/* Subroutine */ int dgelsd_(integer *m, integer *n, integer *nrhs, doublereal *a, integer *lda, doublereal *b, integer *ldb, doublereal * s, doublereal *rcond, integer *rank, doublereal *work, integer *lwork, integer *iwork, integer *info) { /* System generated locals */ integer a_dim1, a_offset, b_dim1, b_offset, i__1, i__2, i__3, i__4; /* Builtin functions */ double log(doublereal); /* Local variables */ static doublereal anrm, bnrm; static integer itau, nlvl, iascl, ibscl; static doublereal sfmin; static integer minmn, maxmn, itaup, itauq, mnthr, nwork; extern /* Subroutine */ int dlabad_(doublereal *, doublereal *); static integer ie, il; extern /* Subroutine */ int dgebrd_(integer *, integer *, doublereal *, integer *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, integer *, integer *); extern doublereal dlamch_(char *); static integer mm; extern doublereal dlange_(char *, integer *, integer *, doublereal *, integer *, doublereal *); extern /* Subroutine */ int dgelqf_(integer *, integer *, doublereal *, integer *, doublereal *, doublereal *, integer *, integer *), dlalsd_(char *, integer *, integer *, integer *, doublereal *, doublereal *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *, integer *), dlascl_(char *, integer *, integer *, doublereal *, doublereal *, integer *, integer *, doublereal *, integer *, integer *), dgeqrf_( integer *, integer *, doublereal *, integer *, doublereal *, doublereal *, integer *, integer *), dlacpy_(char *, integer *, integer *, doublereal *, integer *, doublereal *, 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 doublereal bignum; extern /* Subroutine */ int dormbr_(char *, char *, char *, integer *, integer *, integer *, doublereal *, integer *, doublereal *, doublereal *, integer *, doublereal *, integer *, integer *); static integer wlalsd; extern /* Subroutine */ int dormlq_(char *, char *, integer *, integer *, integer *, doublereal *, integer *, doublereal *, doublereal *, integer *, doublereal *, integer *, integer *); static integer ldwork; extern /* Subroutine */ int dormqr_(char *, char *, integer *, integer *, integer *, doublereal *, integer *, doublereal *, doublereal *, integer *, doublereal *, integer *, integer *); static integer minwrk, maxwrk; static doublereal smlnum; static logical lquery; static integer smlsiz; static doublereal eps; #define a_ref(a_1,a_2) a[(a_2)*a_dim1 + a_1] #define b_ref(a_1,a_2) b[(a_2)*b_dim1 + a_1] /* -- LAPACK driver routine (version 3.0) -- Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., Courant Institute, Argonne National Lab, and Rice University October 31, 1999 Purpose ======= DGELSD computes the minimum-norm solution to a real linear least squares problem: minimize 2-norm(| b - A*x |) using the singular value decomposition (SVD) of A. A is an M-by-N matrix which may be rank-deficient. 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. The problem is solved in three steps: (1) Reduce the coefficient matrix A to bidiagonal form with Householder transformations, reducing the original problem into a "bidiagonal least squares problem" (BLS) (2) Solve the BLS using a divide and conquer approach. (3) Apply back all the Householder tranformations to solve the original least squares problem. The effective rank of A is determined by treating as zero those singular values which are less than RCOND times the largest singular value. The divide and conquer algorithm makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none. Arguments ========= M (input) INTEGER The number of rows of A. M >= 0. N (input) INTEGER The number of columns of 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) DOUBLE PRECISION array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit, A has been destroyed. 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 M-by-NRHS right hand side matrix B. On exit, B is overwritten by the N-by-NRHS solution matrix X. If m >= n and RANK = n, the residual sum-of-squares for the solution in the i-th column is given by the sum of squares of elements n+1:m in that column. LDB (input) INTEGER The leading dimension of the array B. LDB >= max(1,max(M,N)). S (output) DOUBLE PRECISION array, dimension (min(M,N)) The singular values of A in decreasing order. The condition number of A in the 2-norm = S(1)/S(min(m,n)). RCOND (input) DOUBLE PRECISION RCOND is used to determine the effective rank of A. Singular values S(i) <= RCOND*S(1) are treated as zero. If RCOND < 0, machine precision is used instead. RANK (output) INTEGER The effective rank of A, i.e., the number of singular values which are greater than RCOND*S(1). WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK (input) INTEGER The dimension of the array WORK. LWORK must be at least 1. The exact minimum amount of workspace needed depends on M, N and NRHS. As long as LWORK is at least 12*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2, if M is greater than or equal to N or 12*M + 2*M*SMLSIZ + 8*M*NLVL + M*NRHS + (SMLSIZ+1)**2, if M is less than N, the code will execute correctly. SMLSIZ is returned by ILAENV and is equal to the maximum size of the subproblems at the bottom of the computation tree (usually about 25), and NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 ) For good performance, LWORK should generally be larger. 