int sbdsdc_(char *uplo, char *compq, int *n, float *d__,
float *e, float *u, int *ldu, float *vt, int *ldvt, float *q,
int *iq, float *work, int *iwork, int *info)
{
/* System generated locals */
int u_dim1, u_offset, vt_dim1, vt_offset, i__1, i__2;
float r__1;
/* Builtin functions */
double r_sign(float *, float *), log(double);
/* Local variables */
int i__, j, k;
float p, r__;
int z__, ic, ii, kk;
float cs;
int is, iu;
float sn;
int nm1;
float eps;
int ivt, difl, difr, ierr, perm, mlvl, sqre;
extern int lsame_(char *, char *);
int poles;
extern int slasr_(char *, char *, char *, int *,
int *, float *, float *, float *, int *);
int iuplo, nsize, start;
extern int scopy_(int *, float *, int *, float *,
int *), sswap_(int *, float *, int *, float *, int *
), slasd0_(int *, int *, float *, float *, float *, int *
, float *, int *, int *, int *, float *, int *);
extern double slamch_(char *);
extern int slasda_(int *, int *, int *,
int *, float *, float *, float *, int *, float *, int *,
float *, float *, float *, float *, int *, int *, int *,
int *, float *, float *, float *, float *, int *, int *),
xerbla_(char *, int *);
extern int ilaenv_(int *, char *, char *, int *, int *,
int *, int *);
extern int slascl_(char *, int *, int *, float *,
float *, int *, int *, float *, int *, int *);
int givcol;
extern int slasdq_(char *, int *, int *, int
*, int *, int *, float *, float *, float *, int *, float *
, int *, float *, int *, float *, int *);
int icompq;
extern int slaset_(char *, int *, int *, float *,
float *, float *, int *), slartg_(float *, float *, float *
, float *, float *);
float orgnrm;
int givnum;
extern double slanst_(char *, int *, float *, float *);
int givptr, qstart, smlsiz, wstart, smlszp;
/* -- LAPACK routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* SBDSDC computes the singular value decomposition (SVD) of a float */
/* N-by-N (upper or lower) bidiagonal matrix B: B = U * S * VT, */
/* using a divide and conquer method, where S is a diagonal matrix */
/* with non-negative diagonal elements (the singular values of B), and */
/* U and VT are orthogonal matrices of left and right singular vectors, */
/* respectively. SBDSDC can be used to compute all singular values, */
/* and optionally, singular vectors or singular vectors in compact form. */
/* This code 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. See SLASD3 for details. */
/* The code currently calls SLASDQ if singular values only are desired. */
/* However, it can be slightly modified to compute singular values */
/* using the divide and conquer method. */
/* Arguments */
/* ========= */
/* UPLO (input) CHARACTER*1 */
/* = 'U': B is upper bidiagonal. */
/* = 'L': B is lower bidiagonal. */
/* COMPQ (input) CHARACTER*1 */
/* Specifies whether singular vectors are to be computed */
/* as follows: */
/* = 'N': Compute singular values only; */
/* = 'P': Compute singular values and compute singular */
/* vectors in compact form; */
/* = 'I': Compute singular values and singular vectors. */
/* N (input) INTEGER */
/* The order of the matrix B. N >= 0. */
/* D (input/output) REAL array, dimension (N) */
/* On entry, the n diagonal elements of the bidiagonal matrix B. */
/* On exit, if INFO=0, the singular values of B. */
/* E (input/output) REAL array, dimension (N-1) */
/* On entry, the elements of E contain the offdiagonal */
/* elements of the bidiagonal matrix whose SVD is desired. */
/* On exit, E has been destroyed. */
/* U (output) REAL array, dimension (LDU,N) */
/* If COMPQ = 'I', then: */
/* On exit, if INFO = 0, U contains the left singular vectors */
/* of the bidiagonal matrix. */
/* For other values of COMPQ, U is not referenced. */
/* LDU (input) INTEGER */
/* The leading dimension of the array U. LDU >= 1. */
/* If singular vectors are desired, then LDU >= MAX( 1, N ). */
/* VT (output) REAL array, dimension (LDVT,N) */
/* If COMPQ = 'I', then: */
/* On exit, if INFO = 0, VT' contains the right singular */
/* vectors of the bidiagonal matrix. */
/* For other values of COMPQ, VT is not referenced. */
/* LDVT (input) INTEGER */
/* The leading dimension of the array VT. LDVT >= 1. */
/* If singular vectors are desired, then LDVT >= MAX( 1, N ). */
/* Q (output) REAL array, dimension (LDQ) */
/* If COMPQ = 'P', then: */
/* On exit, if INFO = 0, Q and IQ contain the left */
/* and right singular vectors in a compact form, */
/* requiring O(N log N) space instead of 2*N**2. */
/* In particular, Q contains all the REAL data in */
/* LDQ >= N*(11 + 2*SMLSIZ + 8*INT(LOG_2(N/(SMLSIZ+1)))) */
/* words of memory, where 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). */
/* For other values of COMPQ, Q is not referenced. */
/* IQ (output) INTEGER array, dimension (LDIQ) */
/* If COMPQ = 'P', then: */
/* On exit, if INFO = 0, Q and IQ contain the left */
/* and right singular vectors in a compact form, */
/* requiring O(N log N) space instead of 2*N**2. */
/* In particular, IQ contains all INTEGER data in */
/* LDIQ >= N*(3 + 3*INT(LOG_2(N/(SMLSIZ+1)))) */
/* words of memory, where 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). */
/* For other values of COMPQ, IQ is not referenced. */
/* WORK (workspace) REAL array, dimension (MAX(1,LWORK)) */
/* If COMPQ = 'N' then LWORK >= (4 * N). */
/* If COMPQ = 'P' then LWORK >= (6 * N). */
/* If COMPQ = 'I' then LWORK >= (3 * N**2 + 4 * N). */
/* IWORK (workspace) INTEGER array, dimension (8*N) */
/* INFO (output) INTEGER */
/* = 0: successful exit. */
/* < 0: if INFO = -i, the i-th argument had an illegal value. */
/* > 0: The algorithm failed to compute an singular value. */
/* The update process of divide and conquer failed. */
/* Further Details */
/* =============== */
/* Based on contributions by */
/* Ming Gu and Huan Ren, Computer Science Division, University of */
/* California at Berkeley, USA */
/* ===================================================================== */
/* Changed dimension statement in comment describing E from (N) to */
/* (N-1). Sven, 17 Feb 05. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
--d__;
--e;
u_dim1 = *ldu;
u_offset = 1 + u_dim1;
u -= u_offset;
vt_dim1 = *ldvt;
vt_offset = 1 + vt_dim1;
vt -= vt_offset;
--q;
--iq;
--work;
--iwork;
/* Function Body */
*info = 0;
iuplo = 0;
if (lsame_(uplo, "U")) {
iuplo = 1;
}
if (lsame_(uplo, "L")) {
iuplo = 2;
}
if (lsame_(compq, "N")) {
icompq = 0;
} else if (lsame_(compq, "P")) {
icompq = 1;
} else if (lsame_(compq, "I")) {
icompq = 2;
} else {
icompq = -1;
}
if (iuplo == 0) {
*info = -1;
} else if (icompq < 0) {
*info = -2;
} else if (*n < 0) {
*info = -3;
} else if (*ldu < 1 || icompq == 2 && *ldu < *n) {
*info = -7;
} else if (*ldvt < 1 || icompq == 2 && *ldvt < *n) {
*info = -9;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SBDSDC", &i__1);
return 0;
}
/* Quick return if possible */
if (*n == 0) {
return 0;
}
smlsiz = ilaenv_(&c__9, "SBDSDC", " ", &c__0, &c__0, &c__0, &c__0);
if (*n == 1) {
if (icompq == 1) {
q[1] = r_sign(&c_b15, &d__[1]);
q[smlsiz * *n + 1] = 1.f;
} else if (icompq == 2) {
u[u_dim1 + 1] = r_sign(&c_b15, &d__[1]);
vt[vt_dim1 + 1] = 1.f;
}
d__[1] = ABS(d__[1]);
return 0;
}
nm1 = *n - 1;
/* If matrix lower bidiagonal, rotate to be upper bidiagonal */
/* by applying Givens rotations on the left */
wstart = 1;
qstart = 3;
if (icompq == 1) {
scopy_(n, &d__[1], &c__1, &q[1], &c__1);
i__1 = *n - 1;
scopy_(&i__1, &e[1], &c__1, &q[*n + 1], &c__1);
}
if (iuplo == 2) {
qstart = 5;
wstart = (*n << 1) - 1;
i__1 = *n - 1;
for (i__ = 1; i__ <= i__1; ++i__) {
slartg_(&d__[i__], &e[i__], &cs, &sn, &r__);
d__[i__] = r__;
e[i__] = sn * d__[i__ + 1];
d__[i__ + 1] = cs * d__[i__ + 1];
if (icompq == 1) {
q[i__ + (*n << 1)] = cs;
q[i__ + *n * 3] = sn;
} else if (icompq == 2) {
work[i__] = cs;
work[nm1 + i__] = -sn;
}
/* L10: */
}
}
/* If ICOMPQ = 0, use SLASDQ to compute the singular values. */
if (icompq == 0) {
slasdq_("U", &c__0, n, &c__0, &c__0, &c__0, &d__[1], &e[1], &vt[
vt_offset], ldvt, &u[u_offset], ldu, &u[u_offset], ldu, &work[
wstart], info);
goto L40;
}
/* If N is smaller than the minimum divide size SMLSIZ, then solve */
/* the problem with another solver. */
if (*n <= smlsiz) {
if (icompq == 2) {
slaset_("A", n, n, &c_b29, &c_b15, &u[u_offset], ldu);
slaset_("A", n, n, &c_b29, &c_b15, &vt[vt_offset], ldvt);
slasdq_("U", &c__0, n, n, n, &c__0, &d__[1], &e[1], &vt[vt_offset]
, ldvt, &u[u_offset], ldu, &u[u_offset], ldu, &work[
wstart], info);
} else if (icompq == 1) {
iu = 1;
ivt = iu + *n;
slaset_("A", n, n, &c_b29, &c_b15, &q[iu + (qstart - 1) * *n], n);
slaset_("A", n, n, &c_b29, &c_b15, &q[ivt + (qstart - 1) * *n], n);
slasdq_("U", &c__0, n, n, n, &c__0, &d__[1], &e[1], &q[ivt + (
qstart - 1) * *n], n, &q[iu + (qstart - 1) * *n], n, &q[
iu + (qstart - 1) * *n], n, &work[wstart], info);
}
goto L40;
}
if (icompq == 2) {
slaset_("A", n, n, &c_b29, &c_b15, &u[u_offset], ldu);
