#include "blaswrap.h" #include "f2c.h" /* 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) { /* -- LAPACK driver routine (version 3.1) -- Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. November 2006 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. ===================================================================== Test the input arguments. Parameter adjustments */ /* Table of constant values */ static integer c__1 = 1; static integer c_n1 = -1; static doublereal c_b33 = 0.; static integer c__0 = 0; /* System generated locals */ integer a_dim1, a_offset, b_dim1, b_offset, i__1, i__2; /* Local variables */ static integer i__, j, nb, mn; static doublereal anrm, bnrm; static integer brow; static logical tpsd; static integer iascl, ibscl; extern logical lsame_(char *, char *); static integer wsize; static 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 *, ftnlen, ftnlen); static integer scllen; static 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 *); static doublereal smlnum; static logical lquery; extern /* Subroutine */ int dtrtrs_(char *, char *, char *, integer *, integer *, doublereal *, integer *, doublereal *, integer *, integer *); 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, (ftnlen)6, (ftnlen)1); if (tpsd) { /* Computing MAX */ i__1 = nb, i__2 = ilaenv_(&c__1, "DORMQR", "LN", m, nrhs, n, & c_n1, (ftnlen)6, (ftnlen)2); nb = max(i__1,i__2); } else { /* Computing MAX */ i__1 = nb, i__2 = ilaenv_(&c__1, "DORMQR", "LT", m, nrhs, n, & c_n1, (ftnlen)6, (ftnlen)2); nb = max(i__1,i__2); } } else { nb = ilaenv_(&c__1, "DGELQF", " ", m, n, &c_n1, &c_n1, (ftnlen)6, (ftnlen)1); if (tpsd) { /* Computing MAX */ i__1 = nb, i__2 = ilaenv_(&c__1, "DORMLQ", "LT", n, nrhs, m, & c_n1, (ftnlen)6, (ftnlen)2); nb = max(i__1,i__2); } else { /* Computing MAX */ i__1 = nb, i__2 = ilaenv_(&c__1, "DORMLQ", "LN", n, nrhs, m, & c_n1, (ftnlen)6, (ftnlen)2); 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_ */