/* 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)
{
/* 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 */
integer ie, il, mm;
real eps, anrm, bnrm;
integer itau, nlvl, iascl, ibscl;
real sfmin;
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 real 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 real 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 *);
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 *);
integer ldwork;
extern /* Subroutine */
int cunmqr_(char *, char *, integer *, integer *, integer *, complex *, integer *, complex *, complex *, integer *, complex *, integer *, integer *);
integer liwork, minwrk, maxwrk;
real smlnum;
integer lrwork;
logical lquery;
integer nrwork, smlsiz;
/* -- LAPACK driver routine (version 3.4.0) -- */
/* -- LAPACK is a software package provided by Univ. of Tennessee, -- */
/* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */
/* November 2011 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. 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;
--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);
mnthr = ilaenv_(&c__6, "CGELSD", " ", m, n, nrhs, &c_n1);
/* 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); // , expr subst
maxwrk = max(i__1,i__2);
/* Computing MAX */
i__1 = maxwrk;
i__2 = *nrhs * ilaenv_(&c__1, "CUNMQR", "LC", m, nrhs, n, &c_n1); // , expr subst
maxwrk = max(i__1,i__2);
}
if (*m >= *n)
{
/* Path 1 - overdetermined or exactly determined. */
/* Computing MAX */
/* Computing 2nd power */
i__3 = smlsiz + 1;
i__1 = i__3 * i__3;
i__2 = *n * (*nrhs + 1) + (*nrhs << 1); // , expr subst
lrwork = *n * 10 + (*n << 1) * smlsiz + (*n << 3) * nlvl + smlsiz * 3 * *nrhs + max(i__1,i__2);
/* Computing MAX */
i__1 = maxwrk;
i__2 = (*n << 1) + (mm + *n) * ilaenv_(&c__1, "CGEBRD", " ", &mm, n, &c_n1, &c_n1); // , expr subst
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); // , expr subst
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); // , expr subst
maxwrk = max(i__1,i__2);
/* Computing MAX */
i__1 = maxwrk;
i__2 = (*n << 1) + *n * *nrhs; // , expr subst
maxwrk = max(i__1,i__2);
/* Computing MAX */
i__1 = (*n << 1) + mm;
i__2 = (*n << 1) + *n * *nrhs; // , expr subst
minwrk = max(i__1,i__2);
}
if (*n > *m)
{
/* Computing MAX */
/* Computing 2nd power */
i__3 = smlsiz + 1;
i__1 = i__3 * i__3;
i__2 = *n * (*nrhs + 1) + (*nrhs << 1); // , expr subst
lrwork = *m * 10 + (*m << 1) * smlsiz + (*m << 3) * nlvl + smlsiz * 3 * *nrhs + max(i__1,i__2);
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);
/* Computing MAX */
i__1 = maxwrk;
i__2 = *m * *m + (*m << 2) + (*m << 1) * ilaenv_(&c__1, "CGEBRD", " ", m, m, &c_n1, &c_n1); // , expr subst
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); // , expr subst
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); // , expr subst
maxwrk = max(i__1,i__2);
if (*nrhs > 1)
{
/* Computing MAX */
i__1 = maxwrk;
i__2 = *m * *m + *m + *m * *nrhs; // , expr subst
maxwrk = max(i__1,i__2);
}
else
{
/* Computing MAX */
i__1 = maxwrk;
i__2 = *m * *m + (*m << 1); // , expr subst
maxwrk = max(i__1,i__2);
}
/* Computing MAX */
i__1 = maxwrk;
i__2 = *m * *m + (*m << 2) + *m * *nrhs; // , expr subst
maxwrk = max(i__1,i__2);
/* XXX: Ensure the Path 2a case below is triggered. The workspace */
/* calculation should use queries for all routines eventually. */
/* 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; // ; expr subst
i__1 = maxwrk;
i__2 = (*m << 2) + *m * *m + max(i__3,i__4) ; // , expr subst
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);
/* Computing MAX */
i__1 = maxwrk;
i__2 = (*m << 1) + *nrhs * ilaenv_(&c__1, "CUNMBR", "QLC", m, nrhs, m, &c_n1); // , expr subst
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); // , expr subst
maxwrk = max(i__1,i__2);
/* Computing MAX */
i__1 = maxwrk;
i__2 = (*m << 1) + *m * *nrhs; // , expr subst
maxwrk = max(i__1,i__2);
}
/* Computing MAX */
i__1 = (*m << 1) + *n;
i__2 = (*m << 1) + *m * *nrhs; // , expr subst
minwrk = max(i__1,i__2);
}
}
minwrk = min(minwrk,maxwrk);
work[1].r = (real) maxwrk;
work[1].i = 0.f; // , expr subst
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; // ; expr subst
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; // ; expr subst
i__1 = (*m << 2) + *m * *lda + max(i__3,i__4);
i__2 = *m * *lda + *m + *m * *nrhs; // , expr subst
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; // , expr subst
iwork[1] = liwork;
rwork[1] = (real) lrwork;
return 0;
/* End of CGELSD */
}
/* 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)
{
/* System generated locals */
integer a_dim1, a_offset, b_dim1, b_offset, i__1, i__2, i__3, i__4;
real r__1;
complex q__1;
/* Local variables */
static real anrm, bnrm;
static integer itau, iascl, ibscl;
static real sfmin;
static integer minmn, maxmn, itaup, itauq, mnthr, nwork, ie, il;
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 *);
static integer mm;
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 minwrk, maxwrk;
static real smlnum;
static logical lquery;
static integer nrwork, smlsiz;
static real eps;
#define a_subscr(a_1,a_2) (a_2)*a_dim1 + a_1
#define a_ref(a_1,a_2) a[a_subscr(a_1,a_2)]
#define b_subscr(a_1,a_2) (a_2)*b_dim1 + a_1
#define b_ref(a_1,a_2) b[b_subscr(a_1,a_2)]
/* -- 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
=======
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 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 (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 WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
RWORK (workspace) REAL array, dimension at least
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 )
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;
--rwork;
--iwork;
/* Function Body */
*info = 0;
minmn = min(*m,*n);
maxmn = max(*m,*n);
mnthr = ilaenv_(&c__6, "CGELSD", " ", 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, "CGELSD", " ", &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;
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 * 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 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, "CUN"
"MBR", "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) {
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);
r__1 = (real) maxwrk;
q__1.r = r__1, q__1.i = 0.f;
work[1].r = q__1.r, work[1].i = q__1.i;
if (*lwork < minwrk && ! lquery) {
*info = -12;
}
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CGELSD", &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 = 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_b81, &c_b81, &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_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.
(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_ref(2, 1), 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_ref(*m + 1, 1), 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:
r__1 = (real) maxwrk;
q__1.r = r__1, q__1.i = 0.f;
work[1].r = q__1.r, work[1].i = q__1.i;
return 0;
/* End of CGELSD */
} /* cgelsd_ */