slaset_("A", n, n, &c_b29, &c_b15, &vt[vt_offset], ldvt);
}
/* Scale. */
orgnrm = slanst_("M", n, &d__[1], &e[1]);
if (orgnrm == 0.f) {
return 0;
}
slascl_("G", &c__0, &c__0, &orgnrm, &c_b15, n, &c__1, &d__[1], n, &ierr);
slascl_("G", &c__0, &c__0, &orgnrm, &c_b15, &nm1, &c__1, &e[1], &nm1, &
ierr);
eps = slamch_("Epsilon");
mlvl = (int) (log((float) (*n) / (float) (smlsiz + 1)) / log(2.f)) + 1;
smlszp = smlsiz + 1;
if (icompq == 1) {
iu = 1;
ivt = smlsiz + 1;
difl = ivt + smlszp;
difr = difl + mlvl;
z__ = difr + (mlvl << 1);
ic = z__ + mlvl;
is = ic + 1;
poles = is + 1;
givnum = poles + (mlvl << 1);
k = 1;
givptr = 2;
perm = 3;
givcol = perm + mlvl;
}
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
if ((r__1 = d__[i__], ABS(r__1)) < eps) {
d__[i__] = r_sign(&eps, &d__[i__]);
}
/* L20: */
}
start = 1;
sqre = 0;
i__1 = nm1;
for (i__ = 1; i__ <= i__1; ++i__) {
if ((r__1 = e[i__], ABS(r__1)) < eps || i__ == nm1) {
/* Subproblem found. First determine its size and then */
/* apply divide and conquer on it. */
if (i__ < nm1) {
/* A subproblem with E(I) small for I < NM1. */
nsize = i__ - start + 1;
} else if ((r__1 = e[i__], ABS(r__1)) >= eps) {
/* A subproblem with E(NM1) not too small but I = NM1. */
nsize = *n - start + 1;
} else {
/* A subproblem with E(NM1) small. This implies an */
/* 1-by-1 subproblem at D(N). Solve this 1-by-1 problem */
/* first. */
nsize = i__ - start + 1;
if (icompq == 2) {
u[*n + *n * u_dim1] = r_sign(&c_b15, &d__[*n]);
vt[*n + *n * vt_dim1] = 1.f;
} else if (icompq == 1) {
q[*n + (qstart - 1) * *n] = r_sign(&c_b15, &d__[*n]);
q[*n + (smlsiz + qstart - 1) * *n] = 1.f;
}
d__[*n] = (r__1 = d__[*n], ABS(r__1));
}
if (icompq == 2) {
slasd0_(&nsize, &sqre, &d__[start], &e[start], &u[start +
start * u_dim1], ldu, &vt[start + start * vt_dim1],
ldvt, &smlsiz, &iwork[1], &work[wstart], info);
} else {
slasda_(&icompq, &smlsiz, &nsize, &sqre, &d__[start], &e[
start], &q[start + (iu + qstart - 2) * *n], n, &q[
start + (ivt + qstart - 2) * *n], &iq[start + k * *n],
&q[start + (difl + qstart - 2) * *n], &q[start + (
difr + qstart - 2) * *n], &q[start + (z__ + qstart -
2) * *n], &q[start + (poles + qstart - 2) * *n], &iq[
start + givptr * *n], &iq[start + givcol * *n], n, &
iq[start + perm * *n], &q[start + (givnum + qstart -
2) * *n], &q[start + (ic + qstart - 2) * *n], &q[
start + (is + qstart - 2) * *n], &work[wstart], &
iwork[1], info);
if (*info != 0) {
return 0;
}
}
start = i__ + 1;
}
/* L30: */
}
/* Unscale */
slascl_("G", &c__0, &c__0, &c_b15, &orgnrm, n, &c__1, &d__[1], n, &ierr);
L40:
/* Use Selection Sort to minimize swaps of singular vectors */
i__1 = *n;
for (ii = 2; ii <= i__1; ++ii) {
i__ = ii - 1;
kk = i__;
p = d__[i__];
i__2 = *n;
for (j = ii; j <= i__2; ++j) {
if (d__[j] > p) {
kk = j;
p = d__[j];
}
/* L50: */
}
if (kk != i__) {
d__[kk] = d__[i__];
d__[i__] = p;
if (icompq == 1) {
iq[i__] = kk;
} else if (icompq == 2) {
sswap_(n, &u[i__ * u_dim1 + 1], &c__1, &u[kk * u_dim1 + 1], &
c__1);
sswap_(n, &vt[i__ + vt_dim1], ldvt, &vt[kk + vt_dim1], ldvt);
}
} else if (icompq == 1) {
iq[i__] = i__;
}
/* L60: */
}
/* If ICOMPQ = 1, use IQ(N,1) as the indicator for UPLO */
if (icompq == 1) {
if (iuplo == 1) {
iq[*n] = 1;
} else {
iq[*n] = 0;
}
}
/* If B is lower bidiagonal, update U by those Givens rotations */
/* which rotated B to be upper bidiagonal */
if (iuplo == 2 && icompq == 2) {
slasr_("L", "V", "B", n, n, &work[1], &work[*n], &u[u_offset], ldu);
}
return 0;
/* End of SBDSDC */
} /* sbdsdc_ */
/* Subroutine */ int slasda_(integer *icompq, integer *smlsiz, integer *n,
integer *sqre, real *d__, real *e, real *u, integer *ldu, real *vt,
integer *k, real *difl, real *difr, real *z__, real *poles, integer *
givptr, integer *givcol, integer *ldgcol, integer *perm, real *givnum,
real *c__, real *s, real *work, integer *iwork, integer *info)
{
/* System generated locals */
integer givcol_dim1, givcol_offset, perm_dim1, perm_offset, difl_dim1,
difl_offset, difr_dim1, difr_offset, givnum_dim1, givnum_offset,
poles_dim1, poles_offset, u_dim1, u_offset, vt_dim1, vt_offset,
z_dim1, z_offset, i__1, i__2;
/* Builtin functions */
integer pow_ii(integer *, integer *);
/* Local variables */
integer i__, j, m, i1, ic, lf, nd, ll, nl, vf, nr, vl, im1, ncc, nlf, nrf,
vfi, iwk, vli, lvl, nru, ndb1, nlp1, lvl2, nrp1;
real beta;
integer idxq, nlvl;
real alpha;
integer inode, ndiml, ndimr, idxqi, itemp, sqrei;
extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *,
integer *), slasd6_(integer *, integer *, integer *, integer *,
real *, real *, real *, real *, real *, integer *, integer *,
integer *, integer *, integer *, real *, integer *, real *, real *
, real *, real *, integer *, real *, real *, real *, integer *,
integer *);
integer nwork1, nwork2;
extern /* Subroutine */ int xerbla_(char *, integer *), slasdq_(
char *, integer *, integer *, integer *, integer *, integer *,
real *, real *, real *, integer *, real *, integer *, real *,
integer *, real *, integer *), slasdt_(integer *, integer
*, integer *, integer *, integer *, integer *, integer *),
slaset_(char *, integer *, integer *, real *, real *, real *,
integer *);
integer smlszp;
/* -- LAPACK auxiliary routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* Using a divide and conquer approach, SLASDA computes the singular */
/* value decomposition (SVD) of a real upper bidiagonal N-by-M matrix */
/* B with diagonal D and offdiagonal E, where M = N + SQRE. The */
/* algorithm computes the singular values in the SVD B = U * S * VT. */
/* The orthogonal matrices U and VT are optionally computed in */
/* compact form. */
/* A related subroutine, SLASD0, computes the singular values and */
/* the singular vectors in explicit form. */
/* Arguments */
/* ========= */
/* ICOMPQ (input) INTEGER */
/* Specifies whether singular vectors are to be computed */
/* in compact form, as follows */
/* = 0: Compute singular values only. */
/* = 1: Compute singular vectors of upper bidiagonal */
/* matrix in compact form. */
/* SMLSIZ (input) INTEGER */
/* The maximum size of the subproblems at the bottom of the */
/* computation tree. */
/* N (input) INTEGER */
/* The row dimension of the upper bidiagonal matrix. This is */
/* also the dimension of the main diagonal array D. */
/* SQRE (input) INTEGER */
/* Specifies the column dimension of the bidiagonal matrix. */
/* = 0: The bidiagonal matrix has column dimension M = N; */
/* = 1: The bidiagonal matrix has column dimension M = N + 1. */
/* D (input/output) REAL array, dimension ( N ) */
/* On entry D contains the main diagonal of the bidiagonal */
/* matrix. On exit D, if INFO = 0, contains its singular values. */
/* E (input) REAL array, dimension ( M-1 ) */
/* Contains the subdiagonal entries of the bidiagonal matrix. */
/* On exit, E has been destroyed. */
/* U (output) REAL array, */
/* dimension ( LDU, SMLSIZ ) if ICOMPQ = 1, and not referenced */
/* if ICOMPQ = 0. If ICOMPQ = 1, on exit, U contains the left */
/* singular vector matrices of all subproblems at the bottom */
/* level. */
/* LDU (input) INTEGER, LDU = > N. */
/* The leading dimension of arrays U, VT, DIFL, DIFR, POLES, */
/* GIVNUM, and Z. */
/* VT (output) REAL array, */
/* dimension ( LDU, SMLSIZ+1 ) if ICOMPQ = 1, and not referenced */
/* if ICOMPQ = 0. If ICOMPQ = 1, on exit, VT' contains the right */
/* singular vector matrices of all subproblems at the bottom */
/* level. */
/* K (output) INTEGER array, dimension ( N ) */
/* if ICOMPQ = 1 and dimension 1 if ICOMPQ = 0. */
/* If ICOMPQ = 1, on exit, K(I) is the dimension of the I-th */
/* secular equation on the computation tree. */
/* DIFL (output) REAL array, dimension ( LDU, NLVL ), */
/* where NLVL = floor(log_2 (N/SMLSIZ))). */
/* DIFR (output) REAL array, */
/* dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1 and */
/* dimension ( N ) if ICOMPQ = 0. */
/* If ICOMPQ = 1, on exit, DIFL(1:N, I) and DIFR(1:N, 2 * I - 1) */
/* record distances between singular values on the I-th */
/* level and singular values on the (I -1)-th level, and */
/* DIFR(1:N, 2 * I ) contains the normalizing factors for */
/* the right singular vector matrix. See SLASD8 for details. */
/* Z (output) REAL array, */
/* dimension ( LDU, NLVL ) if ICOMPQ = 1 and */
/* dimension ( N ) if ICOMPQ = 0. */
/* The first K elements of Z(1, I) contain the components of */
/* the deflation-adjusted updating row vector for subproblems */
/* on the I-th level. */
/* POLES (output) REAL array, */
/* dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1, and not referenced */
/* if ICOMPQ = 0. If ICOMPQ = 1, on exit, POLES(1, 2*I - 1) and */
/* POLES(1, 2*I) contain the new and old singular values */
/* involved in the secular equations on the I-th level. */
/* GIVPTR (output) INTEGER array, */
/* dimension ( N ) if ICOMPQ = 1, and not referenced if */
/* ICOMPQ = 0. If ICOMPQ = 1, on exit, GIVPTR( I ) records */
