503 lines
14 KiB
C
503 lines
14 KiB
C
#include "clapack.h"
|
|
|
|
/* Table of constant values */
|
|
|
|
static integer c__1 = 1;
|
|
static integer c_n1 = -1;
|
|
static doublereal c_b33 = 0.;
|
|
static integer c__0 = 0;
|
|
|
|
/* Subroutine */ int dgels_(char *trans, integer *m, integer *n, integer *
|
|
nrhs, doublereal *a, integer *lda, doublereal *b, integer *ldb,
|
|
doublereal *work, integer *lwork, integer *info)
|
|
{
|
|
/* System generated locals */
|
|
integer a_dim1, a_offset, b_dim1, b_offset, i__1, i__2;
|
|
|
|
/* Local variables */
|
|
integer i__, j, nb, mn;
|
|
doublereal anrm, bnrm;
|
|
integer brow;
|
|
logical tpsd;
|
|
integer iascl, ibscl;
|
|
extern logical lsame_(char *, char *);
|
|
integer wsize;
|
|
doublereal rwork[1];
|
|
extern /* Subroutine */ int dlabad_(doublereal *, doublereal *);
|
|
extern doublereal dlamch_(char *), dlange_(char *, integer *,
|
|
integer *, doublereal *, integer *, doublereal *);
|
|
extern /* Subroutine */ int dgelqf_(integer *, integer *, doublereal *,
|
|
integer *, doublereal *, doublereal *, integer *, integer *),
|
|
dlascl_(char *, integer *, integer *, doublereal *, doublereal *,
|
|
integer *, integer *, doublereal *, integer *, integer *),
|
|
dgeqrf_(integer *, integer *, doublereal *, integer *,
|
|
doublereal *, doublereal *, integer *, integer *), dlaset_(char *,
|
|
integer *, integer *, doublereal *, doublereal *, doublereal *,
|
|
integer *), xerbla_(char *, integer *);
|
|
extern integer ilaenv_(integer *, char *, char *, integer *, integer *,
|
|
integer *, integer *);
|
|
integer scllen;
|
|
doublereal bignum;
|
|
extern /* Subroutine */ int dormlq_(char *, char *, integer *, integer *,
|
|
integer *, doublereal *, integer *, doublereal *, doublereal *,
|
|
integer *, doublereal *, integer *, integer *),
|
|
dormqr_(char *, char *, integer *, integer *, integer *,
|
|
doublereal *, integer *, doublereal *, doublereal *, integer *,
|
|
doublereal *, integer *, integer *);
|
|
doublereal smlnum;
|
|
logical lquery;
|
|
extern /* Subroutine */ int dtrtrs_(char *, char *, char *, integer *,
|
|
integer *, doublereal *, integer *, doublereal *, integer *,
|
|
integer *);
|
|
|
|
|
|
/* -- LAPACK driver routine (version 3.1) -- */
|
|
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
|
|
/* November 2006 */
|
|
|
|
/* .. Scalar Arguments .. */
|
|
/* .. */
|
|
/* .. Array Arguments .. */
|
|
/* .. */
|
|
|
|
/* Purpose */
|
|
/* ======= */
|
|
|
|
/* DGELS solves overdetermined or underdetermined real linear systems */
|
|
/* involving an M-by-N matrix A, or its transpose, using a QR or LQ */
|
|
/* factorization of A. It is assumed that A has full rank. */
|
|
|
|
/* The following options are provided: */
|
|
|
|
/* 1. If TRANS = 'N' and m >= n: find the least squares solution of */
|
|
/* an overdetermined system, i.e., solve the least squares problem */
|
|
/* minimize || B - A*X ||. */
|
|
|
|