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. IWORK (workspace) INTEGER array, dimension (LIWORK) LIWORK >= 3 * MINMN * NLVL + 11 * MINMN, where MINMN = MIN( M,N ). INFO (output) INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value. > 0: the algorithm for computing the SVD failed to converge; if INFO = i, i off-diagonal elements of an intermediate bidiagonal form did not converge to zero. Further Details =============== Based on contributions by Ming Gu and Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA Osni Marques, LBNL/NERSC, USA ===================================================================== Test the input arguments. Parameter adjustments */ a_dim1 = *lda; a_offset = 1 + a_dim1 * 1; a -= a_offset; b_dim1 = *ldb; b_offset = 1 + b_dim1 * 1; b -= b_offset; --s; --work; --iwork; /* Function Body */ *info = 0; minmn = min(*m,*n); maxmn = max(*m,*n); mnthr = ilaenv_(&c__6, "DGELSD", " ", m, n, nrhs, &c_n1, (ftnlen)6, ( ftnlen)1); lquery = *lwork == -1; if (*m < 0) { *info = -1; } else if (*n < 0) { *info = -2; } else if (*nrhs < 0) { *info = -3; } else if (*lda < max(1,*m)) { *info = -5; } else if (*ldb < max(1,maxmn)) { *info = -7; } smlsiz = ilaenv_(&c__9, "DGELSD", " ", &c__0, &c__0, &c__0, &c__0, ( ftnlen)6, (ftnlen)1); /* Compute workspace. (Note: Comments in the code beginning "Workspace:" describe the minimal amount of workspace needed at that point in the code, as well as the preferred amount for good performance. NB refers to the optimal block size for the immediately following subroutine, as returned by ILAENV.) */ minwrk = 1; minmn = max(1,minmn); /* Computing MAX */ i__1 = (integer) (log((doublereal) minmn / (doublereal) (smlsiz + 1)) / log(2.)) + 1; nlvl = max(i__1,0); if (*info == 0) { maxwrk = 0; mm = *m; if (*m >= *n && *m >= mnthr) { /* Path 1a - overdetermined, with many more rows than columns. */ mm = *n; /* Computing MAX */ i__1 = maxwrk, i__2 = *n + *n * ilaenv_(&c__1, "DGEQRF", " ", m, n, &c_n1, &c_n1, (ftnlen)6, (ftnlen)1); maxwrk = max(i__1,i__2); /* Computing MAX */ i__1 = maxwrk, i__2 = *n + *nrhs * ilaenv_(&c__1, "DORMQR", "LT", m, nrhs, n, &c_n1, (ftnlen)6, (ftnlen)2); maxwrk = max(i__1,i__2); } if (*m >= *n) { /* Path 1 - overdetermined or exactly determined. Computing MAX */ i__1 = maxwrk, i__2 = *n * 3 + (mm + *n) * ilaenv_(&c__1, "DGEBRD" , " ", &mm, n, &c_n1, &c_n1, (ftnlen)6, (ftnlen)1); maxwrk = max(i__1,i__2); /* Computing MAX */ i__1 = maxwrk, i__2 = *n * 3 + *nrhs * ilaenv_(&c__1, "DORMBR", "QLT", &mm, nrhs, n, &c_n1, (ftnlen)6, (ftnlen)3); maxwrk = max(i__1,i__2); /* Computing MAX */ i__1 = maxwrk, i__2 = *n * 3 + (*n - 1) * ilaenv_(&c__1, "DORMBR", "PLN", n, nrhs, n, &c_n1, (ftnlen)6, (ftnlen)3); maxwrk = max(i__1,i__2); /* Computing 2nd power */ i__1 = smlsiz + 1; wlalsd = *n * 9 + (*n << 1) * smlsiz + (*n << 3) * nlvl + *n * * nrhs + i__1 * i__1; /* Computing MAX */ i__1 = maxwrk, i__2 = *n * 3 + wlalsd; maxwrk = max(i__1,i__2); /* Computing MAX */ i__1 = *n * 3 + mm, i__2 = *n * 3 + *nrhs, i__1 = max(i__1,i__2), i__2 = *n * 3 + wlalsd; minwrk = max(i__1,i__2); } if (*n > *m) { /* Computing 2nd power */ i__1 = smlsiz + 1; wlalsd = *m * 9 + (*m << 1) * smlsiz + (*m << 3) * nlvl + *m * * nrhs + i__1 * i__1; if (*n >= mnthr) { /* Path 2a - underdetermined, with many more columns than rows. */ maxwrk = *m + *m * ilaenv_(&c__1, "DGELQF", " ", m, n, &c_n1, &c_n1, (ftnlen)6, (ftnlen)1); /* Computing MAX */ i__1 = maxwrk, i__2 = *m * *m + (*m << 2) + (*m << 1) * ilaenv_(&c__1, "DGEBRD", " ", m, m, &c_n1, &c_n1, ( ftnlen)6, (ftnlen)1); maxwrk = max(i__1,i__2); /* Computing MAX */ i__1 = maxwrk, i__2 = *m * *m + (*m << 2) + *nrhs * ilaenv_(& c__1, "DORMBR", "QLT", m, nrhs, m, &c_n1, (ftnlen)6, ( ftnlen)3); maxwrk = max(i__1,i__2); /* Computing MAX */ i__1 = maxwrk, i__2 = *m * *m + (*m << 2) + (*m - 1) * ilaenv_(&c__1, "DORMBR", "PLN", m, nrhs, m, &c_n1, ( ftnlen)6, (ftnlen)3); maxwrk = max(i__1,i__2); if (*nrhs > 1) { /* Computing MAX */ i__1 = maxwrk, i__2 = *m * *m + *m + *m * *nrhs; maxwrk = max(i__1,i__2); } else { /* Computing MAX */ i__1 = maxwrk, i__2 = *m * *m + (*m << 1); maxwrk = max(i__1,i__2); } /* Computing MAX */ i__1 = maxwrk, i__2 = *m + *nrhs * ilaenv_(&c__1, "DORMLQ", "LT", n, nrhs, m, &c_n1, (ftnlen)6, (ftnlen)2); maxwrk = max(i__1,i__2); /* Computing MAX */ i__1 = maxwrk, i__2 = *m * *m + (*m << 2) + wlalsd; maxwrk = max(i__1,i__2); } else { /* Path 2 - remaining underdetermined cases. */ maxwrk = *m * 3 + (*n + *m) * ilaenv_(&c__1, "DGEBRD", " ", m, n, &c_n1, &c_n1, (ftnlen)6, (ftnlen)1); /* Computing MAX */ i__1 = maxwrk, i__2 = *m * 3 + *nrhs * ilaenv_(&c__1, "DORMBR" , "QLT", m, nrhs, n, &c_n1, (ftnlen)6, (ftnlen)3); maxwrk = max(i__1,i__2); /* Computing MAX */ i__1 = maxwrk, i__2 = *m * 3 + *m * ilaenv_(&c__1, "DORMBR", "PLN", n, nrhs, m, &c_n1, (ftnlen)6, (ftnlen)3); maxwrk = max(i__1,i__2); /* Computing MAX */ i__1 = maxwrk, i__2 = *m * 3 + wlalsd; maxwrk = max(i__1,i__2); } /* Computing MAX */ i__1 = *m * 3 + *nrhs, i__2 = *m * 3 + *m, i__1 = max(i__1,i__2), i__2 = *m * 3 + wlalsd; minwrk = max(i__1,i__2); } minwrk = min(minwrk,maxwrk); work[1] = (doublereal) maxwrk; if (*lwork < minwrk && ! lquery) { *info = -12; } } if (*info != 0) { i__1 = -(*info); xerbla_("DGELSD", &i__1); return 0; } else if (lquery) { goto L10; } /* Quick return if possible. */ if (*m == 0 || *n == 0) { *rank = 0; return 0; } /* Get machine parameters. */ eps = dlamch_("P"); sfmin = dlamch_("S"); smlnum = sfmin / eps; bignum = 1. / smlnum; dlabad_(&smlnum, &bignum); /* Scale A if max entry outside range [SMLNUM,BIGNUM]. */ anrm = dlange_("M", m, n, &a[a_offset], lda, &work[1]); 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_b82, &c_b82, &b[b_offset], ldb); dlaset_("F", &minmn, &c__1, &c_b82, &c_b82, &s[1], &c__1); *rank = 0; goto L10; } /* Scale B if max entry outside range [SMLNUM,BIGNUM]. */ bnrm = dlange_("M", m, nrhs, &b[b_offset], ldb, &work[1]); ibscl = 0; if (bnrm > 0. && bnrm < smlnum) { /* Scale matrix norm up to SMLNUM. */ dlascl_("G", &c__0, &c__0, &bnrm, &smlnum, m, 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, m, nrhs, &b[b_offset], ldb, info); 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_ref(*m + 1, 1), 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_ref(2, 1), 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; if (*n >= mnthr && *lwork >= (*m << 2) + *m * *m + max(i__1,i__2)) { /* 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; 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_ref(*m + 1, 1), 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_ */
/* Subroutine */ int derrbd_(char *path, integer *nunit) { /* Format strings */ static char fmt_9999[] = "(1x,a3,\002 routines passed the tests of the e" "rror exits\002,\002 (\002,i3,\002 tests done)\002)"; static char fmt_9998[] = "(\002 *** \002,a3,\002 routines failed the tes" "ts of the error \002,\002exits ***\002)"; /* Builtin functions */ integer s_wsle(cilist *), e_wsle(void); /* Subroutine */ int s_copy(char *, char *, ftnlen, ftnlen); integer s_wsfe(cilist *), do_fio(integer *, char *, ftnlen), e_wsfe(void); /* Local variables */ doublereal a[16] /* was [4][4] */, d__[4], e[4]; integer i__, j; doublereal q[16] /* was [4][4] */, u[16] /* was [4][4] */, v[16] /* was [4][4] */, w[4]; char c2[2]; integer iq[16] /* was [4][4] */, iw[4], nt; doublereal tp[4], tq[4]; integer info; extern /* Subroutine */ int dgebd2_(integer *, integer *, doublereal *, integer *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, integer *), dbdsdc_(char *, char *, integer *, doublereal *, doublereal *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *, integer *), dgebrd_(integer *, integer *, doublereal *, integer *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, integer *, integer *); extern logical lsamen_(integer *, char *, char *); extern /* Subroutine */ int dbdsqr_(char *, integer *, integer *, integer *, integer *, doublereal *, doublereal *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *), dorgbr_(char *, integer *, integer *, integer *, doublereal *, integer *, doublereal *, doublereal *, integer *, integer *), chkxer_(char *, integer *, integer *, logical *, logical *), dormbr_(char *, char *, char *, integer *, integer *, integer *, doublereal *, integer *, doublereal *, doublereal *, integer *, doublereal *, integer *, integer *); /* Fortran I/O blocks */ static cilist io___1 = { 0, 0, 0, 0, 0 }; static cilist io___18 = { 0, 0, 0, fmt_9999, 0 }; static cilist io___19 = { 0, 0, 0, fmt_9998, 0 }; /* -- LAPACK test routine (version 3.1) -- */ /* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */ /* November 2006 */ /* .. Scalar Arguments .. */ /* .. */ /* Purpose */ /* ======= */ /* DERRBD tests the error exits for DGEBRD, DORGBR, DORMBR, DBDSQR and */ /* DBDSDC. */ /* Arguments */ /* ========= */ /* PATH (input) CHARACTER*3 */ /* The LAPACK path name for the routines to be tested. */ /* NUNIT (input) INTEGER */ /* The unit number for output. */ /* ===================================================================== */ /* .. Parameters .. */ /* .. */ /* .. Local Scalars .. */ /* .. */ /* .. Local Arrays .. */ /* .. */ /* .. External Functions .. */ /* .. */ /* .. External Subroutines .. */ /* .. */ /* .. Scalars in Common .. */ /* .. */ /* .. Common blocks .. */ /* .. */ /* .. Intrinsic Functions .. */ /* .. */ /* .. Executable Statements .. */ infoc_1.nout = *nunit; io___1.ciunit = infoc_1.nout; s_wsle(&io___1); e_wsle(); s_copy(c2, path + 1, (ftnlen)2, (ftnlen)2); /* Set the variables to innocuous values. */ for (j = 1; j <= 4; ++j) { for (i__ = 1; i__ <= 4; ++i__) { a[i__ + (j << 2) - 5] = 1. / (doublereal) (i__ + j); /* L10: */ } /* L20: */ } infoc_1.ok = TRUE_; nt = 0; /* Test error exits of the SVD routines. */ if (lsamen_(&c__2, c2, "BD")) { /* DGEBRD */ s_copy(srnamc_1.srnamt, "DGEBRD", (ftnlen)6, (ftnlen)6); infoc_1.infot = 1; dgebrd_(&c_n1, &c__0, a, &c__1, d__, e, tq, tp, w, &c__1, &info); chkxer_("DGEBRD", &infoc_1.infot, &infoc_1.nout, &infoc_1.lerr, & infoc_1.ok); infoc_1.infot = 2; dgebrd_(&c__0, &c_n1, a, &c__1, d__, e, tq, tp, w, &c__1, &info); chkxer_("DGEBRD", &infoc_1.infot, &infoc_1.nout, &infoc_1.lerr, & infoc_1.ok); infoc_1.infot = 4; dgebrd_(&c__2, &c__1, a, &c__1, d__, e, tq, tp, w, &c__2, &info); chkxer_("DGEBRD", &infoc_1.infot, &infoc_1.nout, &infoc_1.lerr, & infoc_1.ok); infoc_1.infot = 10; dgebrd_(&c__2, &c__1, a, &c__2, d__, e, tq, tp, w, &c__1, &info); chkxer_("DGEBRD", &infoc_1.infot, &infoc_1.nout, &infoc_1.lerr, & infoc_1.ok); nt += 4; /* DGEBD2 */ s_copy(srnamc_1.srnamt, "DGEBD2", (ftnlen)6, (ftnlen)6); infoc_1.infot = 