/* the number of Givens rotations performed on the I-th */
/* problem on the computation tree. */
/* GIVCOL (output) INTEGER array, */
/* dimension ( LDGCOL, 2 * NLVL ) if ICOMPQ = 1, and not */
/* referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I, */
/* GIVCOL(1, 2 *I - 1) and GIVCOL(1, 2 *I) record the locations */
/* of Givens rotations performed on the I-th level on the */
/* computation tree. */
/* LDGCOL (input) INTEGER, LDGCOL = > N. */
/* The leading dimension of arrays GIVCOL and PERM. */
/* PERM (output) INTEGER array, dimension ( LDGCOL, NLVL ) */
/* if ICOMPQ = 1, and not referenced */
/* if ICOMPQ = 0. If ICOMPQ = 1, on exit, PERM(1, I) records */
/* permutations done on the I-th level of the computation tree. */
/* GIVNUM (output) REAL array, */
/* dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1, and not */
/* referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I, */
/* GIVNUM(1, 2 *I - 1) and GIVNUM(1, 2 *I) record the C- and S- */
/* values of Givens rotations performed on the I-th level on */
/* the computation tree. */
/* C (output) REAL array, */
/* dimension ( N ) if ICOMPQ = 1, and dimension 1 if ICOMPQ = 0. */
/* If ICOMPQ = 1 and the I-th subproblem is not square, on exit, */
/* C( I ) contains the C-value of a Givens rotation related to */
/* the right null space of the I-th subproblem. */
/* S (output) REAL array, dimension ( N ) if */
/* ICOMPQ = 1, and dimension 1 if ICOMPQ = 0. If ICOMPQ = 1 */
/* and the I-th subproblem is not square, on exit, S( I ) */
/* contains the S-value of a Givens rotation related to */
/* the right null space of the I-th subproblem. */
/* WORK (workspace) REAL array, dimension */
/* (6 * N + (SMLSIZ + 1)*(SMLSIZ + 1)). */
/* IWORK (workspace) INTEGER array, dimension (7*N). */
/* INFO (output) INTEGER */
/* = 0: successful exit. */
/* < 0: if INFO = -i, the i-th argument had an illegal value. */
/* > 0: if INFO = 1, an singular value did not converge */
/* Further Details */
/* =============== */
/* Based on contributions by */
/* Ming Gu and Huan Ren, Computer Science Division, University of */
/* California at Berkeley, USA */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
--d__;
--e;
givnum_dim1 = *ldu;
givnum_offset = 1 + givnum_dim1;
givnum -= givnum_offset;
poles_dim1 = *ldu;
poles_offset = 1 + poles_dim1;
poles -= poles_offset;
z_dim1 = *ldu;
z_offset = 1 + z_dim1;
z__ -= z_offset;
difr_dim1 = *ldu;
difr_offset = 1 + difr_dim1;
difr -= difr_offset;
difl_dim1 = *ldu;
difl_offset = 1 + difl_dim1;
difl -= difl_offset;
vt_dim1 = *ldu;
vt_offset = 1 + vt_dim1;
vt -= vt_offset;
u_dim1 = *ldu;
u_offset = 1 + u_dim1;
u -= u_offset;
--k;
--givptr;
perm_dim1 = *ldgcol;
perm_offset = 1 + perm_dim1;
perm -= perm_offset;
givcol_dim1 = *ldgcol;
givcol_offset = 1 + givcol_dim1;
givcol -= givcol_offset;
--c__;
--s;
--work;
--iwork;
/* Function Body */
*info = 0;
if (*icompq < 0 || *icompq > 1) {
*info = -1;
} else if (*smlsiz < 3) {
*info = -2;
} else if (*n < 0) {
*info = -3;
} else if (*sqre < 0 || *sqre > 1) {
*info = -4;
} else if (*ldu < *n + *sqre) {
*info = -8;
} else if (*ldgcol < *n) {
*info = -17;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SLASDA", &i__1);
return 0;
}
m = *n + *sqre;
/* If the input matrix is too small, call SLASDQ to find the SVD. */
if (*n <= *smlsiz) {
if (*icompq == 0) {
slasdq_("U", sqre, n, &c__0, &c__0, &c__0, &d__[1], &e[1], &vt[
vt_offset], ldu, &u[u_offset], ldu, &u[u_offset], ldu, &
work[1], info);
} else {
slasdq_("U", sqre, n, &m, n, &c__0, &d__[1], &e[1], &vt[vt_offset]
, ldu, &u[u_offset], ldu, &u[u_offset], ldu, &work[1],
info);
}
return 0;
}
/* Book-keeping and set up the computation tree. */
inode = 1;
ndiml = inode + *n;
ndimr = ndiml + *n;
idxq = ndimr + *n;
iwk = idxq + *n;
ncc = 0;
nru = 0;
smlszp = *smlsiz + 1;
vf = 1;
vl = vf + m;
nwork1 = vl + m;
nwork2 = nwork1 + smlszp * smlszp;
slasdt_(n, &nlvl, &nd, &iwork[inode], &iwork[ndiml], &iwork[ndimr],
smlsiz);
/* for the nodes on bottom level of the tree, solve */
/* their subproblems by SLASDQ. */
ndb1 = (nd + 1) / 2;
i__1 = nd;
for (i__ = ndb1; i__ <= i__1; ++i__) {
/* IC : center row of each node */
/* NL : number of rows of left subproblem */
/* NR : number of rows of right subproblem */
/* NLF: starting row of the left subproblem */
/* NRF: starting row of the right subproblem */
i1 = i__ - 1;
ic = iwork[inode + i1];
nl = iwork[ndiml + i1];
nlp1 = nl + 1;
nr = iwork[ndimr + i1];
nlf = ic - nl;
nrf = ic + 1;
idxqi = idxq + nlf - 2;
vfi = vf + nlf - 1;
vli = vl + nlf - 1;
sqrei = 1;
if (*icompq == 0) {
slaset_("A", &nlp1, &nlp1, &c_b11, &c_b12, &work[nwork1], &smlszp);
slasdq_("U", &sqrei, &nl, &nlp1, &nru, &ncc, &d__[nlf], &e[nlf], &
work[nwork1], &smlszp, &work[nwork2], &nl, &work[nwork2],
&nl, &work[nwork2], info);
itemp = nwork1 + nl * smlszp;
scopy_(&nlp1, &work[nwork1], &c__1, &work[vfi], &c__1);
scopy_(&nlp1, &work[itemp], &c__1, &work[vli], &c__1);
} else {
slaset_("A", &nl, &nl, &c_b11, &c_b12, &u[nlf + u_dim1], ldu);
slaset_("A", &nlp1, &nlp1, &c_b11, &c_b12, &vt[nlf + vt_dim1],
ldu);
slasdq_("U", &sqrei, &nl, &nlp1, &nl, &ncc, &d__[nlf], &e[nlf], &
vt[nlf + vt_dim1], ldu, &u[nlf + u_dim1], ldu, &u[nlf +
u_dim1], ldu, &work[nwork1], info);
scopy_(&nlp1, &vt[nlf + vt_dim1], &c__1, &work[vfi], &c__1);
scopy_(&nlp1, &vt[nlf + nlp1 * vt_dim1], &c__1, &work[vli], &c__1)
;
}
if (*info != 0) {
return 0;
}
i__2 = nl;
for (j = 1; j <= i__2; ++j) {
iwork[idxqi + j] = j;
/* L10: */
}
if (i__ == nd && *sqre == 0) {
sqrei = 0;
} else {
sqrei = 1;
}
idxqi += nlp1;
vfi += nlp1;
vli += nlp1;
nrp1 = nr + sqrei;
if (*icompq == 0) {
slaset_("A", &nrp1, &nrp1, &c_b11, &c_b12, &work[nwork1], &smlszp);
slasdq_("U", &sqrei, &nr, &nrp1, &nru, &ncc, &d__[nrf], &e[nrf], &
work[nwork1], &smlszp, &work[nwork2], &nr, &work[nwork2],
&nr, &work[nwork2], info);
itemp = nwork1 + (nrp1 - 1) * smlszp;
scopy_(&nrp1, &work[nwork1], &c__1, &work[vfi], &c__1);
scopy_(&nrp1, &work[itemp], &c__1, &work[vli], &c__1);
} else {
slaset_("A", &nr, &nr, &c_b11, &c_b12, &u[nrf + u_dim1], ldu);
slaset_("A", &nrp1, &nrp1, &c_b11, &c_b12, &vt[nrf + vt_dim1],
ldu);
slasdq_("U", &sqrei, &nr, &nrp1, &nr, &ncc, &d__[nrf], &e[nrf], &
vt[nrf + vt_dim1], ldu, &u[nrf + u_dim1], ldu, &u[nrf +
u_dim1], ldu, &work[nwork1], info);
scopy_(&nrp1, &vt[nrf + vt_dim1], &c__1, &work[vfi], &c__1);
scopy_(&nrp1, &vt[nrf + nrp1 * vt_dim1], &c__1, &work[vli], &c__1)
;
}
if (*info != 0) {
return 0;
}
i__2 = nr;
for (j = 1; j <= i__2; ++j) {
iwork[idxqi + j] = j;
/* L20: */
}
/* L30: */
}
/* Now conquer each subproblem bottom-up. */
j = pow_ii(&c__2, &nlvl);
for (lvl = nlvl; lvl >= 1; --lvl) {
lvl2 = (lvl << 1) - 1;
/* Find the first node LF and last node LL on */
/* the current level LVL. */
if (lvl == 1) {
lf = 1;
ll = 1;
} else {
i__1 = lvl - 1;
lf = pow_ii(&c__2, &i__1);
ll = (lf << 1) - 1;
}
i__1 = ll;
for (i__ = lf; i__ <= i__1; ++i__) {
im1 = i__ - 1;
ic = iwork[inode + im1];
nl = iwork[ndiml + im1];
nr = iwork[ndimr + im1];
nlf = ic - nl;
nrf = ic + 1;
if (i__ == ll) {
sqrei = *sqre;
} else {
sqrei = 1;
}
vfi = vf + nlf - 1;
vli = vl + nlf - 1;
idxqi = idxq + nlf - 1;
alpha = d__[ic];
beta = e[ic];
if (*icompq == 0) {
slasd6_(icompq, &nl, &nr, &sqrei, &d__[nlf], &work[vfi], &
work[vli], &alpha, &beta, &iwork[idxqi], &perm[
perm_offset], &givptr[1], &givcol[givcol_offset],
ldgcol, &givnum[givnum_offset], ldu, &poles[
poles_offset], &difl[difl_offset], &difr[difr_offset],
&z__[z_offset], &k[1], &c__[1], &s[1], &work[nwork1],
&iwork[iwk], info);
} else {
--j;
slasd6_(icompq, &nl, &nr, &sqrei, &d__[nlf], &work[vfi], &
work[vli], &alpha, &beta, &iwork[idxqi], &perm[nlf +
lvl * perm_dim1], &givptr[j], &givcol[nlf + lvl2 *
givcol_dim1], ldgcol, &givnum[nlf + lvl2 *
givnum_dim1], ldu, &poles[nlf + lvl2 * poles_dim1], &
difl[nlf + lvl * difl_dim1], &difr[nlf + lvl2 *
difr_dim1], &z__[nlf + lvl * z_dim1], &k[j], &c__[j],
&s[j], &work[nwork1], &iwork[iwk], info);
}
if (*info != 0) {
return 0;
}
/* L40: */
}
/* L50: */
}
return 0;
/* End of SLASDA */
} /* slasda_ */
/* Subroutine */
int slasd0_(integer *n, integer *sqre, real *d__, real *e, real *u, integer *ldu, real *vt, integer *ldvt, integer *smlsiz, integer *iwork, real *work, integer *info)
{
/* System generated locals */
integer u_dim1, u_offset, vt_dim1, vt_offset, i__1, i__2;
/* Builtin functions */
integer pow_ii(integer *, integer *);
/* Local variables */
integer i__, j, m, i1, ic, lf, nd, ll, nl, nr, im1, ncc, nlf, nrf, iwk, lvl, ndb1, nlp1, nrp1;
real beta;
integer idxq, nlvl;
real alpha;
integer inode, ndiml, idxqc, ndimr, itemp, sqrei;
extern /* Subroutine */
int slasd1_(integer *, integer *, integer *, real *, real *, real *, real *, integer *, real *, integer *, integer * , integer *, real *, integer *), xerbla_(char *, integer *), slasdq_(char *, integer *, integer *, integer *, integer *, integer *, real *, real *, real *, integer *, real *, integer * , real *, integer *, real *, integer *), slasdt_(integer * , integer *, integer *, integer *, integer *, integer *, integer * );
