#include "blaswrap.h" #include "f2c.h" /* Subroutine */ int cgelsd_(integer *m, integer *n, integer *nrhs, complex * a, integer *lda, complex *b, integer *ldb, real *s, real *rcond, integer *rank, complex *work, integer *lwork, real *rwork, integer * iwork, integer *info) { /* -- LAPACK driver routine (version 3.1) -- Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. November 2006 Purpose ======= CGELSD 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 tranformations, 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 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) COMPLEX 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) COMPLEX 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 the modulus of elements n+1:m in that column. LDB (input) INTEGER The leading dimension of the array B. LDB >= max(1,M,N). S (output) REAL 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) REAL 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) COMPLEX 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 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 2 * N + N * NRHS if M is greater than or equal to N or 2 * M + M * NRHS if M is less than N, the code will execute correctly. 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 array WORK and the minimum sizes of the arrays RWORK and IWORK, and returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK is issued by XERBLA. RWORK (workspace) REAL array, dimension (MAX(1,LRWORK)) LRWORK >= 10*N + 2*N*SMLSIZ + 8*N*NLVL + 3*SMLSIZ*NRHS + (SMLSIZ+1)**2 if M is greater than or equal to N or 10*M + 2*M*SMLSIZ + 8*M*NLVL + 3*SMLSIZ*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 ) On exit, if INFO = 0, RWORK(1) returns the minimum LRWORK. IWORK (workspace) INTEGER array, dimension (MAX(1,LIWORK)) LIWORK >= max(1, 3*MINMN*NLVL + 11*MINMN), where MINMN = MIN( M,N ). On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK. 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 */ /* Table of constant values */ static complex c_b1 = {0.f,0.f}; static integer c__9 = 9; static integer c__0 = 0; static integer c__6 = 6; static integer c_n1 = -1; static integer c__1 = 1; static real c_b80 = 0.f; /* 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 integer ie, il, mm; static real eps, anrm, bnrm; static integer itau, nlvl, iascl, ibscl; static real sfmin; static integer minmn, maxmn, itaup, itauq, mnthr, nwork; extern /* Subroutine */ int cgebrd_(integer *, integer *, complex *, integer *, real *, real *, complex *, complex *, complex *, integer *, integer *), slabad_(real *, real *); extern doublereal clange_(char *, integer *, integer *, complex *, integer *, real *); extern /* Subroutine */ int cgelqf_(integer *, integer *, complex *, integer *, complex *, complex *, integer *, integer *), clalsd_( char *, integer *, integer *, integer *, real *, real *, complex * , integer *, real *, integer *, complex *, real *, integer *, integer *), clascl_(char *, integer *, integer *, real *, real *, integer *, integer *, complex *, integer *, integer *), cgeqrf_(integer *, integer *, complex *, integer *, complex *, complex *, integer *, integer *); extern doublereal slamch_(char *); extern /* Subroutine */ int clacpy_(char *, integer *, integer *, complex *, integer *, complex *, integer *), claset_(char *, integer *, integer *, complex *, complex *, complex *, integer *), xerbla_(char *, integer *); extern integer ilaenv_(integer *, char *, char *, integer *, integer *, integer *, integer *, ftnlen, ftnlen); static real bignum; extern /* Subroutine */ int slascl_(char *, integer *, integer *, real *, real *, integer *, integer *, real *, integer *, integer *), cunmbr_(char *, char *, char *, integer *, integer *, integer *, complex *, integer *, complex *, complex *, integer *, complex *, integer *, integer *), slaset_( char *, integer *, integer *, real *, real *, real *, integer *), cunmlq_(char *, char *, integer *, integer *, integer *, complex *, integer *, complex *, complex *, integer *, complex *, integer *, integer *); static integer ldwork; extern /* Subroutine */ int cunmqr_(char *, char *, integer *, integer *, integer *, complex *, integer *, complex *, complex *, integer *, complex *, integer *, integer *); static integer liwork, minwrk, maxwrk; static real smlnum; static integer lrwork; static logical lquery; static integer nrwork, smlsiz; a_dim1 = *lda; a_offset = 1 + a_dim1; a -= a_offset; b_dim1 = *ldb; b_offset = 1 + b_dim1; b -= b_offset; --s; --work; --rwork; --iwork; /* Function Body */ *info = 0; minmn = min(*m,*n); maxmn = max(*m,*n); 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.) */ if (*info == 0) { minwrk = 1; maxwrk = 1; liwork = 1; lrwork = 1; if (minmn > 0) { smlsiz = ilaenv_(&c__9, "CGELSD", " ", &c__0, &c__0, &c__0, &c__0, (ftnlen)6, (ftnlen)1); mnthr = ilaenv_(&c__6, "CGELSD", " ", m, n, nrhs, &c_n1, (ftnlen) 6, (ftnlen)1); /* Computing MAX */ i__1 = (integer) (log((real) minmn / (real) (smlsiz + 1)) / log( 2.f)) + 1; nlvl = max(i__1,0); liwork = minmn * 3 * nlvl + minmn * 11; 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 * ilaenv_(&c__1, "CGEQRF", " ", m, n, &c_n1, &c_n1, (ftnlen)6, (ftnlen)1); maxwrk = max(i__1,i__2); /* Computing MAX */ i__1 = maxwrk, i__2 = *nrhs * ilaenv_(&c__1, "CUNMQR", "LC", 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 2nd power */ i__1 = smlsiz + 1; lrwork = *n * 10 + (*n << 1) * smlsiz + (*n << 3) * nlvl + smlsiz * 3 * *nrhs + i__1 * i__1; /* Computing MAX */ i__1 = maxwrk, i__2 = (*n << 1) + (mm + *n) * ilaenv_(&c__1, "CGEBRD", " ", &mm, n, &c_n1, &c_n1, (ftnlen)6, (ftnlen)1); maxwrk = max(i__1,i__2); /* Computing MAX */ i__1 = maxwrk, i__2 = (*n << 1) + *nrhs * ilaenv_(&c__1, "CUNMBR", "QLC", &mm, nrhs, n, &c_n1, (ftnlen)6, (ftnlen)3); maxwrk = max(i__1,i__2); /* Computing MAX */ i__1 = maxwrk, i__2 = (*n << 1) + (*n - 1) * ilaenv_(&c__1, "CUNMBR", "PLN", n, nrhs, n, &c_n1, (ftnlen)6, (ftnlen)3); maxwrk = max(i__1,i__2); /* Computing MAX */ i__1 = maxwrk, i__2 = (*n << 1) + *n * *nrhs; maxwrk = max(i__1,i__2); /* Computing MAX */ i__1 = (*n << 1) + mm, i__2 = (*n << 1) + *n * *nrhs; minwrk = max(i__1,i__2); } if (*n > *m) { /* Computing 2nd power */ i__1 = smlsiz + 1; lrwork = *m * 10 + (*m << 1) * smlsiz + (*m << 3) * nlvl + smlsiz * 3 * *nrhs + i__1 * i__1; if (*n >= mnthr) { /* Path 2a - underdetermined, with many more columns than rows. */ maxwrk = *m + *m * ilaenv_(&c__1, "CGELQF", " ", 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, "CGEBRD", " ", 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, "CUNMBR", "QLC", 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, "CUNMLQ", "LC", n, nrhs, m, &c_n1, (ftnlen)6, (ftnlen)2); 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 * *m + (*m << 2) + *m * *nrhs; maxwrk = max(i__1,i__2); } else { /* Path 2 - underdetermined. */ maxwrk = (*m << 1) + (*n + *m) * ilaenv_(&c__1, "CGEBRD", " ", m, n, &c_n1, &c_n1, (ftnlen)6, (ftnlen)1); /* Computing MAX */ i__1 = maxwrk, i__2 = (*m << 1) + *nrhs * ilaenv_(&c__1, "CUNMBR", "QLC", m, nrhs, m, &c_n1, (ftnlen)6, (ftnlen)3); maxwrk = max(i__1,i__2); /* Computing MAX */ i__1 = maxwrk, i__2 = (*m << 1) + *m * ilaenv_(&c__1, "CUNMBR", "PLN", n, nrhs, m, &c_n1, (ftnlen)6, (ftnlen)3); maxwrk = max(i__1,i__2); /* Computing MAX */ i__1 = maxwrk, i__2 = (*m << 1) + *m * *nrhs; maxwrk = max(i__1,i__2); } /* Computing MAX */ i__1 = (*m << 1) + *n, i__2 = (*m << 1) + *m * *nrhs; minwrk = max(i__1,i__2); } } minwrk = min(minwrk,maxwrk); work[1].r = (real) maxwrk, work[1].i = 0.f; iwork[1] = liwork; rwork[1] = (real) lrwork; if (*lwork < minwrk && ! lquery) { *info = -12; } } if (*info != 0) { i__1 = -(*info); xerbla_("CGELSD", &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 = slamch_("P"); sfmin = slamch_("S"); smlnum = sfmin / eps; bignum = 1.f / smlnum; slabad_(&smlnum, &bignum); /* Scale A if max entry outside range [SMLNUM,BIGNUM]. */ anrm = clange_("M", m, n, &a[a_offset], lda, &rwork[1]); iascl = 0; if (anrm > 0.f && anrm < smlnum) { /* Scale matrix norm up to SMLNUM */ clascl_("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. */ clascl_("G", &c__0, &c__0, &anrm, &bignum, m, n, &a[a_offset], lda, info); iascl = 2; } else if (anrm == 0.f) { /* Matrix all zero. Return zero solution. */ i__1 = max(*m,*n); claset_("F", &i__1, nrhs, &c_b1, &c_b1, &b[b_offset], ldb); slaset_("F", &minmn, &c__1, &c_b80, &c_b80, &s[1], &c__1); *rank = 0; goto L10; } /* Scale B if max entry outside range [SMLNUM,BIGNUM]. */ bnrm = clange_("M", m, nrhs, &b[b_offset], ldb, &rwork[1]); ibscl = 0; if (bnrm > 0.f && bnrm < smlnum) { /* Scale matrix norm up to SMLNUM. */ clascl_("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. */ clascl_("G", &c__0, &c__0, &bnrm, &bignum, m, nrhs, &b[b_offset], ldb, info); ibscl = 2; } /* If M < N make sure B(M+1:N,:) = 0 */ if (*m < *n) { i__1 = *n - *m; claset_("F", &i__1, nrhs, &c_b1, &c_b1, &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. (RWorkspace: need N) (CWorkspace: need N, prefer N*NB) */ i__1 = *lwork - nwork + 1; cgeqrf_(m, n, &a[a_offset], lda, &work[itau], &work[nwork], &i__1, info); /* Multiply B by transpose(Q). (RWorkspace: need N) (CWorkspace: need NRHS, prefer NRHS*NB) */ i__1 = *lwork - nwork + 1; cunmqr_("L", "C", 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; claset_("L", &i__1, &i__2, &c_b1, &c_b1, &a[a_dim1 + 2], lda); } } itauq = 1; itaup = itauq + *n; nwork = itaup + *n; ie = 1; nrwork = ie + *n; /* Bidiagonalize R in A. (RWorkspace: need N) (CWorkspace: need 2*N+MM, prefer 2*N+(MM+N)*NB) */ i__1 = *lwork - nwork + 1; cgebrd_(&mm, n, &a[a_offset], lda, &s[1], &rwork[ie], &work[itauq], & work[itaup], &work[nwork], &i__1, info); /* Multiply B by transpose of left bidiagonalizing vectors of R. (CWorkspace: need 2*N+NRHS, prefer 2*N+NRHS*NB) */ i__1 = *lwork - nwork + 1; cunmbr_("Q", "L", "C", &mm, nrhs, n, &a[a_offset], lda, &work[itauq], &b[b_offset], ldb, &work[nwork], &i__1, info); /* Solve the bidiagonal least squares problem. */ clalsd_("U", &smlsiz, n, nrhs, &s[1], &rwork[ie], &b[b_offset], ldb, rcond, rank, &work[nwork], &rwork[nrwork], &iwork[1], info); if (*info != 0) { goto L10; } /* Multiply B by right bidiagonalizing vectors of R. */ i__1 = *lwork - nwork + 1; cunmbr_("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. (CWorkspace: need 2*M, prefer M+M*NB) */ i__1 = *lwork - nwork + 1; cgelqf_(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. */ clacpy_("L", m, m, &a[a_offset], lda, &work[il], &ldwork); i__1 = *m - 1; i__2 = *m - 1; claset_("U", &i__1, &i__2, &c_b1, &c_b1, &work[il + ldwork], & ldwork); itauq = il + ldwork * *m; itaup = itauq + *m; nwork = itaup + *m; ie = 1; nrwork = ie + *m; /* Bidiagonalize L in WORK(IL). (RWorkspace: need M) (CWorkspace: need M*M+4*M, prefer M*M+4*M+2*M*NB) */ i__1 = *lwork - nwork + 1; cgebrd_(m, m, &work[il], &ldwork, &s[1], &rwork[ie], &work[itauq], &work[itaup], &work[nwork], &i__1, info); /* Multiply B by transpose of left bidiagonalizing vectors of L. (CWorkspace: need M*M+4*M+NRHS, prefer M*M+4*M+NRHS*NB) */ i__1 = *lwork - nwork + 1; cunmbr_("Q", "L", "C", m, nrhs, m, &work[il], &ldwork, &work[ itauq], &b[b_offset], ldb, &work[nwork], &i__1, info); /* Solve the bidiagonal least squares problem. */ clalsd_("U", &smlsiz, m, nrhs, &s[1], &rwork[ie], &b[b_offset], ldb, rcond, rank, &work[nwork], &rwork[nrwork], &iwork[1], info); if (*info != 0) { goto L10; } /* Multiply B by right bidiagonalizing vectors of L. */ i__1 = *lwork - nwork + 1; cunmbr_("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; claset_("F", &i__1, nrhs, &c_b1, &c_b1, &b[*m + 1 + b_dim1], ldb); nwork = itau + *m; /* Multiply transpose(Q) by B. (CWorkspace: need NRHS, prefer NRHS*NB) */ i__1 = *lwork - nwork + 1; cunmlq_("L", "C", n, nrhs, m, &a[a_offset], lda, &work[itau], &b[ b_offset], ldb, &work[nwork], &i__1, info); } else { /* Path 2 - remaining underdetermined cases. */ itauq = 1; itaup = itauq + *m; nwork = itaup + *m; ie = 1; nrwork = ie + *m; /* Bidiagonalize A. (RWorkspace: need M) (CWorkspace: need 2*M+N, prefer 2*M+(M+N)*NB) */ i__1 = *lwork - nwork + 1; cgebrd_(m, n, &a[a_offset], lda, &s[1], &rwork[ie], &work[itauq], &work[itaup], &work[nwork], &i__1, info); /* Multiply B by transpose of left bidiagonalizing vectors. (CWorkspace: need 2*M+NRHS, prefer 2*M+NRHS*NB) */ i__1 = *lwork - nwork + 1; cunmbr_("Q", "L", "C", m, nrhs, n, &a[a_offset], lda, &work[itauq] , &b[b_offset], ldb, &work[nwork], &i__1, info); /* Solve the bidiagonal least squares problem. */ clalsd_("L", &smlsiz, m, nrhs, &s[1], &rwork[ie], &b[b_offset], ldb, rcond, rank, &work[nwork], &rwork[nrwork], &iwork[1], info); if (*info != 0) { goto L10; } /* Multiply B by right bidiagonalizing vectors of A. */ i__1 = *lwork - nwork + 1; cunmbr_("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) { clascl_("G", &c__0, &c__0, &anrm, &smlnum, n, nrhs, &b[b_offset], ldb, info); slascl_("G", &c__0, &c__0, &smlnum, &anrm, &minmn, &c__1, &s[1], & minmn, info); } else if (iascl == 2) { clascl_("G", &c__0, &c__0, &anrm, &bignum, n, nrhs, &b[b_offset], ldb, info); slascl_("G", &c__0, &c__0, &bignum, &anrm, &minmn, &c__1, &s[1], & minmn, info); } if (ibscl == 1) { clascl_("G", &c__0, &c__0, &smlnum, &bnrm, n, nrhs, &b[b_offset], ldb, info); } else if (ibscl == 2) { clascl_("G", &c__0, &c__0, &bignum, &bnrm, n, nrhs, &b[b_offset], ldb, info); } L10: work[1].r = (real) maxwrk, work[1].i = 0.f; iwork[1] = liwork; rwork[1] = (real) lrwork; return 0; /* End of CGELSD */ } /* cgelsd_ */