1; dgebd2_(&c_n1, &c__0, a, &c__1, d__, e, tq, tp, w, &info); chkxer_("DGEBD2", &infoc_1.infot, &infoc_1.nout, &infoc_1.lerr, & infoc_1.ok); infoc_1.infot = 2; dgebd2_(&c__0, &c_n1, a, &c__1, d__, e, tq, tp, w, &info); chkxer_("DGEBD2", &infoc_1.infot, &infoc_1.nout, &infoc_1.lerr, & infoc_1.ok); infoc_1.infot = 4; dgebd2_(&c__2, &c__1, a, &c__1, d__, e, tq, tp, w, &info); chkxer_("DGEBD2", &infoc_1.infot, &infoc_1.nout, &infoc_1.lerr, & infoc_1.ok); nt += 3; /* DORGBR */ s_copy(srnamc_1.srnamt, "DORGBR", (ftnlen)6, (ftnlen)6); infoc_1.infot = 1; dorgbr_("/", &c__0, &c__0, &c__0, a, &c__1, tq, w, &c__1, &info); chkxer_("DORGBR", &infoc_1.infot, &infoc_1.nout, &infoc_1.lerr, & infoc_1.ok); infoc_1.infot = 2; dorgbr_("Q", &c_n1, &c__0, &c__0, a, &c__1, tq, w, &c__1, &info); chkxer_("DORGBR", &infoc_1.infot, &infoc_1.nout, &infoc_1.lerr, & infoc_1.ok); infoc_1.infot = 3; dorgbr_("Q", &c__0, &c_n1, &c__0, a, &c__1, tq, w, &c__1, &info); chkxer_("DORGBR", &infoc_1.infot, &infoc_1.nout, &infoc_1.lerr, & infoc_1.ok); infoc_1.infot = 3; dorgbr_("Q", &c__0, &c__1, &c__0, a, &c__1, tq, w, &c__1, &info); chkxer_("DORGBR", &infoc_1.infot, &infoc_1.nout, &infoc_1.lerr, & infoc_1.ok); infoc_1.infot = 3; dorgbr_("Q", &c__1, &c__0, &c__1, a, &c__1, tq, w, &c__1, &info); chkxer_("DORGBR", &infoc_1.infot, &infoc_1.nout, &infoc_1.lerr, & infoc_1.ok); infoc_1.infot = 3; dorgbr_("P", &c__1, &c__0, &c__0, a, &c__1, tq, w, &c__1, &info); chkxer_("DORGBR", &infoc_1.infot, &infoc_1.nout, &infoc_1.lerr, & infoc_1.ok); infoc_1.infot = 3; dorgbr_("P", &c__0, &c__1, &c__1, a, &c__1, tq, w, &c__1, &info); chkxer_("DORGBR", &infoc_1.infot, &infoc_1.nout, &infoc_1.lerr, & infoc_1.ok); infoc_1.infot = 4; dorgbr_("Q", &c__0, &c__0, &c_n1, a, &c__1, tq, w, &c__1, &info); chkxer_("DORGBR", &infoc_1.infot, &infoc_1.nout, &infoc_1.lerr, & infoc_1.ok); infoc_1.infot = 6; dorgbr_("Q", &c__2, &c__1, &c__1, a, &c__1, tq, w, &c__1, &info); chkxer_("DORGBR", &infoc_1.infot, &infoc_1.nout, &infoc_1.lerr, & infoc_1.ok); infoc_1.infot = 9; dorgbr_("Q", &c__2, &c__2, &c__1, a, &c__2, tq, w, &c__1, &info); chkxer_("DORGBR", &infoc_1.infot, &infoc_1.nout, &infoc_1.lerr, & infoc_1.ok); nt += 10; /* DORMBR */ s_copy(srnamc_1.srnamt, "DORMBR", (ftnlen)6, (ftnlen)6); infoc_1.infot = 1; dormbr_("/", "L", "T", &c__0, &c__0, &c__0, a, &c__1, tq, u, &c__1, w, &c__1, &info); chkxer_("DORMBR", &infoc_1.infot, &infoc_1.nout, &infoc_1.lerr, & infoc_1.ok); infoc_1.infot = 2; dormbr_("Q", "/", "T", &c__0, &c__0, &c__0, a, &c__1, tq, u, &c__1, w, &c__1, &info); chkxer_("DORMBR", &infoc_1.infot, &infoc_1.nout, &infoc_1.lerr, & infoc_1.ok); infoc_1.infot = 3; dormbr_("Q", "L", "/", &c__0, &c__0, &c__0, a, &c__1, tq, u, &c__1, w, &c__1, &info); chkxer_("DORMBR", &infoc_1.infot, &infoc_1.nout, &infoc_1.lerr, & infoc_1.ok); infoc_1.infot = 4; dormbr_("Q", "L", "T", &c_n1, &c__0, &c__0, a, &c__1, tq, u, &c__1, w, &c__1, &info); chkxer_("DORMBR", &infoc_1.infot, &infoc_1.nout, &infoc_1.lerr, & infoc_1.ok); infoc_1.infot = 5; dormbr_("Q", "L", "T", &c__0, &c_n1, &c__0, a, &c__1, tq, u, &c__1, w, &c__1, &info); chkxer_("DORMBR", &infoc_1.infot, &infoc_1.nout, &infoc_1.lerr, & infoc_1.ok); infoc_1.infot = 6; dormbr_("Q", "L", "T", &c__0, &c__0, &c_n1, a, &c__1, tq, u, &c__1, w, &c__1, &info); chkxer_("DORMBR", &infoc_1.infot, &infoc_1.nout, &infoc_1.lerr, & infoc_1.ok); infoc_1.infot = 8; dormbr_("Q", "L", "T", &c__2, &c__0, &c__0, a, &c__1, tq, u, &c__2, w, &c__1, &info); chkxer_("DORMBR", &infoc_1.infot, &infoc_1.nout, &infoc_1.lerr, & infoc_1.ok); infoc_1.infot = 8; dormbr_("Q", "R", "T", &c__0, &c__2, &c__0, a, &c__1, tq, u, &c__1, w, &c__1, &info); chkxer_("DORMBR", &infoc_1.infot, &infoc_1.nout, &infoc_1.lerr, & infoc_1.ok); infoc_1.infot = 8; dormbr_("P", "L", "T", &c__2, &c__0, &c__2, a, &c__1, tq, u, &c__2, w, &c__1, &info); chkxer_("DORMBR", &infoc_1.infot, &infoc_1.nout, &infoc_1.lerr, & infoc_1.ok); infoc_1.infot = 8; dormbr_("P", "R", "T", &c__0, &c__2, &c__2, a, &c__1, tq, u, &c__1, w, &c__1, &info); chkxer_("DORMBR", &infoc_1.infot, &infoc_1.nout, &infoc_1.lerr, & infoc_1.ok); infoc_1.infot = 11; dormbr_("Q", "R", "T", &c__2, &c__0, &c__0, a, &c__1, tq, u, &c__1, w, &c__1, &info); chkxer_("DORMBR", &infoc_1.infot, &infoc_1.nout, &infoc_1.lerr, & infoc_1.ok); infoc_1.infot = 13; dormbr_("Q", "L", "T", &c__0, &c__2, &c__0, a, &c__1, tq, u, &c__1, w, &c__1, &info); chkxer_("DORMBR", &infoc_1.infot, &infoc_1.nout, &infoc_1.lerr, & infoc_1.ok); infoc_1.infot = 13; dormbr_("Q", "R", "T", &c__2, &c__0, &c__0, a, &c__1, tq, u, &c__2, w, &c__1, &info); chkxer_("DORMBR", &infoc_1.infot, &infoc_1.nout, &infoc_1.lerr, & infoc_1.ok); nt += 13; /* DBDSQR */ s_copy(srnamc_1.srnamt, "DBDSQR", (ftnlen)6, (ftnlen)6); infoc_1.infot = 1; dbdsqr_("/", &c__0, &c__0, &c__0, &c__0, d__, e, v, &c__1, u, &c__1, a, &c__1, w, &info); chkxer_("DBDSQR", &infoc_1.infot, &infoc_1.nout, &infoc_1.lerr, & infoc_1.ok); infoc_1.infot = 2; dbdsqr_("U", &c_n1, &c__0, &c__0, &c__0, d__, e, v, &c__1, u, &c__1, a, &c__1, w, &info); chkxer_("DBDSQR", &infoc_1.infot, &infoc_1.nout, &infoc_1.lerr, & infoc_1.ok); infoc_1.infot = 3; dbdsqr_("U", &c__0, &c_n1, &c__0, &c__0, d__, e, v, &c__1, u, &c__1, a, &c__1, w, &info); chkxer_("DBDSQR", &infoc_1.infot, &infoc_1.nout, &infoc_1.lerr, & infoc_1.ok); infoc_1.infot = 