/* -- LAPACK auxiliary routine (version 3.4.2) -- */
/* -- LAPACK is a software package provided by Univ. of Tennessee, -- */
/* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */
/* September 2012 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* ===================================================================== */
/* .. Local Scalars .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
--d__;
--e;
u_dim1 = *ldu;
u_offset = 1 + u_dim1;
u -= u_offset;
vt_dim1 = *ldvt;
vt_offset = 1 + vt_dim1;
vt -= vt_offset;
--iwork;
--work;
/* Function Body */
*info = 0;
if (*n < 0)
{
*info = -1;
}
else if (*sqre < 0 || *sqre > 1)
{
*info = -2;
}
m = *n + *sqre;
if (*ldu < *n)
{
*info = -6;
}
else if (*ldvt < m)
{
*info = -8;
}
else if (*smlsiz < 3)
{
*info = -9;
}
if (*info != 0)
{
i__1 = -(*info);
xerbla_("SLASD0", &i__1);
return 0;
}
/* If the input matrix is too small, call SLASDQ to find the SVD. */
if (*n <= *smlsiz)
{
slasdq_("U", sqre, n, &m, n, &c__0, &d__[1], &e[1], &vt[vt_offset], ldvt, &u[u_offset], ldu, &u[u_offset], ldu, &work[1], info);
return 0;
}
/* Set up the computation tree. */
inode = 1;
ndiml = inode + *n;
ndimr = ndiml + *n;
idxq = ndimr + *n;
iwk = idxq + *n;
slasdt_(n, &nlvl, &nd, &iwork[inode], &iwork[ndiml], &iwork[ndimr], smlsiz);
/* For the nodes on bottom level of the tree, solve */
/* their subproblems by SLASDQ. */
ndb1 = (nd + 1) / 2;
ncc = 0;
i__1 = nd;
for (i__ = ndb1;
i__ <= i__1;
++i__)
{
/* IC : center row of each node */
/* NL : number of rows of left subproblem */
/* NR : number of rows of right subproblem */
/* NLF: starting row of the left subproblem */
/* NRF: starting row of the right subproblem */
i1 = i__ - 1;
ic = iwork[inode + i1];
nl = iwork[ndiml + i1];
nlp1 = nl + 1;
nr = iwork[ndimr + i1];
nrp1 = nr + 1;
nlf = ic - nl;
nrf = ic + 1;
sqrei = 1;
slasdq_("U", &sqrei, &nl, &nlp1, &nl, &ncc, &d__[nlf], &e[nlf], &vt[ nlf + nlf * vt_dim1], ldvt, &u[nlf + nlf * u_dim1], ldu, &u[ nlf + nlf * u_dim1], ldu, &work[1], info);
if (*info != 0)
{
return 0;
}
itemp = idxq + nlf - 2;
i__2 = nl;
for (j = 1;
j <= i__2;
++j)
{
iwork[itemp + j] = j;
/* L10: */
}
if (i__ == nd)
{
sqrei = *sqre;
}
else
{
sqrei = 1;
}
nrp1 = nr + sqrei;
slasdq_("U", &sqrei, &nr, &nrp1, &nr, &ncc, &d__[nrf], &e[nrf], &vt[ nrf + nrf * vt_dim1], ldvt, &u[nrf + nrf * u_dim1], ldu, &u[ nrf + nrf * u_dim1], ldu, &work[1], info);
if (*info != 0)
{
return 0;
}
itemp = idxq + ic;
i__2 = nr;
for (j = 1;
j <= i__2;
++j)
{
iwork[itemp + j - 1] = j;
/* L20: */
}
/* L30: */
}
/* Now conquer each subproblem bottom-up. */
for (lvl = nlvl;
lvl >= 1;
--lvl)
{
/* Find the first node LF and last node LL on the */
/* current level LVL. */
if (lvl == 1)
{
lf = 1;
ll = 1;
}
else
{
i__1 = lvl - 1;
lf = pow_ii(&c__2, &i__1);
ll = (lf << 1) - 1;
}
i__1 = ll;
for (i__ = lf;
i__ <= i__1;
++i__)
{
im1 = i__ - 1;
ic = iwork[inode + im1];
nl = iwork[ndiml + im1];
nr = iwork[ndimr + im1];
nlf = ic - nl;
if (*sqre == 0 && i__ == ll)
{
sqrei = *sqre;
}
else
{
sqrei = 1;
}
idxqc = idxq + nlf - 1;
alpha = d__[ic];
beta = e[ic];
slasd1_(&nl, &nr, &sqrei, &d__[nlf], &alpha, &beta, &u[nlf + nlf * u_dim1], ldu, &vt[nlf + nlf * vt_dim1], ldvt, &iwork[ idxqc], &iwork[iwk], &work[1], info);
if (*info != 0)
{
return 0;
}
/* L40: */
}
/* L50: */
}
return 0;
/* End of SLASD0 */
}
/* Subroutine */ int slalsd_(char *uplo, integer *smlsiz, integer *n, integer
*nrhs, real *d__, real *e, real *b, integer *ldb, real *rcond,
integer *rank, real *work, integer *iwork, integer *info)
{
/* System generated locals */
integer b_dim1, b_offset, i__1, i__2;
real r__1;
/* Builtin functions */
double log(doublereal), r_sign(real *, real *);
/* Local variables */
static integer difl, difr, perm, nsub, nlvl, sqre, bxst;
extern /* Subroutine */ int srot_(integer *, real *, integer *, real *,
integer *, real *, real *);
static integer c__, i__, j, k;
static real r__;
static integer s, u, z__;
extern /* Subroutine */ int sgemm_(char *, char *, integer *, integer *,
integer *, real *, real *, integer *, real *, integer *, real *,
real *, integer *);
static integer poles, sizei, nsize;
extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *,
integer *);
static integer nwork, icmpq1, icmpq2;
extern doublereal sopbl3_(char *, integer *, integer *, integer *)
;
static real cs;
static integer bx;
static real sn;
static integer st;
extern /* Subroutine */ int slasda_(integer *, integer *, integer *,
integer *, real *, real *, real *, integer *, real *, integer *,
real *, real *, real *, real *, integer *, integer *, integer *,
integer *, real *, real *, real *, real *, integer *, integer *);
extern doublereal slamch_(char *);
static integer vt;
extern /* Subroutine */ int xerbla_(char *, integer *), slalsa_(
integer *, integer *, integer *, integer *, real *, integer *,
real *, integer *, real *, integer *, real *, integer *, real *,
real *, real *, real *, integer *, integer *, integer *, integer *
, real *, real *, real *, real *, integer *, integer *), slascl_(
char *, integer *, integer *, real *, real *, integer *, integer *
, real *, integer *, integer *);
static integer givcol;
extern integer isamax_(integer *, real *, integer *);
extern /* Subroutine */ int slasdq_(char *, integer *, integer *, integer
*, integer *, integer *, real *, real *, real *, integer *, real *
, integer *, real *, integer *, real *, integer *),
slacpy_(char *, integer *, integer *, real *, integer *, real *,
integer *), slartg_(real *, real *, real *, real *, real *
), slaset_(char *, integer *, integer *, real *, real *, real *,
integer *);
static real orgnrm;
static integer givnum;
extern doublereal slanst_(char *, integer *, real *, real *);
extern /* Subroutine */ int slasrt_(char *, integer *, real *, integer *);
static integer givptr, nm1, smlszp, st1;
static real eps;
static integer iwk;
static real tol;
#define b_ref(a_1,a_2) b[(a_2)*b_dim1 + a_1]
/* -- LAPACK routine (instrumented to count ops, version 3.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
October 31, 1999
Purpose
=======
SLALSD uses the singular value decomposition of A to solve the least
squares problem of finding X to minimize the Euclidean norm of each
column of A*X-B, where A is N-by-N upper bidiagonal, and X and B
are N-by-NRHS. The solution X overwrites B.
The singular values of A smaller than RCOND times the largest
singular value are treated as zero in solving the least squares
problem; in this case a minimum norm solution is returned.
The actual singular values are returned in D in ascending order.
This code 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 XMP, Cray YMP, Cray C 90, or Cray 2.
It could conceivably fail on hexadecimal or decimal machines
without guard digits, but we know of none.
Arguments
=========
UPLO (input) CHARACTER*1
= 'U': D and E define an upper bidiagonal matrix.
= 'L': D and E define a lower bidiagonal matrix.
SMLSIZ (input) INTEGER
The maximum size of the subproblems at the bottom of the
computation tree.
N (input) INTEGER
The dimension of the bidiagonal matrix. N >= 0.
NRHS (input) INTEGER
The number of columns of B. NRHS must be at least 1.
D (input/output) REAL array, dimension (N)
On entry D contains the main diagonal of the bidiagonal
matrix. On exit, if INFO = 0, D contains its singular values.
E (input) REAL array, dimension (N-1)
Contains the super-diagonal entries of the bidiagonal matrix.
On exit, E has been destroyed.
B (input/output) REAL array, dimension (LDB,NRHS)
On input, B contains the right hand sides of the least
squares problem. On output, B contains the solution X.
LDB (input) INTEGER
The leading dimension of B in the calling subprogram.
LDB must be at least max(1,N).
RCOND (input) REAL
The singular values of A less than or equal to RCOND times
the largest singular value are treated as zero in solving
the least squares problem. If RCOND is negative,
machine precision is used instead.
For example, if diag(S)*X=B were the least squares problem,
where diag(S) is a diagonal matrix of singular values, the
solution would be X(i) = B(i) / S(i) if S(i) is greater than
RCOND*max(S), and X(i) = 0 if S(i) is less than or equal to
RCOND*max(S).
RANK (output) INTEGER
The number of singular values of A greater than RCOND times
the largest singular value.
WORK (workspace) REAL array, dimension at least
(9*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2),
where NLVL = max(0, INT(log_2 (N/(SMLSIZ+1))) + 1).
IWORK (workspace) INTEGER array, dimension at least
(3 * N * NLVL + 11 * N)
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: The algorithm failed to compute an singular value while
working on the submatrix lying in rows and columns
INFO/(N+1) through MOD(INFO,N+1).
=====================================================================
Test the input parameters.
