwork - nwork + 1;
|
|
dormbr_("Q", "L", "T", m, nrhs, n, &a[a_offset], lda, &work[itauq]
|
|
, &b[b_offset], ldb, &work[nwork], &i__1, info);
|
|
|
|
/* Solve the bidiagonal least squares problem. */
|
|
|
|
dlalsd_("L", &smlsiz, m, nrhs, &s[1], &work[ie], &b[b_offset],
|
|
ldb, rcond, rank, &work[nwork], &iwork[1], info);
|
|
if (*info != 0) {
|
|
goto L10;
|
|
}
|
|
|
|
/* Multiply B by right bidiagonalizing vectors of A. */
|
|
|
|
i__1 = *lwork - nwork + 1;
|
|
dormbr_("P", "L", "N", n, nrhs, m, &a[a_offset], lda, &work[itaup]
|
|
, &b[b_offset], ldb, &work[nwork], &i__1, info);
|
|
|
|
}
|
|
}
|
|
|
|
/* Undo scaling. */
|
|
|
|
if (iascl == 1) {
|
|
dlascl_("G", &c__0, &c__0, &anrm, &smlnum, n, nrhs, &b[b_offset], ldb,
|
|
info);
|
|
dlascl_("G", &c__0, &c__0, &smlnum, &anrm, &minmn, &c__1, &s[1], &
|
|
minmn, info);
|
|
} else if (iascl == 2) {
|
|
dlascl_("G", &c__0, &c__0, &anrm, &bignum, n, nrhs, &b[b_offset], ldb,
|
|
info);
|
|
dlascl_("G", &c__0, &c__0, &bignum, &anrm, &minmn, &c__1, &s[1], &
|
|
minmn, info);
|
|
}
|
|
if (ibscl == 1) {
|
|
dlascl_("G", &c__0, &c__0, &smlnum, &bnrm, n, nrhs, &b[b_offset], ldb,
|
|
info);
|
|
} else if (ibscl == 2) {
|
|
dlascl_("G", &c__0, &c__0, &bignum, &bnrm, n, nrhs, &b[b_offset], ldb,
|
|
info);
|
|
}
|
|
|
|
L10:
|
|
work[1] = (doublereal) maxwrk;
|
|
return 0;
|
|
|
|
/* End of DGELSD */
|
|
|
|
} /* dgelsd_ */
|