4; dbdsqr_("U", &c__0, &c__0, &c_n1, &c__0, d__, e, v, &c__1, u, &c__1, a, &c__1, w, &info); chkxer_("DBDSQR", &infoc_1.infot, &infoc_1.nout, &infoc_1.lerr, & infoc_1.ok); infoc_1.infot = 5; dbdsqr_("U", &c__0, &c__0, &c__0, &c_n1, d__, e, v, &c__1, u, &c__1, a, &c__1, w, &info); chkxer_("DBDSQR", &infoc_1.infot, &infoc_1.nout, &infoc_1.lerr, & infoc_1.ok); infoc_1.infot = 9; dbdsqr_("U", &c__2, &c__1, &c__0, &c__0, d__, e, v, &c__1, u, &c__1, a, &c__1, w, &info); chkxer_("DBDSQR", &infoc_1.infot, &infoc_1.nout, &infoc_1.lerr, & infoc_1.ok); infoc_1.infot = 11; dbdsqr_("U", &c__0, &c__0, &c__2, &c__0, d__, e, v, &c__1, u, &c__1, a, &c__1, w, &info); chkxer_("DBDSQR", &infoc_1.infot, &infoc_1.nout, &infoc_1.lerr, & infoc_1.ok); infoc_1.infot = 13; dbdsqr_("U", &c__2, &c__0, &c__0, &c__1, d__, e, v, &c__1, u, &c__1, a, &c__1, w, &info); chkxer_("DBDSQR", &infoc_1.infot, &infoc_1.nout, &infoc_1.lerr, & infoc_1.ok); nt += 8; /* DBDSDC */ s_copy(srnamc_1.srnamt, "DBDSDC", (ftnlen)6, (ftnlen)6); infoc_1.infot = 1; dbdsdc_("/", "N", &c__0, d__, e, u, &c__1, v, &c__1, q, iq, w, iw, & info); chkxer_("DBDSDC", &infoc_1.infot, &infoc_1.nout, &infoc_1.lerr, & infoc_1.ok); infoc_1.infot = 2; dbdsdc_("U", "/", &c__0, d__, e, u, &c__1, v, &c__1, q, iq, w, iw, & info); chkxer_("DBDSDC", &infoc_1.infot, &infoc_1.nout, &infoc_1.lerr, & infoc_1.ok); infoc_1.infot = 3; dbdsdc_("U", "N", &c_n1, d__, e, u, &c__1, v, &c__1, q, iq, w, iw, & info); chkxer_("DBDSDC", &infoc_1.infot, &infoc_1.nout, &infoc_1.lerr, & infoc_1.ok); infoc_1.infot = 7; dbdsdc_("U", "I", &c__2, d__, e, u, &c__1, v, &c__1, q, iq, w, iw, & info); chkxer_("DBDSDC", &infoc_1.infot, &infoc_1.nout, &infoc_1.lerr, & infoc_1.ok); infoc_1.infot = 9; dbdsdc_("U", "I", &c__2, d__, e, u, &c__2, v, &c__1, q, iq, w, iw, & info); chkxer_("DBDSDC", &infoc_1.infot, &infoc_1.nout, &infoc_1.lerr, & infoc_1.ok); nt += 5; } /* Print a summary line. */ if (infoc_1.ok) { io___18.ciunit = infoc_1.nout; s_wsfe(&io___18); do_fio(&c__1, path, (ftnlen)3); do_fio(&c__1, (char *)&nt, (ftnlen)sizeof(integer)); e_wsfe(); } else { io___19.ciunit = infoc_1.nout; s_wsfe(&io___19); do_fio(&c__1, path, (ftnlen)3); e_wsfe(); } return 0; /* End of DERRBD */ } /* derrbd_ */
/* Subroutine */ int dgelss_(integer *m, integer *n, integer *nrhs, doublereal *a, integer *lda, doublereal *b, integer *ldb, doublereal * s, doublereal *rcond, integer *rank, doublereal *work, integer *lwork, integer *info) { /* System generated locals */ integer a_dim1, a_offset, b_dim1, b_offset, i__1, i__2, i__3, i__4; doublereal d__1; /* Local variables */ static doublereal anrm, bnrm; static integer itau; static doublereal vdum[1]; static integer i__; extern /* Subroutine */ int dgemm_(char *, char *, integer *, integer *, integer *, doublereal *, doublereal *, integer *, doublereal *, integer *, doublereal *, doublereal *, integer *); static integer iascl, ibscl; extern /* Subroutine */ int dgemv_(char *, integer *, integer *, doublereal *, doublereal *, integer *, doublereal *, integer *, doublereal *, doublereal *, integer *), drscl_(integer *, doublereal *, doublereal *, integer *); static integer chunk; static doublereal sfmin; static integer minmn; extern /* Subroutine */ int dcopy_(integer *, doublereal *, integer *, doublereal *, integer *); static integer maxmn, itaup, itauq, mnthr, iwork; extern /* Subroutine */ int dlabad_(doublereal *, doublereal *); static integer bl, ie, il; extern /* Subroutine */ int dgebrd_(integer *, integer *, doublereal *, integer *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, integer *, integer *); extern doublereal dlamch_(char *); static integer mm; extern doublereal dlange_(char *, integer *, integer *, doublereal *, integer *, doublereal *); static integer bdspac; 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 *), dlacpy_(char *, integer *, integer *, doublereal *, integer *, doublereal *, integer *), dlaset_(char *, integer *, integer *, doublereal *, doublereal *, doublereal *, integer *), xerbla_(char *, integer *), dbdsqr_(char *, integer *, integer *, integer *, integer *, doublereal *, doublereal *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *), dorgbr_(char *, integer *, integer *, integer *, doublereal *, integer *, doublereal *, doublereal *, integer *, integer *); static doublereal bignum; extern integer ilaenv_(integer *, char *, char *, integer *, integer *, integer *, integer *, ftnlen, ftnlen); extern /* Subroutine */ int dormbr_(char *, char *, char *, integer *, integer *, integer *, doublereal *, integer *, doublereal *, doublereal *, integer *, doublereal *, integer *, integer *), dormlq_(char *, char *, integer *, integer *, integer *, doublereal *, integer *, doublereal *, doublereal *, integer *, doublereal *, integer *, integer *); static integer ldwork; extern /* Subroutine */ int dormqr_(char *, char *, integer *, integer *, integer *, doublereal *, integer *, doublereal *, doublereal *, integer *, doublereal *, integer *, integer *); static integer minwrk, maxwrk; static doublereal smlnum; static logical lquery; static doublereal eps, thr; #define a_ref(a_1,a_2) a[(a_2)*a_dim1 + a_1] #define b_ref(a_1,a_2) b[(a_2)*b_dim1 + a_1] /* -- LAPACK driver routine (version 3.0) -- Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., Courant Institute, Argonne National Lab, and Rice University October 31, 1999 Purpose ======= DGELSS computes the minimum norm solution to a real linear least squares problem: Minimize 2-norm(| b - A*x |). using the singular value decomposition (SVD) of A. A is an M-by-N matrix which may be rank-deficient. 