Parameter adjustments */
--d__;
--e;
b_dim1 = *ldb;
b_offset = 1 + b_dim1 * 1;
b -= b_offset;
--work;
--iwork;
/* Function Body */
*info = 0;
if (*n < 0) {
*info = -3;
} else if (*nrhs < 1) {
*info = -4;
} else if (*ldb < 1 || *ldb < *n) {
*info = -8;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SLALSD", &i__1);
return 0;
}
eps = slamch_("Epsilon");
/* Set up the tolerance. */
if (*rcond <= 0.f || *rcond >= 1.f) {
*rcond = eps;
}
*rank = 0;
/* Quick return if possible. */
if (*n == 0) {
return 0;
} else if (*n == 1) {
if (d__[1] == 0.f) {
slaset_("A", &c__1, nrhs, &c_b6, &c_b6, &b[b_offset], ldb);
} else {
*rank = 1;
latime_1.ops += (real) (*nrhs << 1);
slascl_("G", &c__0, &c__0, &d__[1], &c_b11, &c__1, nrhs, &b[
b_offset], ldb, info);
d__[1] = dabs(d__[1]);
}
return 0;
}
/* Rotate the matrix if it is lower bidiagonal. */
if (*(unsigned char *)uplo == 'L') {
latime_1.ops += (real) ((*n - 1) * 6);
i__1 = *n - 1;
for (i__ = 1; i__ <= i__1; ++i__) {
slartg_(&d__[i__], &e[i__], &cs, &sn, &r__);
d__[i__] = r__;
e[i__] = sn * d__[i__ + 1];
d__[i__ + 1] = cs * d__[i__ + 1];
if (*nrhs == 1) {
latime_1.ops += 6.f;
srot_(&c__1, &b_ref(i__, 1), &c__1, &b_ref(i__ + 1, 1), &c__1,
&cs, &sn);
} else {
work[(i__ << 1) - 1] = cs;
work[i__ * 2] = sn;
}
/* L10: */
}
if (*nrhs > 1) {
latime_1.ops += (real) ((*n - 1) * 6 * *nrhs);
i__1 = *nrhs;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = *n - 1;
for (j = 1; j <= i__2; ++j) {
cs = work[(j << 1) - 1];
sn = work[j * 2];
srot_(&c__1, &b_ref(j, i__), &c__1, &b_ref(j + 1, i__), &
c__1, &cs, &sn);
/* L20: */
}
/* L30: */
}
}
}
/* Scale. */
nm1 = *n - 1;
orgnrm = slanst_("M", n, &d__[1], &e[1]);
if (orgnrm == 0.f) {
slaset_("A", n, nrhs, &c_b6, &c_b6, &b[b_offset], ldb);
return 0;
}
latime_1.ops += (real) (*n + nm1);
slascl_("G", &c__0, &c__0, &orgnrm, &c_b11, n, &c__1, &d__[1], n, info);
slascl_("G", &c__0, &c__0, &orgnrm, &c_b11, &nm1, &c__1, &e[1], &nm1,
info);
/* If N is smaller than the minimum divide size SMLSIZ, then solve
the problem with another solver. */
if (*n <= *smlsiz) {
nwork = *n * *n + 1;
slaset_("A", n, n, &c_b6, &c_b11, &work[1], n);
slasdq_("U", &c__0, n, n, &c__0, nrhs, &d__[1], &e[1], &work[1], n, &
work[1], n, &b[b_offset], ldb, &work[nwork], info);
if (*info != 0) {
return 0;
}
latime_1.ops += 1.f;
tol = *rcond * (r__1 = d__[isamax_(n, &d__[1], &c__1)], dabs(r__1));
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
if (d__[i__] <= tol) {
slaset_("A", &c__1, nrhs, &c_b6, &c_b6, &b_ref(i__, 1), ldb);
} else {
latime_1.ops += (real) (*nrhs);
slascl_("G", &c__0, &c__0, &d__[i__], &c_b11, &c__1, nrhs, &
b_ref(i__, 1), ldb, info);
++(*rank);
}
/* L40: */
}
latime_1.ops += sopbl3_("SGEMM ", n, nrhs, n);
sgemm_("T", "N", n, nrhs, n, &c_b11, &work[1], n, &b[b_offset], ldb, &
c_b6, &work[nwork], n);
slacpy_("A", n, nrhs, &work[nwork], n, &b[b_offset], ldb);
/* Unscale. */
latime_1.ops += (real) (*n + *n * *nrhs);
slascl_("G", &c__0, &c__0, &c_b11, &orgnrm, n, &c__1, &d__[1], n,
info);
slasrt_("D", n, &d__[1], info);
slascl_("G", &c__0, &c__0, &orgnrm, &c_b11, n, nrhs, &b[b_offset],
ldb, info);
return 0;
}
/* Book-keeping and setting up some constants. */
nlvl = (integer) (log((real) (*n) / (real) (*smlsiz + 1)) / log(2.f)) + 1;
smlszp = *smlsiz + 1;
u = 1;
vt = *smlsiz * *n + 1;
difl = vt + smlszp * *n;
difr = difl + nlvl * *n;
z__ = difr + (nlvl * *n << 1);
c__ = z__ + nlvl * *n;
s = c__ + *n;
poles = s + *n;
givnum = poles + (nlvl << 1) * *n;
bx = givnum + (nlvl << 1) * *n;
nwork = bx + *n * *nrhs;
sizei = *n + 1;
k = sizei + *n;
givptr = k + *n;
perm = givptr + *n;
givcol = perm + nlvl * *n;
iwk = givcol + (nlvl * *n << 1);
st = 1;
sqre = 0;
icmpq1 = 1;
icmpq2 = 0;
nsub = 0;
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
if ((r__1 = d__[i__], dabs(r__1)) < eps) {
d__[i__] = r_sign(&eps, &d__[i__]);
}
/* L50: */
}
i__1 = nm1;
for (i__ = 1; i__ <= i__1; ++i__) {
if ((r__1 = e[i__], dabs(r__1)) < eps || i__ == nm1) {
++nsub;
iwork[nsub] = st;
/* Subproblem found. First determine its size and then
apply divide and conquer on it. */
if (i__ < nm1) {
/* A subproblem with E(I) small for I < NM1. */
nsize = i__ - st + 1;
iwork[sizei + nsub - 1] = nsize;
} else if ((r__1 = e[i__], dabs(r__1)) >= eps) {
/* A subproblem with E(NM1) not too small but I = NM1. */
nsize = *n - st + 1;
iwork[sizei + nsub - 1] = nsize;
} else {
/* A subproblem with E(NM1) small. This implies an
1-by-1 subproblem at D(N), which is not solved
explicitly. */
nsize = i__ - st + 1;
iwork[sizei + nsub - 1] = nsize;
++nsub;
iwork[nsub] = *n;
iwork[sizei + nsub - 1] = 1;
scopy_(nrhs, &b_ref(*n, 1), ldb, &work[bx + nm1], n);
}
st1 = st - 1;
if (nsize == 1) {
/* This is a 1-by-1 subproblem and is not solved
explicitly. */
scopy_(nrhs, &b_ref(st, 1), ldb, &work[bx + st1], n);
} else if (nsize <= *smlsiz) {
/* This is a small subproblem and is solved by SLASDQ. */
slaset_("A", &nsize, &nsize, &c_b6, &c_b11, &work[vt + st1],
n);
slasdq_("U", &c__0, &nsize, &nsize, &c__0, nrhs, &d__[st], &e[
st], &work[vt + st1], n, &work[nwork], n, &b_ref(st,
1), ldb, &work[nwork], info);
if (*info != 0) {
return 0;
}
slacpy_("A", &nsize, nrhs, &b_ref(st, 1), ldb, &work[bx + st1]
, n);
} else {
/* A large problem. Solve it using divide and conquer. */
slasda_(&icmpq1, smlsiz, &nsize, &sqre, &d__[st], &e[st], &
work[u + st1], n, &work[vt + st1], &iwork[k + st1], &
work[difl + st1], &work[difr + st1], &work[z__ + st1],
&work[poles + st1], &iwork[givptr + st1], &iwork[
givcol + st1], n, &iwork[perm + st1], &work[givnum +
st1], &work[c__ + st1], &work[s + st1], &work[nwork],
&iwork[iwk], info);
if (*info != 0) {
return 0;
}
bxst = bx + st1;
slalsa_(&icmpq2, smlsiz, &nsize, nrhs, &b_ref(st, 1), ldb, &
work[bxst], n, &work[u + st1], n, &work[vt + st1], &
iwork[k + st1], &work[difl + st1], &work[difr + st1],
&work[z__ + st1], &work[poles + st1], &iwork[givptr +
st1], &iwork[givcol + st1], n, &iwork[perm + st1], &
work[givnum + st1], &work[c__ + st1], &work[s + st1],
&work[nwork], &iwork[iwk], info);
if (*info != 0) {
return 0;
}
}
st = i__ + 1;
}
/* L60: */
}
/* Apply the singular values and treat the tiny ones as zero. */
tol = *rcond * (r__1 = d__[isamax_(n, &d__[1], &c__1)], dabs(r__1));
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
/* Some of the elements in D can be negative because 1-by-1
subproblems were not solved explicitly. */
if ((r__1 = d__[i__], dabs(r__1)) <= tol) {
slaset_("A", &c__1, nrhs, &c_b6, &c_b6, &work[bx + i__ - 1], n);
} else {
++(*rank);
latime_1.ops += (real) (*nrhs);
slascl_("G", &c__0, &c__0, &d__[i__], &c_b11, &c__1, nrhs, &work[
bx + i__ - 1], n, info);
}
d__[i__] = (r__1 = d__[i__], dabs(r__1));
/* L70: */
}
/* Now apply back the right singular vectors. */
icmpq2 = 1;
i__1 = nsub;
for (i__ = 1; i__ <= i__1; ++i__) {
st = iwork[i__];
st1 = st - 1;
nsize = iwork[sizei + i__ - 1];
bxst = bx + st1;
if (nsize == 1) {
scopy_(nrhs, &work[bxst], n, &b_ref(st, 1), ldb);
} else if (nsize <= *smlsiz) {
latime_1.ops += sopbl3_("SGEMM ", &nsize, nrhs, &nsize)
;
sgemm_("T", "N", &nsize, nrhs, &nsize, &c_b11, &work[vt + st1], n,
&work[bxst], n, &c_b6, &b_ref(st, 1), ldb);
} else {
slalsa_(&icmpq2, smlsiz, &nsize, nrhs, &work[bxst], n, &b_ref(st,
1), ldb, &work[u + st1], n, &work[vt + st1], &iwork[k +
st1], &work[difl + st1], &work[difr + st1], &work[z__ +
st1], &work[poles + st1], &iwork[givptr + st1], &iwork[
givcol + st1], n, &iwork[perm + st1], &work[givnum + st1],
&work[c__ + st1], &work[s + st1], &work[nwork], &iwork[
iwk], info);
if (*info != 0) {
return 0;
}
}
/* L80: */
}
/* Unscale and sort the singular values. */
latime_1.ops += (real) (*n + *n * *nrhs);
slascl_("G", &c__0, &c__0, &c_b11, &orgnrm, n, &c__1, &d__[1], n, info);
slasrt_("D", n, &d__[1], info);
slascl_("G", &c__0, &c__0, &orgnrm, &c_b11, n, nrhs, &b[b_offset], ldb,
info);
return 0;
/* End of SLALSD */
} /* slalsd_ */
/* Subroutine */ int slasd0_(integer *n, integer *sqre, real *d__, real *e,
real *u, integer *ldu, real *vt, integer *ldvt, integer *smlsiz,
integer *iwork, real *work, integer *info)
{
/* System generated locals */
integer u_dim1, u_offset, vt_dim1, vt_offset, i__1, i__2;
/* Builtin functions */
integer pow_ii(integer *, integer *);
/* Local variables */
integer i__, j, m, i1, ic, lf, nd, ll, nl, nr, im1, ncc, nlf, nrf, iwk,
lvl, ndb1, nlp1, nrp1;
real beta;
integer idxq, nlvl;
real alpha;
integer inode, ndiml, idxqc, ndimr, itemp, sqrei;
extern /* Subroutine */ int slasd1_(integer *, integer *, integer *, real
*, real *, real *, real *, integer *, real *, integer *, integer *