/* 2. If TRANS = 'N' and m < n: find the minimum norm solution of */
|
|
/* an underdetermined system A * X = B. */
|
|
|
|
/* 3. If TRANS = 'T' and m >= n: find the minimum norm solution of */
|
|
/* an undetermined system A**T * X = B. */
|
|
|
|
/* 4. If TRANS = 'T' and m < n: find the least squares solution of */
|
|
/* an overdetermined system, i.e., solve the least squares problem */
|
|
/* minimize || B - A**T * X ||. */
|
|
|
|
/* Several right hand side vectors b and solution vectors x can be */
|
|
/* handled in a single call; they are stored as the columns of the */
|
|
/* M-by-NRHS right hand side matrix B and the N-by-NRHS solution */
|
|
/* matrix X. */
|
|
|
|
/* Arguments */
|
|
/* ========= */
|
|
|
|
/* TRANS (input) CHARACTER*1 */
|
|
/* = 'N': the linear system involves A; */
|
|
/* = 'T': the linear system involves A**T. */
|
|
|
|
/* M (input) INTEGER */
|
|
/* The number of rows of the matrix A. M >= 0. */
|
|
|
|
/* N (input) INTEGER */
|
|
/* The number of columns of the matrix A. N >= 0. */
|
|
|
|
/* NRHS (input) INTEGER */
|
|
/* The number of right hand sides, i.e., the number of */
|
|
/* columns of the matrices B and X. NRHS >=0. */
|
|
|
|
/* A (input/output) DOUBLE PRECISION array, dimension (LDA,N) */
|
|
/* On entry, the M-by-N matrix A. */
|
|
/* On exit, */
|
|
/* if M >= N, A is overwritten by details of its QR */
|
|
/* factorization as returned by DGEQRF; */
|
|
/* if M < N, A is overwritten by details of its LQ */
|
|
/* factorization as returned by DGELQF. */
|
|
|
|
/* LDA (input) INTEGER */
|
|
/* The leading dimension of the array A. LDA >= max(1,M). */
|
|
|
|
/* B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) */
|
|
/* On entry, the matrix B of right hand side vectors, stored */
|
|
/* columnwise; B is M-by-NRHS if TRANS = 'N', or N-by-NRHS */
|
|
/* if TRANS = 'T'. */
|
|
/* On exit, if INFO = 0, B is overwritten by the solution */
|
|
/* vectors, stored columnwise: */
|
|
/* if TRANS = 'N' and m >= n, rows 1 to n of B contain the least */
|
|
/* squares solution vectors; the residual sum of squares for the */
|
|
/* solution in each column is given by the sum of squares of */
|
|
/* elements N+1 to M in that column; */
|
|
/* if TRANS = 'N' and m < n, rows 1 to N of B contain the */
|
|
/* minimum norm solution vectors; */
|
|
/* if TRANS = 'T' and m >= n, rows 1 to M of B contain the */
|
|
/* minimum norm solution vectors; */
|
|
/* if TRANS = 'T' and m < n, rows 1 to M of B contain the */
|
|
/* least squares solution vectors; the residual sum of squares */
|
|
/* for the solution in each column is given by the sum of */
|
|
/* squares of elements M+1 to N in that column. */
|
|
|
|
/* LDB (input) INTEGER */
|
|
/* The leading dimension of the array B. LDB >= MAX(1,M,N). */
|
|
|
|
/* WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK)) */
|
|
/* On exit, if INFO = 0, WORK(1) returns the optimal LWORK. */
|
|
|
|
/* LWORK (input) INTEGER */