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. The effective rank of A is determined by treating as zero those singular values which are less than RCOND times the largest singular value. Arguments ========= 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, the first min(m,n) rows of A are overwritten with its right singular vectors, stored rowwise. 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 M-by-NRHS right hand side matrix B. On exit, B is overwritten by the N-by-NRHS solution matrix X. If m >= n and RANK = n, the residual sum-of-squares for the solution in the i-th column is given by the sum of squares of elements n+1:m in that column. LDB (input) INTEGER The leading dimension of the array B. LDB >= max(1,max(M,N)). S (output) DOUBLE PRECISION array, dimension (min(M,N)) The singular values of A in decreasing order. The condition number of A in the 2-norm = S(1)/S(min(m,n)). RCOND (input) DOUBLE PRECISION RCOND is used to determine the effective rank of A. Singular values S(i) <= RCOND*S(1) are treated as zero. If RCOND < 0, machine precision is used instead. RANK (output) INTEGER The effective rank of A, i.e., the number of singular values which are greater than RCOND*S(1). WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK (input) INTEGER The dimension of the array WORK. LWORK >= 1, and also: LWORK >= 3*min(M,N) + max( 2*min(M,N), max(M,N), NRHS ) For good performance, LWORK should generally be larger. 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: the algorithm for computing the SVD failed to converge; if INFO = i, i off-diagonal elements of an intermediate bidiagonal form did not converge to zero. ===================================================================== Test the input arguments Parameter adjustments */ a_dim1 = *lda; a_offset = 1 + a_dim1 * 1; a -= a_offset; b_dim1 = *ldb; b_offset = 1 + b_dim1 * 1; b -= b_offset; --s; --work; /* Function Body */ *info = 0; minmn = min(*m,*n); maxmn = max(*m,*n); mnthr = ilaenv_(&c__6, "DGELSS", " ", m, n, nrhs, &c_n1, (ftnlen)6, ( ftnlen)1); lquery = *lwork == -1; if (*m < 0) { *info = -1; } else if (*n < 0) { *info = -2; } else if (*nrhs < 0) { *info = -3; } else if (*lda < max(1,*m)) { *info = -5; } else if (*ldb < max(1,maxmn)) { *info = -7; } /* Compute workspace (Note: Comments in the code beginning "Workspace:" describe the minimal amount of workspace needed at that point in the code, as well as the preferred amount for good performance. NB refers to the optimal block size for the immediately following subroutine, as returned by ILAENV.) */ minwrk = 1; if (*info == 0 && (*lwork >= 1 || lquery)) { maxwrk = 0; mm = *m; if (*m >= *n && *m >= mnthr) { /* Path 1a - overdetermined, with many more rows than columns */ mm = *n; /* Computing MAX */ i__1 = maxwrk, i__2 = *n + *n * ilaenv_(&c__1, "DGEQRF", " ", m, n, &c_n1, &c_n1, (ftnlen)6, (ftnlen)1); maxwrk = max(i__1,i__2); /* Computing MAX */ i__1 = maxwrk, i__2 = *n + *nrhs * ilaenv_(&c__1, "DORMQR", "LT", m, nrhs, n, &c_n1, (ftnlen)6, (ftnlen)2); maxwrk = max(i__1,i__2); } if (*m >= *n) { /* Path 1 - overdetermined or exactly determined Compute workspace needed for DBDSQR Computing MAX */ i__1 = 1, i__2 = *n * 5; bdspac = max(i__1,i__2); /* Computing MAX */ i__1 = maxwrk, i__2 = *n * 3 + (mm + *n) * ilaenv_(&c__1, "DGEBRD" , " ", &mm, n, &c_n1, &c_n1, (ftnlen)6, (ftnlen)1); maxwrk = max(i__1,i__2); /* Computing MAX */ i__1 = maxwrk, i__2 = *n * 3 + *nrhs * ilaenv_(&c__1, "DORMBR", "QLT", &mm, nrhs, n, &c_n1, (ftnlen)6, (ftnlen)3); maxwrk = max(i__1,i__2); /* Computing MAX */ i__1 = maxwrk, i__2 = *n * 3 + (*n - 1) * ilaenv_(&c__1, "DORGBR", "P", n, n, n, &c_n1, (ftnlen)6, (ftnlen)1); maxwrk = max(i__1,i__2); maxwrk = max(maxwrk,bdspac); /* Computing MAX */ i__1 = maxwrk, i__2 = *n * *nrhs; maxwrk = max(i__1,i__2); /* Computing MAX */ i__1 = *n * 3 + mm, i__2 = *n * 3 + *nrhs, i__1 = max(i__1,i__2); minwrk = max(i__1,bdspac); maxwrk = max(minwrk,maxwrk); } if (*n > *m) { /* Compute workspace needed for DBDSQR Computing MAX */ i__1 = 1, i__2 = *m * 5; bdspac = max(i__1,i__2); /* Computing MAX */ i__1 = *m * 3 + *nrhs, i__2 = *m * 3 + *n, i__1 = max(i__1,i__2); minwrk = max(i__1,bdspac); if (*n >= mnthr) { /* Path 2a - underdetermined, with many more columns than rows */ maxwrk = *m + *m * ilaenv_(&c__1, "DGELQF", " ", m, n, &c_n1, &c_n1, (ftnlen)6, (ftnlen)1); /* Computing MAX */ i__1 = maxwrk, i__2 = *m * *m + (*m << 2) + (*m << 1) * ilaenv_(&c__1, "DGEBRD", " ", m, m, &c_n1, &c_n1, ( ftnlen)6, (ftnlen)1); maxwrk = max(i__1,i__2); /* Computing MAX */ i__1 = maxwrk, i__2 = *m * *m + (*m << 2) + *nrhs * ilaenv_(& c__1, "DORMBR", "QLT", m, nrhs, m, &c_n1, (ftnlen)6, ( ftnlen)3); maxwrk = max(i__1,i__2); /* Computing MAX */ i__1 = maxwrk, i__2 = *m * *m + (*m << 2) + (*m - 1) * ilaenv_(&c__1, "DORGBR", "P", m, m, m, &c_n1, (ftnlen) 6, (ftnlen)1); maxwrk = max(i__1,i__2); /* Computing MAX */ i__1 = maxwrk, i__2 = *m * *m + *m + bdspac; maxwrk = max(i__1,i__2); if (*nrhs > 1) { /* Computing MAX */ i__1 = maxwrk, i__2 = *m * *m + *m + *m * *nrhs; maxwrk = max(i__1,i__2); } else { /* Computing MAX */ i__1 = maxwrk, i__2 = *m * *m + (*m << 1); maxwrk = max(i__1,i__2); } /* Computing MAX */ i__1 = maxwrk, i__2 = *m + *nrhs * ilaenv_(&c__1, "DORMLQ", "LT", n, nrhs, m, &c_n1, (ftnlen)6, (ftnlen)2); maxwrk = max(i__1,i__2); } else { /* Path 2 - underdetermined */ maxwrk = *m * 3 + (*n + *m) * ilaenv_(&c__1, "DGEBRD", " ", m, n, &c_n1, &c_n1, (ftnlen)6, (ftnlen)1); /* Computing MAX */ i__1 = maxwrk, i__2 = *m * 3 + *nrhs * ilaenv_(&c__1, "DORMBR" , "QLT", m, nrhs, m, &c_n1, (ftnlen)6, (ftnlen)3); maxwrk = max(i__1,i__2); /* Computing MAX */ i__1 = maxwrk, i__2 = *m * 3 + *m * ilaenv_(&c__1, "DORGBR", "P", m, n, m, &c_n1, (ftnlen)6, (ftnlen)1); maxwrk = max(i__1,i__2); maxwrk = max(maxwrk,bdspac); /* Computing MAX */ i__1 = maxwrk, i__2 = *n * *nrhs; maxwrk = max(i__1,i__2); } } maxwrk = max(minwrk,maxwrk); work[1] = (doublereal) maxwrk; } minwrk = max(minwrk,1); if (*lwork < minwrk && ! lquery) { *info = -12; } if (*info != 0) { i__1 = -(*info); xerbla_("DGELSS", &i__1); return 0; } else if (lquery) { return 0; } /* Quick return if possible */ if (*m == 0 || *n == 0) { *rank = 0; return 0; } /* Get machine parameters */ eps = dlamch_("P"); sfmin = dlamch_("S"); smlnum = sfmin / eps; bignum = 1. / smlnum; dlabad_(&smlnum, &bignum); /* Scale A if max element outside range [SMLNUM,BIGNUM] */ anrm = dlange_("M", m, n, &a[a_offset], lda, &work[1]); 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_b74, &c_b74, &b[b_offset], ldb); dlaset_("F", &minmn, &c__1, &c_b74, &c_b74, &s[1], &c__1); *rank = 0; goto L70; } /* Scale B if max element outside range [SMLNUM,BIGNUM] */ bnrm = dlange_("M", m, nrhs, &b[b_offset], ldb, &work[1]); ibscl = 0; if (bnrm > 0. && bnrm < smlnum) { /* Scale matrix norm up to SMLNUM */ dlascl_("G", &c__0, &c__0, &bnrm, &smlnum, m, 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, m, nrhs, &b[b_offset], ldb, info); ibscl = 2; } /* 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; iwork = itau + *n; /* Compute A=Q*R (Workspace: need 2*N, prefer N+N*NB) */ i__1 = *lwork - iwork + 1; dgeqrf_(m, n, &a[a_offset], lda, &work[itau], &work[iwork], &i__1, info); /* Multiply B by transpose(Q) (Workspace: need N+NRHS, prefer N+NRHS*NB) */ i__1 = *lwork - iwork + 1; dormqr_("L", "T", m, nrhs, n, &a[a_offset], lda, &work[itau], &b[ b_offset], ldb, &work[iwork], &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_b74, &c_b74, &a_ref(2, 1), lda); } } ie = 1; itauq = ie + *n; itaup = itauq + *n; iwork = itaup + *n; /* Bidiagonalize R in A (Workspace: need 3*N+MM, prefer 3*N+(MM+N)*NB) */ i__1 = *lwork - iwork + 1; dgebrd_(&mm, n, &a[a_offset], lda, &s[1], &work[ie], &work[itauq], & work[itaup], &work[iwork], &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 - iwork + 1; dormbr_("Q", "L", "T", &mm, nrhs, n, &a[a_offset], lda, &work[itauq], &b[b_offset], ldb, &work[iwork], &i__1, info); /* Generate right bidiagonalizing vectors of R in A (Workspace: need 4*N-1, prefer 3*N+(N-1)*NB) */ i__1 = *lwork - iwork + 1; dorgbr_("P", n, n, n, &a[a_offset], lda, &work[itaup], &work[iwork], & i__1, info); iwork = ie + *n; /* Perform bidiagonal QR iteration multiply B by transpose of left singular vectors compute right singular vectors in A (Workspace: need BDSPAC) */ dbdsqr_("U", n, n, &c__0, nrhs, &s[1], &work[ie], &a[a_offset], lda, vdum, &c__1, &b[b_offset], ldb, &work[iwork], info) ; if (*info != 0) { goto L70; } /* Multiply B by reciprocals of singular values Computing MAX */ d__1 = *rcond * s[1]; thr = max(d__1,sfmin); if (*rcond < 0.) { /* Computing MAX */ d__1 = eps * s[1]; thr = max(d__1,sfmin); } *rank = 0; i__1 = *n; for (i__ = 1; i__ <= i__1; ++i__) { if (s[i__] > thr) { drscl_(nrhs, &s[i__], &b_ref(i__, 1), ldb); ++(*rank); } else { dlaset_("F", &c__1, nrhs, &c_b74, &c_b74, &b_ref(i__, 1), ldb); } /* L10: */ } /* Multiply B by right singular vectors (Workspace: need N, prefer N*NRHS) */ if (*lwork >= *ldb * *nrhs && *nrhs > 1) { dgemm_("T", "N", n, nrhs, n, &c_b108, &a[a_offset], lda, &b[ b_offset], ldb, &c_b74, &work[1], ldb); dlacpy_("G", n, nrhs, &work[1], ldb, &b[b_offset], ldb) ; } else if (*nrhs > 1) { chunk = *lwork / *n; i__1 = *nrhs; i__2 = chunk; for (i__ = 1; i__2 < 0 ? i__ >= i__1 : i__ <= i__1; i__ += i__2) { /* Computing MIN */ i__3 = *nrhs - i__ + 1; bl = min(i__3,chunk); dgemm_("T", "N", n, &bl, n, &c_b108, &a[a_offset], lda, & b_ref(1, i__), ldb, &c_b74, &work[1], n); dlacpy_("G", n, &bl, &work[1], n, &b_ref(1, i__), ldb); /* L20: */ } } else { dgemv_("T", n, n, &c_b108, &a[a_offset], lda, &b[b_offset], &c__1, &c_b74, &work[1], &c__1); dcopy_(n, &work[1], &c__1, &b[b_offset], &c__1); } } else /* if(complicated condition) */ { /* Computing MAX */ i__2 = *m, i__1 = (*m << 1) - 4, i__2 = max(i__2,i__1), i__2 = max( i__2,*nrhs), i__1 = *n - *m * 3; if (*n >= mnthr && *lwork >= (*m << 2) + *m * *m + max(i__2,i__1)) { /* 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__2 = (*m << 2) + *m * *lda + max(i__3,i__4), i__1 = *m * *lda + *m + *m * *nrhs; if (*lwork >= max(i__2,i__1)) { ldwork = *lda; } itau = 1; iwork = *m + 1; /* Compute A=L*Q (Workspace: need 2*M, prefer M+M*NB) */ i__2 = *lwork - iwork + 1; dgelqf_(m, n, &a[a_offset], lda, &work[itau], &work[iwork], &i__2, info); il = iwork; /* Copy L to WORK(IL), zeroing out above it */ dlacpy_("L", m, m, &a[a_offset], lda, &work[il], &ldwork); i__2 = *m - 1; i__1 = *m - 1; dlaset_("U", &i__2, &i__1, &c_b74, &c_b74, &work[il + ldwork], & ldwork); ie = il + ldwork * *m; itauq = ie + *m; itaup = itauq + *m; iwork = itaup + *m; /* Bidiagonalize L in WORK(IL) (Workspace: need M*M+5*M, prefer M*M+4*M+2*M*NB) */ i__2 = *lwork - iwork + 1; dgebrd_(m, m, &work[il], &ldwork, &s[1], &work[ie], &work[itauq], &work[itaup], &work[iwork], &i__2, 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__2 = *lwork - iwork + 1; dormbr_("Q", "L", "T", m, nrhs, m, &work[il], &ldwork, &work[ itauq], &b[b_offset], ldb, &work[iwork], &i__2, info); /* Generate right bidiagonalizing vectors of R in WORK(IL) (Workspace: need M*M+5*M-1, prefer M*M+4*M+(M-1)*NB) */ i__2 = *lwork - iwork + 1; dorgbr_("P", m, m, m, &work[il], &ldwork, &work[itaup], &work[ iwork], &i__2, info); iwork = ie + *m; /* Perform bidiagonal QR iteration, computing right singular vectors of L in WORK(IL) and multiplying B by transpose of left singular vectors (Workspace: need M*M+M+BDSPAC) */ dbdsqr_("U", m, m, &c__0, nrhs, &s[1], &work[ie], &work[il], & ldwork, &a[a_offset], lda, &b[b_offset], ldb, &work[iwork] , info); if (*info != 0) { goto L70; } /* Multiply B by reciprocals of singular values Computing MAX */ d__1 = *rcond * s[1]; thr = max(d__1,sfmin); if (*rcond < 0.) { /* Computing MAX */ d__1 = eps * s[1]; thr = max(d__1,sfmin); } *rank = 0; i__2 = *m; for (i__ = 1; i__ <= i__2; ++i__) { if (s[i__] > thr) { drscl_(nrhs, &s[i__], &b_ref(i__, 1), ldb); ++(*rank); } else { dlaset_("F", &c__1, nrhs, &c_b74, &c_b74, &b_ref(i__, 1), ldb); } /* L30: */ } iwork = ie; /* Multiply B by right singular vectors of L in WORK(IL) (Workspace: need M*M+2*M, prefer M*M+M+M*NRHS) */ if (*lwork >= *ldb * *nrhs + iwork - 1 && *nrhs > 1) { dgemm_("T", "N", m, nrhs, m, &c_b108, &work[il], &ldwork, &b[ b_offset], ldb, &c_b74, &work[iwork], ldb); dlacpy_("G", m, nrhs, &work[iwork], ldb, &b[b_offset], ldb); } else if (*nrhs > 1) { chunk = (*lwork - iwork + 1) / *m; i__2 = *nrhs; i__1 = chunk; for (i__ = 1; i__1 < 0 ? i__ >= i__2 : i__ <= i__2; i__ += i__1) { /* Computing MIN */ i__3 = *nrhs - i__ + 1; bl = min(i__3,chunk); dgemm_("T", "N", m, &bl, m, &c_b108, &work[il], &ldwork, & b_ref(1, i__), ldb, &c_b74, &work[iwork], n); dlacpy_("G", m, &bl, &work[iwork], n, &b_ref(1, i__), ldb); /* L40: */ } } else { dgemv_("T", m, m, &c_b108, &work[il], &ldwork, &b_ref(1, 1), & c__1, &c_b74, &work[iwork], &c__1); dcopy_(m, &work[iwork], &c__1, &b_ref(1, 1), &c__1); } /* Zero out below first M rows of B */ i__1 = *n - *m; dlaset_("F", &i__1, nrhs, &c_b74, &c_b74, &b_ref(*m + 1, 1), ldb); iwork = itau + *m; /* Multiply transpose(Q) by B (Workspace: need M+NRHS, prefer M+NRHS*NB) */ i__1 = *lwork - iwork + 1; dormlq_("L", "T", n, nrhs, m, &a[a_offset], lda, &work[itau], &b[ b_offset], ldb, &work[iwork], &i__1, info); } else { /* Path 2 - remaining underdetermined cases */ ie = 1; itauq = ie + *m; itaup = itauq + *m; iwork = itaup + *m; /* Bidiagonalize A (Workspace: need 3*M+N, prefer 3*M+(M+N)*NB) */ i__1 = *lwork - iwork + 1; dgebrd_(m, n, &a[a_offset], lda, &s[1], &work[ie], &work[itauq], & work[itaup], &work[iwork], &i__1, info); /* Multiply B by transpose of left bidiagonalizing vectors (Workspace: need 3*M+NRHS, prefer 3*M+NRHS*NB) */ i__1 = *lwork - iwork + 1; dormbr_("Q", "L", "T", m, nrhs, n, &a[a_offset], lda, &work[itauq] , &b[b_offset], ldb, &work[iwork], &i__1, info); /* Generate right bidiagonalizing vectors in A (Workspace: need 4*M, prefer 3*M+M*NB) */ i__1 = *lwork - iwork + 1; dorgbr_("P", m, n, m, &a[a_offset], lda, &work[itaup], &work[ iwork], &i__1, info); iwork = ie + *m; /* Perform bidiagonal QR iteration, computing right singular vectors of A in A and multiplying B by transpose of left singular vectors (Workspace: need BDSPAC) */ dbdsqr_("L", m, n, &c__0, nrhs, &s[1], &work[ie], &a[a_offset], lda, vdum, &c__1, &b[b_offset], ldb, &work[iwork], info); if (*info != 0) { goto L70; } /* Multiply B by reciprocals of singular values Computing MAX */ d__1 = *rcond * s[1]; thr = max(d__1,sfmin); if (*rcond < 0.) { /* Computing MAX */ d__1 = eps * s[1]; thr = max(d__1,sfmin); } *rank = 0; i__1 = *m; for (i__ = 1; i__ <= i__1; ++i__) { if (s[i__] > thr) { drscl_(nrhs, &s[i__], &b_ref(i__, 1), ldb); ++(*rank); } else { dlaset_("F", &c__1, nrhs, &c_b74, &c_b74, &b_ref(i__, 1), ldb); } /* L50: */ } /* Multiply B by right singular vectors of A (Workspace: need N, prefer N*NRHS) */ if (*lwork >= *ldb * *nrhs && *nrhs > 1) { dgemm_("T", "N", n, nrhs, m, &c_b108, &a[a_offset], lda, &b[ b_offset], ldb, &c_b74, &work[1], ldb); dlacpy_("F", n, nrhs, &work[1], ldb, &b[b_offset], ldb); } else if (*nrhs > 1) { chunk = *lwork / *n; i__1 = *nrhs; i__2 = chunk; for (i__ = 1; i__2 < 0 ? i__ >= i__1 : i__ <= i__1; i__ += i__2) { /* Computing MIN */ i__3 = *nrhs - i__ + 1; bl = min(i__3,chunk); dgemm_("T", "N", n, &bl, m, &c_b108, &a[a_offset], lda, & b_ref(1, i__), ldb, &c_b74, &work[1], n); dlacpy_("F", n, &bl, &work[1], n, &b_ref(1, i__), ldb); /* L60: */ } } else { dgemv_("T", m, n, &c_b108, &a[a_offset], lda, &b[b_offset], & c__1, &c_b74, &work[1], &c__1); dcopy_(n, &work[1], &c__1, &b[b_offset], &c__1); } } } /* 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); } L70: work[1] = (doublereal) maxwrk; return 0; /* End of DGELSS */ } /* dgelss_ */