, integer *, real *, integer *), xerbla_(char *, integer *), slasdq_(char *, integer *, integer *, integer *, integer
*, integer *, real *, real *, real *, integer *, real *, integer *
, real *, integer *, real *, integer *), slasdt_(integer *
, integer *, integer *, integer *, integer *, integer *, integer *
);
/* -- LAPACK auxiliary routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* Using a divide and conquer approach, SLASD0 computes the singular */
/* value decomposition (SVD) of a real upper bidiagonal N-by-M */
/* matrix B with diagonal D and offdiagonal E, where M = N + SQRE. */
/* The algorithm computes orthogonal matrices U and VT such that */
/* B = U * S * VT. The singular values S are overwritten on D. */
/* A related subroutine, SLASDA, computes only the singular values, */
/* and optionally, the singular vectors in compact form. */
/* Arguments */
/* ========= */
/* N (input) INTEGER */
/* On entry, the row dimension of the upper bidiagonal matrix. */
/* This is also the dimension of the main diagonal array D. */
/* SQRE (input) INTEGER */
/* Specifies the column dimension of the bidiagonal matrix. */
/* = 0: The bidiagonal matrix has column dimension M = N; */
/* = 1: The bidiagonal matrix has column dimension M = N+1; */
/* D (input/output) REAL array, dimension (N) */
/* On entry D contains the main diagonal of the bidiagonal */
/* matrix. */
/* On exit D, if INFO = 0, contains its singular values. */
/* E (input) REAL array, dimension (M-1) */
/* Contains the subdiagonal entries of the bidiagonal matrix. */
/* On exit, E has been destroyed. */
/* U (output) REAL array, dimension at least (LDQ, N) */
/* On exit, U contains the left singular vectors. */
/* LDU (input) INTEGER */
/* On entry, leading dimension of U. */
/* VT (output) REAL array, dimension at least (LDVT, M) */
/* On exit, VT' contains the right singular vectors. */
/* LDVT (input) INTEGER */
/* On entry, leading dimension of VT. */
/* SMLSIZ (input) INTEGER */
/* On entry, maximum size of the subproblems at the */
/* bottom of the computation tree. */
/* IWORK (workspace) INTEGER array, dimension (8*N) */
/* WORK (workspace) REAL array, dimension (3*M**2+2*M) */
/* INFO (output) INTEGER */
/* = 0: successful exit. */
/* < 0: if INFO = -i, the i-th argument had an illegal value. */
/* > 0: if INFO = 1, an singular value did not converge */
/* Further Details */
/* =============== */
/* Based on contributions by */
/* Ming Gu and Huan Ren, Computer Science Division, University of */
/* California at Berkeley, USA */
/* ===================================================================== */
/* .. Local Scalars .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
--d__;
--e;
u_dim1 = *ldu;
u_offset = 1 + u_dim1;
u -= u_offset;
vt_dim1 = *ldvt;
vt_offset = 1 + vt_dim1;
vt -= vt_offset;
--iwork;
--work;
/* Function Body */
*info = 0;
if (*n < 0) {
*info = -1;
} else if (*sqre < 0 || *sqre > 1) {
*info = -2;
}
m = *n + *sqre;
if (*ldu < *n) {
*info = -6;
} else if (*ldvt < m) {
*info = -8;
} else if (*smlsiz < 3) {
*info = -9;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SLASD0", &i__1);
return 0;
}
/* If the input matrix is too small, call SLASDQ to find the SVD. */
if (*n <= *smlsiz) {
slasdq_("U", sqre, n, &m, n, &c__0, &d__[1], &e[1], &vt[vt_offset],
ldvt, &u[u_offset], ldu, &u[u_offset], ldu, &work[1], info);
return 0;
}
/* Set up the computation tree. */
inode = 1;
ndiml = inode + *n;
ndimr = ndiml + *n;
idxq = ndimr + *n;
iwk = idxq + *n;
slasdt_(n, &nlvl, &nd, &iwork[inode], &iwork[ndiml], &iwork[ndimr],
smlsiz);
/* For the nodes on bottom level of the tree, solve */
/* their subproblems by SLASDQ. */
ndb1 = (nd + 1) / 2;
ncc = 0;
i__1 = nd;
for (i__ = ndb1; i__ <= i__1; ++i__) {
/* IC : center row of each node */
/* NL : number of rows of left subproblem */
/* NR : number of rows of right subproblem */
/* NLF: starting row of the left subproblem */
/* NRF: starting row of the right subproblem */
i1 = i__ - 1;
ic = iwork[inode + i1];
nl = iwork[ndiml + i1];
nlp1 = nl + 1;
nr = iwork[ndimr + i1];
nrp1 = nr + 1;
nlf = ic - nl;
nrf = ic + 1;
sqrei = 1;
slasdq_("U", &sqrei, &nl, &nlp1, &nl, &ncc, &d__[nlf], &e[nlf], &vt[
nlf + nlf * vt_dim1], ldvt, &u[nlf + nlf * u_dim1], ldu, &u[
nlf + nlf * u_dim1], ldu, &work[1], info);
if (*info != 0) {
return 0;
}
itemp = idxq + nlf - 2;
i__2 = nl;
for (j = 1; j <= i__2; ++j) {
iwork[itemp + j] = j;
/* L10: */
}
if (i__ == nd) {
sqrei = *sqre;
} else {
sqrei = 1;
}
nrp1 = nr + sqrei;
slasdq_("U", &sqrei, &nr, &nrp1, &nr, &ncc, &d__[nrf], &e[nrf], &vt[
nrf + nrf * vt_dim1], ldvt, &u[nrf + nrf * u_dim1], ldu, &u[
nrf + nrf * u_dim1], ldu, &work[1], info);
if (*info != 0) {
return 0;
}
itemp = idxq + ic;
i__2 = nr;
for (j = 1; j <= i__2; ++j) {
iwork[itemp + j - 1] = j;
/* L20: */
}
/* L30: */
}
/* Now conquer each subproblem bottom-up. */
for (lvl = nlvl; lvl >= 1; --lvl) {
/* Find the first node LF and last node LL on the */
/* current level LVL. */
if (lvl == 1) {
lf = 1;
ll = 1;
} else {
i__1 = lvl - 1;
lf = pow_ii(&c__2, &i__1);
ll = (lf << 1) - 1;
}
i__1 = ll;
for (i__ = lf; i__ <= i__1; ++i__) {
im1 = i__ - 1;
ic = iwork[inode + im1];
nl = iwork[ndiml + im1];
nr = iwork[ndimr + im1];
nlf = ic - nl;
if (*sqre == 0 && i__ == ll) {
sqrei = *sqre;
} else {
sqrei = 1;
}
idxqc = idxq + nlf - 1;
alpha = d__[ic];
beta = e[ic];
slasd1_(&nl, &nr, &sqrei, &d__[nlf], &alpha, &beta, &u[nlf + nlf *
u_dim1], ldu, &vt[nlf + nlf * vt_dim1], ldvt, &iwork[
idxqc], &iwork[iwk], &work[1], info);
if (*info != 0) {
return 0;
}
/* L40: */
}
/* L50: */
}
return 0;
/* End of SLASD0 */
} /* slasd0_ */
/* Subroutine */ int clalsd_(char *uplo, integer *smlsiz, integer *n, integer
*nrhs, real *d__, real *e, complex *b, integer *ldb, real *rcond,
integer *rank, complex *work, real *rwork, integer *iwork, integer *
info)
{
/* System generated locals */
integer b_dim1, b_offset, i__1, i__2, i__3, i__4, i__5, i__6;
real r__1;
complex q__1;
/* Builtin functions */
double r_imag(complex *), log(doublereal), r_sign(real *, real *);
/* Local variables */
integer c__, i__, j, k;
real r__;
integer s, u, z__;
real cs;
integer bx;
real sn;
integer st, vt, nm1, st1;
real eps;
integer iwk;
real tol;
integer difl, difr;
real rcnd;
integer jcol, irwb, perm, nsub, nlvl, sqre, bxst, jrow, irwu, jimag,
jreal;
extern /* Subroutine */ int sgemm_(char *, char *, integer *, integer *,
integer *, real *, real *, integer *, real *, integer *, real *,
real *, integer *);
integer irwib;
extern /* Subroutine */ int ccopy_(integer *, complex *, integer *,
complex *, integer *);
integer poles, sizei, irwrb, nsize;
extern /* Subroutine */ int csrot_(integer *, complex *, integer *,
complex *, integer *, real *, real *);
integer irwvt, icmpq1, icmpq2;
extern /* Subroutine */ int clalsa_(integer *, integer *, integer *,
integer *, complex *, integer *, complex *, integer *, real *,
integer *, real *, integer *, real *, real *, real *, real *,
integer *, integer *, integer *, integer *, real *, real *, real *
, real *, integer *, integer *), clascl_(char *, integer *,
integer *, real *, real *, integer *, integer *, complex *,
integer *, integer *);
extern doublereal slamch_(char *);
extern /* Subroutine */ int slasda_(integer *, integer *, integer *,
integer *, real *, real *, real *, integer *, real *, integer *,
real *, real *, real *, real *, integer *, integer *, integer *,
integer *, real *, real *, real *, real *, integer *, integer *),
clacpy_(char *, integer *, integer *, complex *, integer *,
complex *, integer *), claset_(char *, integer *, integer
*, complex *, complex *, complex *, integer *), xerbla_(
char *, integer *), slascl_(char *, integer *, integer *,
real *, real *, integer *, integer *, real *, integer *, integer *
);
extern integer isamax_(integer *, real *, integer *);
integer givcol;
extern /* Subroutine */ int slasdq_(char *, integer *, integer *, integer
*, integer *, integer *, real *, real *, real *, integer *, real *
, integer *, real *, integer *, real *, integer *),
slaset_(char *, integer *, integer *, real *, real *, real *,
integer *), slartg_(real *, real *, real *, real *, real *
);
real orgnrm;
integer givnum;
extern doublereal slanst_(char *, integer *, real *, real *);
extern /* Subroutine */ int slasrt_(char *, integer *, real *, integer *);
integer givptr, nrwork, irwwrk, smlszp;
/* -- LAPACK routine (version 3.1) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* CLALSD uses the singular value decomposition of A to solve the least */
/* squares problem of finding X to minimize the Euclidean norm of each */