|
|
/* The dimension of the array WORK. */
|
|
/* LWORK >= max( 1, MN + max( MN, NRHS ) ). */
|
|
/* For optimal performance, */
|
|
/* LWORK >= max( 1, MN + max( MN, NRHS )*NB ). */
|
|
/* where MN = min(M,N) and NB is the optimum block size. */
|
|
|
|
/* If LWORK = -1, then a workspace query is assumed; the routine */
|
|
/* only calculates the optimal size of the WORK array, returns */
|
|
/* this value as the first entry of the WORK array, and no error */
|
|
/* message related to LWORK is issued by XERBLA. */
|
|
|
|
/* INFO (output) INTEGER */
|
|
/* = 0: successful exit */
|
|
/* < 0: if INFO = -i, the i-th argument had an illegal value */
|
|
/* > 0: if INFO = i, the i-th diagonal element of the */
|
|
/* triangular factor of A is zero, so that A does not have */
|
|
/* full rank; the least squares solution could not be */
|
|
/* computed. */
|
|
|
|
/* ===================================================================== */
|
|
|
|
/* .. Parameters .. */
|
|
/* .. */
|
|
/* .. Local Scalars .. */
|
|
/* .. */
|
|
/* .. Local Arrays .. */
|
|
/* .. */
|
|
/* .. External Functions .. */
|
|
/* .. */
|
|
/* .. External Subroutines .. */
|
|
/* .. */
|
|
/* .. Intrinsic Functions .. */
|
|
/* .. */
|
|
/* .. Executable Statements .. */
|
|
|
|
/* Test the input arguments. */
|
|
|
|
/* Parameter adjustments */
|
|
a_dim1 = *lda;
|
|
a_offset = 1 + a_dim1;
|
|
a -= a_offset;
|
|
b_dim1 = *ldb;
|
|
b_offset = 1 + b_dim1;
|
|
b -= b_offset;
|
|
--work;
|
|
|
|
/* Function Body */
|
|
*info = 0;
|
|
mn = min(*m,*n);
|
|
lquery = *lwork == -1;
|
|
if (! (lsame_(trans, "N") || lsame_(trans, "T"))) {
|
|
*info = -1;
|
|
} else if (*m < 0) {
|
|
*info = -2;
|
|
} else if (*n < 0) {
|
|
*info = -3;
|
|
} else if (*nrhs < 0) {
|
|
*info = -4;
|
|
} else if (*lda < max(1,*m)) {
|
|
*info = -6;
|
|
} else /* if(complicated condition) */ {
|
|
/* Computing MAX */
|
|
i__1 = max(1,*m);
|
|
if (*ldb < max(i__1,*n)) {
|
|
*info = -8;
|
|
} else /* if(complicated condition) */ {
|
|
/* Computing MAX */
|
|
i__1 = 1, i__2 = mn + max(mn,*nrhs);
|
|
if (*lwork < max(i__1,i__2) && ! lquery) {
|
|
*info = -10;
|
|
}
|
|
}
|
|
}
|
|
|
|
/* Figure out optimal block size */
|
|
|
|
if (*info == 0 || *info == -10) {
|
|
|
|
tpsd = TRUE_;
|
|
if (lsame_(trans, "N")) {
|
|
tpsd = FALSE_;
|
|
}
|
|
|
|
if (*m >= *n) {
|
|
nb = ilaenv_(&c__1, "DGEQRF", " ", m, n, &c_n1, &c_n1);
|
|
if (tpsd) {
|
|
/* Computing MAX */
|
|
i__1 = nb, i__2 = ilaenv_(&c__1, "DORMQR", "LN", m, nrhs, n, &
|
|
c_n1);
|
|
nb = max(i__1,i__2);
|
|
} else {
|
|
/* Computing MAX */
|
|
i__1 = nb, i__2 = ilaenv_(&c__1, "DORMQR", "LT", m, nrhs, n, &
|
|
c_n1);
|
|
nb = max(i__1,i__2);
|
|
}
|
|
} else {
|
|
nb = ilaenv_(&c__1, "DGELQF", " ", m, n, &c_n1, &c_n1);
|
|
if (tpsd) {
|
|
/* Computing MAX */
|
|
i__1 = nb, i__2 = ilaenv_(&c__1, "DORMLQ", "LT", n, nrhs, m, &
|
|
c_n1);
|
|
nb = max(i__1,i__2);
|
|
} else {
|
|
/* Computing MAX */
|
|
i__1 = nb, i__2 = ilaenv_(&c__1, "DORMLQ", "LN", n, nrhs, m, &
|
|
c_n1);
|
|
nb = max(i__1,i__2);
|
|
}
|
|
}
|
|
|
|
/* Computing MAX */
|
|
i__1 = 1, i__2 = mn + max(mn,*nrhs) * nb;
|
|
wsize = max(i__1,i__2);
|
|
work[1] = (doublereal) wsize;
|
|
|
|
}
|
|
|
|
if (*info != 0) {
|
|
i__1 = -(*info);
|
|
xerbla_("DGELS ", &i__1);
|
|
return 0;
|
|
} else if (lquery) {
|
|
return 0;
|
|
}
|
|
|
|
/* Quick return if possible */
|
|
|
|
/* Computing MIN */
|
|
i__1 = min(*m,*n);
|
|
if (min(i__1,*nrhs) == 0) {
|
|
i__1 = max(*m,*n);
|
|
dlaset_("Full", &i__1, nrhs, &c_b33, &c_b33, &b[b_offset], ldb);
|
|
return 0;
|
|
}
|
|
|
|
/* Get machine parameters */
|
|
|
|
smlnum = dlamch_("S") / dlamch_("P");
|
|
bignum = 1. / smlnum;
|
|
dlabad_(&smlnum, &bignum);
|
|
|
|
/* Scale A, B if max element outside range [SMLNUM,BIGNUM] */
|
|
|
|
anrm = dlange_("M", m, n, &a[a_offset], lda, rwork);
|
|
iascl = 0;
|
|
if (anrm > 0. && anrm < smlnum) {
|
|
|
|
/* Scale matrix norm up to SMLNUM */
|
|
|
|
dlascl_("G", &c__0, &c__0, &anrm, &smlnum, m, n, &a[a_offset], lda,
|
|
info);
|
|
iascl = 1;
|
|
} else if (anrm > bignum) {
|
|
|
|
/* Scale matrix norm down to BIGNUM */
|
|
|
|
dlascl_("G", &c__0, &c__0, &anrm, &bignum, m, n, &a[a_offset], lda,
|
|
info);
|
|
iascl = 2;
|
|
} else if (anrm == 0.) {
|
|
|
|
/* Matrix all zero. Return zero solution. */
|
|
|
|
i__1 = max(*m,*n);
|
|
dlaset_("F", &i__1, nrhs, &c_b33, &c_b33, &b[b_offset], ldb);
|
|
goto L50;
|
|
}
|
|
|
|
brow = *m;
|
|
if (tpsd) {
|
|
brow = *n;
|
|
}
|
|
bnrm = dlange_("M", &brow, nrhs, &b[b_offset], ldb, rwork);
|
|
ibscl = 0;
|
|
if (bnrm > 0. && bnrm < smlnum) {
|
|
|
|
/* Scale matrix norm up to SMLNUM */
|
|
|
|
dlascl_("G", &c__0, &c__0, &bnrm, &smlnum, &brow, nrhs, &b[b_offset],
|
|
ldb, info);
|
|
ibscl = 1;
|
|
} else if (bnrm > bignum) {
|
|
|
|
/* Scale matrix norm down to BIGNUM */
|
|
|
|
dlascl_("G", &c__0, &c__0, &bnrm, &bignum, &brow, nrhs, &b[b_offset],
|
|
ldb, info);
|
|
ibscl = 2;
|
|
}
|
|
|
|
if (*m >= *n) {
|
|
|
|
/* compute QR factorization of A */
|
|
|
|
i__1 = *lwork - mn;
|
|
dgeqrf_(m, n, &a[a_offset], lda, &work[1], &work[mn + 1], &i__1, info)
|
|
;
|
|
|
|
/* workspace at least N, optimally N*NB */
|
|
|
|
if (! tpsd) {
|
|
|
|
/* Least-Squares Problem min || A * X - B || */
|
|
|
|
/* B(1:M,1:NRHS) := Q' * B(1:M,1:NRHS) */
|
|
|
|
i__1 = *lwork - mn;
|
|
dormqr_("Left", "Transpose", m, nrhs, n, &a[a_offset], lda, &work[
|
|
1], &b[b_offset], ldb, &work[mn + 1], &i__1, info);
|
|
|
|
/* workspace at least NRHS, optimally NRHS*NB */
|
|
|
|
/* B(1:N,1:NRHS) := inv(R) * B(1:N,1:NRHS) */
|
|
|
|
dtrtrs_("Upper", "No transpose", "Non-unit", n, nrhs, &a[a_offset]
|
|
, lda, &b[b_offset], ldb, info);
|
|
|
|
if (*info > 0) {
|
|
return 0;
|
|
}
|
|
|
|
scllen = *n;
|
|
|
|
} else {
|
|
|
|
/* Overdetermined system of equations A' * X = B */
|
|
|
|
/* B(1:N,1:NRHS) := inv(R') * B(1:N,1:NRHS) */