/* column of A*X-B, where A is N-by-N upper bidiagonal, and X and B */
/* are N-by-NRHS. The solution X overwrites B. */
/* The singular values of A smaller than RCOND times the largest */
/* singular value are treated as zero in solving the least squares */
/* problem; in this case a minimum norm solution is returned. */
/* The actual singular values are returned in D in ascending order. */
/* This code 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 XMP, Cray YMP, Cray C 90, or Cray 2. */
/* It could conceivably fail on hexadecimal or decimal machines */
/* without guard digits, but we know of none. */
/* Arguments */
/* ========= */
/* UPLO (input) CHARACTER*1 */
/* = 'U': D and E define an upper bidiagonal matrix. */
/* = 'L': D and E define a lower bidiagonal matrix. */
/* SMLSIZ (input) INTEGER */
/* The maximum size of the subproblems at the bottom of the */
/* computation tree. */
/* N (input) INTEGER */
/* The dimension of the bidiagonal matrix. N >= 0. */
/* NRHS (input) INTEGER */
/* The number of columns of B. NRHS must be at least 1. */
/* D (input/output) REAL array, dimension (N) */
/* On entry D contains the main diagonal of the bidiagonal */
/* matrix. On exit, if INFO = 0, D contains its singular values. */
/* E (input/output) REAL array, dimension (N-1) */
/* Contains the super-diagonal entries of the bidiagonal matrix. */
/* On exit, E has been destroyed. */
/* B (input/output) COMPLEX array, dimension (LDB,NRHS) */
/* On input, B contains the right hand sides of the least */
/* squares problem. On output, B contains the solution X. */
/* LDB (input) INTEGER */
/* The leading dimension of B in the calling subprogram. */
/* LDB must be at least max(1,N). */
/* RCOND (input) REAL */
/* The singular values of A less than or equal to RCOND times */
/* the largest singular value are treated as zero in solving */
/* the least squares problem. If RCOND is negative, */
/* machine precision is used instead. */
/* For example, if diag(S)*X=B were the least squares problem, */
/* where diag(S) is a diagonal matrix of singular values, the */
/* solution would be X(i) = B(i) / S(i) if S(i) is greater than */
/* RCOND*max(S), and X(i) = 0 if S(i) is less than or equal to */
/* RCOND*max(S). */
/* RANK (output) INTEGER */
/* The number of singular values of A greater than RCOND times */
/* the largest singular value. */
/* WORK (workspace) COMPLEX array, dimension (N * NRHS). */
/* RWORK (workspace) REAL array, dimension at least */
/* (9*N + 2*N*SMLSIZ + 8*N*NLVL + 3*SMLSIZ*NRHS + (SMLSIZ+1)**2), */
/* where */
/* NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 ) */
/* IWORK (workspace) INTEGER array, dimension (3*N*NLVL + 11*N). */
/* INFO (output) INTEGER */
/* = 0: successful exit. */
/* < 0: if INFO = -i, the i-th argument had an illegal value. */
/* > 0: The algorithm failed to compute an singular value while */
/* working on the submatrix lying in rows and columns */
/* INFO/(N+1) through MOD(INFO,N+1). */
/* 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 */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
--d__;
--e;
b_dim1 = *ldb;
b_offset = 1 + b_dim1;
b -= b_offset;
--work;
--rwork;
--iwork;
/* Function Body */
*info = 0;
if (*n < 0) {
*info = -3;
} else if (*nrhs < 1) {
*info = -4;
} else if (*ldb < 1 || *ldb < *n) {
*info = -8;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CLALSD", &i__1);
return 0;
}
eps = slamch_("Epsilon");
/* Set up the tolerance. */
if (*rcond <= 0.f || *rcond >= 1.f) {
rcnd = eps;
} else {
rcnd = *rcond;
}
*rank = 0;
/* Quick return if possible. */
if (*n == 0) {
return 0;
} else if (*n == 1) {
if (d__[1] == 0.f) {
claset_("A", &c__1, nrhs, &c_b1, &c_b1, &b[b_offset], ldb);
} else {
*rank = 1;
clascl_("G", &c__0, &c__0, &d__[1], &c_b10, &c__1, nrhs, &b[
b_offset], ldb, info);
d__[1] = dabs(d__[1]);
}
return 0;
}
/* Rotate the matrix if it is lower bidiagonal. */
if (*(unsigned char *)uplo == 'L') {
i__1 = *n - 1;
for (i__ = 1; i__ <= i__1; ++i__) {
slartg_(&d__[i__], &e[i__], &cs, &sn, &r__);
d__[i__] = r__;
e[i__] = sn * d__[i__ + 1];
d__[i__ + 1] = cs * d__[i__ + 1];
if (*nrhs == 1) {
csrot_(&c__1, &b[i__ + b_dim1], &c__1, &b[i__ + 1 + b_dim1], &
c__1, &cs, &sn);
} else {
rwork[(i__ << 1) - 1] = cs;
rwork[i__ * 2] = sn;
}
/* L10: */
}
if (*nrhs > 1) {
i__1 = *nrhs;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = *n - 1;
for (j = 1; j <= i__2; ++j) {
cs = rwork[(j << 1) - 1];
sn = rwork[j * 2];
csrot_(&c__1, &b[j + i__ * b_dim1], &c__1, &b[j + 1 + i__
* b_dim1], &c__1, &cs, &sn);
/* L20: */
}
/* L30: */
}
}
}
/* Scale. */
nm1 = *n - 1;
orgnrm = slanst_("M", n, &d__[1], &e[1]);
if (orgnrm == 0.f) {
claset_("A", n, nrhs, &c_b1, &c_b1, &b[b_offset], ldb);
return 0;
}
slascl_("G", &c__0, &c__0, &orgnrm, &c_b10, n, &c__1, &d__[1], n, info);
slascl_("G", &c__0, &c__0, &orgnrm, &c_b10, &nm1, &c__1, &e[1], &nm1,
info);
/* If N is smaller than the minimum divide size SMLSIZ, then solve */
/* the problem with another solver. */
if (*n <= *smlsiz) {
irwu = 1;
irwvt = irwu + *n * *n;
irwwrk = irwvt + *n * *n;
irwrb = irwwrk;
irwib = irwrb + *n * *nrhs;
irwb = irwib + *n * *nrhs;
slaset_("A", n, n, &c_b35, &c_b10, &rwork[irwu], n);
slaset_("A", n, n, &c_b35, &c_b10, &rwork[irwvt], n);
slasdq_("U", &c__0, n, n, n, &c__0, &d__[1], &e[1], &rwork[irwvt], n,
&rwork[irwu], n, &rwork[irwwrk], &c__1, &rwork[irwwrk], info);
if (*info != 0) {
return 0;
}
/* In the real version, B is passed to SLASDQ and multiplied */
/* internally by Q'. Here B is complex and that product is */
/* computed below in two steps (real and imaginary parts). */
j = irwb - 1;
i__1 = *nrhs;
for (jcol = 1; jcol <= i__1; ++jcol) {
i__2 = *n;
for (jrow = 1; jrow <= i__2; ++jrow) {
++j;
i__3 = jrow + jcol * b_dim1;
rwork[j] = b[i__3].r;
/* L40: */
}
/* L50: */
}
sgemm_("T", "N", n, nrhs, n, &c_b10, &rwork[irwu], n, &rwork[irwb], n,
&c_b35, &rwork[irwrb], n);
j = irwb - 1;
i__1 = *nrhs;
for (jcol = 1; jcol <= i__1; ++jcol) {
i__2 = *n;
for (jrow = 1; jrow <= i__2; ++jrow) {
++j;
rwork[j] = r_imag(&b[jrow + jcol * b_dim1]);
/* L60: */
}
/* L70: */
}
sgemm_("T", "N", n, nrhs, n, &c_b10, &rwork[irwu], n, &rwork[irwb], n,
&c_b35, &rwork[irwib], n);
jreal = irwrb - 1;
jimag = irwib - 1;
i__1 = *nrhs;
for (jcol = 1; jcol <= i__1; ++jcol) {
i__2 = *n;
for (jrow = 1; jrow <= i__2; ++jrow) {
++jreal;
++jimag;
i__3 = jrow + jcol * b_dim1;
i__4 = jreal;
i__5 = jimag;
q__1.r = rwork[i__4], q__1.i = rwork[i__5];
b[i__3].r = q__1.r, b[i__3].i = q__1.i;
/* L80: */
}
/* L90: */
}
tol = rcnd * (r__1 = d__[isamax_(n, &d__[1], &c__1)], dabs(r__1));
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
if (d__[i__] <= tol) {
claset_("A", &c__1, nrhs, &c_b1, &c_b1, &b[i__ + b_dim1], ldb);
} else {
clascl_("G", &c__0, &c__0, &d__[i__], &c_b10, &c__1, nrhs, &b[
i__ + b_dim1], ldb, info);
++(*rank);
}
/* L100: */
}
/* Since B is complex, the following call to SGEMM is performed */
/* in two steps (real and imaginary parts). That is for V * B */
/* (in the real version of the code V' is stored in WORK). */
/* CALL SGEMM( 'T', 'N', N, NRHS, N, ONE, WORK, N, B, LDB, ZERO, */
/* $ WORK( NWORK ), N ) */
j = irwb - 1;
i__1 = *nrhs;
for (jcol = 1; jcol <= i__1; ++jcol) {
i__2 = *n;
for (jrow = 1; jrow <= i__2; ++jrow) {
++j;
i__3 = jrow + jcol * b_dim1;
rwork[j] = b[i__3].r;
/* L110: */
}
/* L120: */
}
sgemm_("T", "N", n, nrhs, n, &c_b10, &rwork[irwvt], n, &rwork[irwb],
n, &c_b35, &rwork[irwrb], n);
j = irwb - 1;
i__1 = *nrhs;
for (jcol = 1; jcol <= i__1; ++jcol) {
i__2 = *n;
for (jrow = 1; jrow <= i__2; ++jrow) {
++j;
rwork[j] = r_imag(&b[jrow + jcol * b_dim1]);
/* L130: */
}
/* L140: */
}
sgemm_("T", "N", n, nrhs, n, &c_b10, &rwork[irwvt], n, &rwork[irwb],
n, &c_b35, &rwork[irwib], n);
jreal = irwrb - 1;
jimag = irwib - 1;
i__1 = *nrhs;
for (jcol = 1; jcol <= i__1; ++jcol) {
i__2 = *n;
for (jrow = 1; jrow <= i__2; ++jrow) {
++jreal;
++jimag;
i__3 = jrow + jcol * b_dim1;
i__4 = jreal;
i__5 = jimag;
q__1.r = rwork[i__4], q__1.i = rwork[i__5];
b[i__3].r = q__1.r, b[i__3].i = q__1.i;
/* L150: */
}
/* L160: */
}
/* Unscale. */
slascl_("G", &c__0, &c__0, &c_b10, &orgnrm, n, &c__1, &d__[1], n,
info);
slasrt_("D", n, &d__[1], info);
clascl_("G", &c__0, &c__0, &orgnrm, &c_b10, n, nrhs, &b[b_offset],
ldb, info);
return 0;
}
/* Book-keeping and setting up some constants. */
nlvl = (integer) (log((real) (*n) / (real) (*smlsiz + 1)) / log(2.f)) + 1;
smlszp = *smlsiz + 1;
u = 1;
vt = *smlsiz * *n + 1;
difl = vt + smlszp * *n;
difr = difl + nlvl * *n;
z__ = difr + (nlvl * *n << 1);
c__ = z__ + nlvl * *n;
s = c__ + *n;
poles = s + *n;
givnum = poles + (nlvl << 1) * *n;