|
|
|
|
dtrtrs_("Upper", "Transpose", "Non-unit", n, nrhs, &a[a_offset],
|
|
lda, &b[b_offset], ldb, info);
|
|
|
|
if (*info > 0) {
|
|
return 0;
|
|
}
|
|
|
|
/* B(N+1:M,1:NRHS) = ZERO */
|
|
|
|
i__1 = *nrhs;
|
|
for (j = 1; j <= i__1; ++j) {
|
|
i__2 = *m;
|
|
for (i__ = *n + 1; i__ <= i__2; ++i__) {
|
|
b[i__ + j * b_dim1] = 0.;
|
|
/* L10: */
|
|
}
|
|
/* L20: */
|
|
}
|
|
|
|
/* B(1:M,1:NRHS) := Q(1:N,:) * B(1:N,1:NRHS) */
|
|
|
|
i__1 = *lwork - mn;
|
|
dormqr_("Left", "No transpose", m, nrhs, n, &a[a_offset], lda, &
|
|
work[1], &b[b_offset], ldb, &work[mn + 1], &i__1, info);
|
|
|
|
/* workspace at least NRHS, optimally NRHS*NB */
|
|
|
|
scllen = *m;
|
|
|
|
}
|
|
|
|
} else {
|
|
|
|
/* Compute LQ factorization of A */
|
|
|
|
i__1 = *lwork - mn;
|
|
dgelqf_(m, n, &a[a_offset], lda, &work[1], &work[mn + 1], &i__1, info)
|
|
;
|
|
|
|
/* workspace at least M, optimally M*NB. */
|
|
|
|
if (! tpsd) {
|
|
|
|
/* underdetermined system of equations A * X = B */
|
|
|
|
/* B(1:M,1:NRHS) := inv(L) * B(1:M,1:NRHS) */
|
|
|
|
dtrtrs_("Lower", "No transpose", "Non-unit", m, nrhs, &a[a_offset]
|
|
, lda, &b[b_offset], ldb, info);
|
|
|
|
if (*info > 0) {
|
|
return 0;
|
|
}
|
|
|
|
/* B(M+1:N,1:NRHS) = 0 */
|
|
|
|
i__1 = *nrhs;
|
|
for (j = 1; j <= i__1; ++j) {
|
|
i__2 = *n;
|
|
for (i__ = *m + 1; i__ <= i__2; ++i__) {
|
|
b[i__ + j * b_dim1] = 0.;
|
|
/* L30: */
|
|
}
|
|
/* L40: */
|
|
}
|
|
|
|
/* B(1:N,1:NRHS) := Q(1:N,:)' * B(1:M,1:NRHS) */
|
|
|
|
i__1 = *lwork - mn;
|
|
dormlq_("Left", "Transpose", n, nrhs, m, &a[a_offset], lda, &work[
|
|
1], &b[b_offset], ldb, &work[mn + 1], &i__1, info);
|
|
|
|
/* workspace at least NRHS, optimally NRHS*NB */
|
|
|
|
scllen = *n;
|
|
|
|
} else {
|
|
|
|
/* overdetermined system min || A' * X - B || */
|
|
|
|
/* B(1:N,1:NRHS) := Q * B(1:N,1:NRHS) */
|
|
|
|
i__1 = *lwork - mn;
|
|
dormlq_("Left", "No transpose", n, nrhs, m, &a[a_offset], lda, &
|
|
work[1], &b[b_offset], ldb, &work[mn + 1], &i__1, info);
|
|
|
|
/* workspace at least NRHS, optimally NRHS*NB */
|
|
|
|
/* B(1:M,1:NRHS) := inv(L') * B(1:M,1:NRHS) */
|
|
|
|
dtrtrs_("Lower", "Transpose", "Non-unit", m, nrhs, &a[a_offset],
|
|
lda, &b[b_offset], ldb, info);
|
|
|
|
if (*info > 0) {
|
|
return 0;
|
|
}
|
|
|
|
scllen = *m;
|
|
|
|
}
|
|
|
|
}
|
|
|
|
/* Undo scaling */
|
|
|
|
if (iascl == 1) {
|
|
dlascl_("G", &c__0, &c__0, &anrm, &smlnum, &scllen, nrhs, &b[b_offset]
|
|
, ldb, info);
|
|
} else if (iascl == 2) {
|
|
dlascl_("G", &c__0, &c__0, &anrm, &bignum, &scllen, nrhs, &b[b_offset]
|
|
, ldb, info);
|
|
}
|
|
if (ibscl == 1) {
|
|
dlascl_("G", &c__0, &c__0, &smlnum, &bnrm, &scllen, nrhs, &b[b_offset]
|
|
, ldb, info);
|
|
} else if (ibscl == 2) {
|
|
dlascl_("G", &c__0, &c__0, &bignum, &bnrm, &scllen, nrhs, &b[b_offset]
|
|
, ldb, info);
|
|
}
|
|
|
|
L50:
|
|
work[1] = (doublereal) wsize;
|
|
|
|
return 0;
|
|
|
|
/* End of DGELS */
|
|
|
|
} /* dgels_ */
|