nrwork = givnum + (nlvl << 1) * *n;
bx = 1;
irwrb = nrwork;
irwib = irwrb + *smlsiz * *nrhs;
irwb = irwib + *smlsiz * *nrhs;
sizei = *n + 1;
k = sizei + *n;
givptr = k + *n;
perm = givptr + *n;
givcol = perm + nlvl * *n;
iwk = givcol + (nlvl * *n << 1);
st = 1;
sqre = 0;
icmpq1 = 1;
icmpq2 = 0;
nsub = 0;
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
if ((r__1 = d__[i__], dabs(r__1)) < eps) {
d__[i__] = r_sign(&eps, &d__[i__]);
}
/* L170: */
}
i__1 = nm1;
for (i__ = 1; i__ <= i__1; ++i__) {
if ((r__1 = e[i__], dabs(r__1)) < eps || i__ == nm1) {
++nsub;
iwork[nsub] = st;
/* Subproblem found. First determine its size and then */
/* apply divide and conquer on it. */
if (i__ < nm1) {
/* A subproblem with E(I) small for I < NM1. */
nsize = i__ - st + 1;
iwork[sizei + nsub - 1] = nsize;
} else if ((r__1 = e[i__], dabs(r__1)) >= eps) {
/* A subproblem with E(NM1) not too small but I = NM1. */
nsize = *n - st + 1;
iwork[sizei + nsub - 1] = nsize;
} else {
/* A subproblem with E(NM1) small. This implies an */
/* 1-by-1 subproblem at D(N), which is not solved */
/* explicitly. */
nsize = i__ - st + 1;
iwork[sizei + nsub - 1] = nsize;
++nsub;
iwork[nsub] = *n;
iwork[sizei + nsub - 1] = 1;
ccopy_(nrhs, &b[*n + b_dim1], ldb, &work[bx + nm1], n);
}
st1 = st - 1;
if (nsize == 1) {
/* This is a 1-by-1 subproblem and is not solved */
/* explicitly. */
ccopy_(nrhs, &b[st + b_dim1], ldb, &work[bx + st1], n);
} else if (nsize <= *smlsiz) {
/* This is a small subproblem and is solved by SLASDQ. */
slaset_("A", &nsize, &nsize, &c_b35, &c_b10, &rwork[vt + st1],
n);
slaset_("A", &nsize, &nsize, &c_b35, &c_b10, &rwork[u + st1],
n);
slasdq_("U", &c__0, &nsize, &nsize, &nsize, &c__0, &d__[st], &
e[st], &rwork[vt + st1], n, &rwork[u + st1], n, &
rwork[nrwork], &c__1, &rwork[nrwork], info)
;
if (*info != 0) {
return 0;
}
/* In the real version, B is passed to SLASDQ and multiplied */
/* internally by Q'. Here B is complex and that product is */
/* computed below in two steps (real and imaginary parts). */
j = irwb - 1;
i__2 = *nrhs;
for (jcol = 1; jcol <= i__2; ++jcol) {
i__3 = st + nsize - 1;
for (jrow = st; jrow <= i__3; ++jrow) {
++j;
i__4 = jrow + jcol * b_dim1;
rwork[j] = b[i__4].r;
/* L180: */
}
/* L190: */
}
sgemm_("T", "N", &nsize, nrhs, &nsize, &c_b10, &rwork[u + st1]
, n, &rwork[irwb], &nsize, &c_b35, &rwork[irwrb], &
nsize);
j = irwb - 1;
i__2 = *nrhs;
for (jcol = 1; jcol <= i__2; ++jcol) {
i__3 = st + nsize - 1;
for (jrow = st; jrow <= i__3; ++jrow) {
++j;
rwork[j] = r_imag(&b[jrow + jcol * b_dim1]);
/* L200: */
}
/* L210: */
}
sgemm_("T", "N", &nsize, nrhs, &nsize, &c_b10, &rwork[u + st1]
, n, &rwork[irwb], &nsize, &c_b35, &rwork[irwib], &
nsize);
jreal = irwrb - 1;
jimag = irwib - 1;
i__2 = *nrhs;
for (jcol = 1; jcol <= i__2; ++jcol) {
i__3 = st + nsize - 1;
for (jrow = st; jrow <= i__3; ++jrow) {
++jreal;
++jimag;
i__4 = jrow + jcol * b_dim1;
i__5 = jreal;
i__6 = jimag;
q__1.r = rwork[i__5], q__1.i = rwork[i__6];
b[i__4].r = q__1.r, b[i__4].i = q__1.i;
/* L220: */
}
/* L230: */
}
clacpy_("A", &nsize, nrhs, &b[st + b_dim1], ldb, &work[bx +
st1], n);
} else {
/* A large problem. Solve it using divide and conquer. */
slasda_(&icmpq1, smlsiz, &nsize, &sqre, &d__[st], &e[st], &
rwork[u + st1], n, &rwork[vt + st1], &iwork[k + st1],
&rwork[difl + st1], &rwork[difr + st1], &rwork[z__ +
st1], &rwork[poles + st1], &iwork[givptr + st1], &
iwork[givcol + st1], n, &iwork[perm + st1], &rwork[
givnum + st1], &rwork[c__ + st1], &rwork[s + st1], &
rwork[nrwork], &iwork[iwk], info);
if (*info != 0) {
return 0;
}
bxst = bx + st1;
clalsa_(&icmpq2, smlsiz, &nsize, nrhs, &b[st + b_dim1], ldb, &
work[bxst], n, &rwork[u + st1], n, &rwork[vt + st1], &
iwork[k + st1], &rwork[difl + st1], &rwork[difr + st1]
, &rwork[z__ + st1], &rwork[poles + st1], &iwork[
givptr + st1], &iwork[givcol + st1], n, &iwork[perm +
st1], &rwork[givnum + st1], &rwork[c__ + st1], &rwork[
s + st1], &rwork[nrwork], &iwork[iwk], info);
if (*info != 0) {
return 0;
}
}
st = i__ + 1;
}
/* L240: */
}
/* Apply the singular values and treat the tiny ones as zero. */
tol = rcnd * (r__1 = d__[isamax_(n, &d__[1], &c__1)], dabs(r__1));
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
/* Some of the elements in D can be negative because 1-by-1 */
/* subproblems were not solved explicitly. */
if ((r__1 = d__[i__], dabs(r__1)) <= tol) {
claset_("A", &c__1, nrhs, &c_b1, &c_b1, &work[bx + i__ - 1], n);
} else {
++(*rank);
clascl_("G", &c__0, &c__0, &d__[i__], &c_b10, &c__1, nrhs, &work[
bx + i__ - 1], n, info);
}
d__[i__] = (r__1 = d__[i__], dabs(r__1));
/* L250: */
}
/* Now apply back the right singular vectors. */
icmpq2 = 1;
i__1 = nsub;
for (i__ = 1; i__ <= i__1; ++i__) {
st = iwork[i__];
st1 = st - 1;
nsize = iwork[sizei + i__ - 1];
bxst = bx + st1;
if (nsize == 1) {
ccopy_(nrhs, &work[bxst], n, &b[st + b_dim1], ldb);
} else if (nsize <= *smlsiz) {
/* Since B and BX are complex, the following call to SGEMM */
/* is performed in two steps (real and imaginary parts). */
/* CALL SGEMM( 'T', 'N', NSIZE, NRHS, NSIZE, ONE, */
/* $ RWORK( VT+ST1 ), N, RWORK( BXST ), N, ZERO, */
/* $ B( ST, 1 ), LDB ) */
j = bxst - *n - 1;
jreal = irwb - 1;
i__2 = *nrhs;
for (jcol = 1; jcol <= i__2; ++jcol) {
j += *n;
i__3 = nsize;
for (jrow = 1; jrow <= i__3; ++jrow) {
++jreal;
i__4 = j + jrow;
rwork[jreal] = work[i__4].r;
/* L260: */
}
/* L270: */
}
sgemm_("T", "N", &nsize, nrhs, &nsize, &c_b10, &rwork[vt + st1],
n, &rwork[irwb], &nsize, &c_b35, &rwork[irwrb], &nsize);
j = bxst - *n - 1;
jimag = irwb - 1;
i__2 = *nrhs;
for (jcol = 1; jcol <= i__2; ++jcol) {
j += *n;
i__3 = nsize;
for (jrow = 1; jrow <= i__3; ++jrow) {
++jimag;
rwork[jimag] = r_imag(&work[j + jrow]);
/* L280: */
}
/* L290: */
}
sgemm_("T", "N", &nsize, nrhs, &nsize, &c_b10, &rwork[vt + st1],
n, &rwork[irwb], &nsize, &c_b35, &rwork[irwib], &nsize);
jreal = irwrb - 1;
jimag = irwib - 1;
i__2 = *nrhs;
for (jcol = 1; jcol <= i__2; ++jcol) {
i__3 = st + nsize - 1;
for (jrow = st; jrow <= i__3; ++jrow) {
++jreal;
++jimag;
i__4 = jrow + jcol * b_dim1;
i__5 = jreal;
i__6 = jimag;
q__1.r = rwork[i__5], q__1.i = rwork[i__6];
b[i__4].r = q__1.r, b[i__4].i = q__1.i;
/* L300: */
}
/* L310: */
}
} else {
clalsa_(&icmpq2, smlsiz, &nsize, nrhs, &work[bxst], n, &b[st +
b_dim1], ldb, &rwork[u + st1], n, &rwork[vt + st1], &
iwork[k + st1], &rwork[difl + st1], &rwork[difr + st1], &
rwork[z__ + st1], &rwork[poles + st1], &iwork[givptr +
st1], &iwork[givcol + st1], n, &iwork[perm + st1], &rwork[
givnum + st1], &rwork[c__ + st1], &rwork[s + st1], &rwork[
nrwork], &iwork[iwk], info);
if (*info != 0) {
return 0;
}
}
/* L320: */
}
/* Unscale and sort the singular values. */
slascl_("G", &c__0, &c__0, &c_b10, &orgnrm, n, &c__1, &d__[1], n, info);
slasrt_("D", n, &d__[1], info);
clascl_("G", &c__0, &c__0, &orgnrm, &c_b10, n, nrhs, &b[b_offset], ldb,
info);
return 0;
/* End of CLALSD */
} /* clalsd_ */
/* Subroutine */ int sbdsdc_(char *uplo, char *compq, integer *n, real *d__,
real *e, real *u, integer *ldu, real *vt, integer *ldvt, real *q,
integer *iq, real *work, integer *iwork, integer *info)
{
/* System generated locals */
integer u_dim1, u_offset, vt_dim1, vt_offset, i__1, i__2;
real r__1;
/* Builtin functions */
double r_sign(real *, real *), log(doublereal);
/* Local variables */
static integer difl, difr, ierr, perm, mlvl, sqre, i__, j, k;
static real p, r__;
static integer z__;
extern logical lsame_(char *, char *);
static integer poles;
extern /* Subroutine */ int slasr_(char *, char *, char *, integer *,
integer *, real *, real *, real *, integer *);
static integer iuplo, nsize, start;
extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *,
integer *), sswap_(integer *, real *, integer *, real *, integer *
), slasd0_(integer *, integer *, real *, real *, real *, integer *
, real *, integer *, integer *, integer *, real *, integer *);
static integer ic, ii, kk;
static real cs;
static integer is, iu;
static real sn;
extern doublereal slamch_(char *);
extern /* Subroutine */ int slasda_(integer *, integer *, integer *,
integer *, real *, real *, real *, integer *, real *, integer *,
real *, real *, real *, real *, integer *, integer *, integer *,
integer *, real *, real *, real *, real *, integer *, integer *),
xerbla_(char *, integer *);
extern integer ilaenv_(integer *, char *, char *, integer *, integer *,
integer *, integer *, ftnlen, ftnlen);
extern /* Subroutine */ int slascl_(char *, integer *, integer *, real *,
real *, integer *, integer *, real *, integer *, integer *);
static integer givcol;
extern /* Subroutine */ int slasdq_(char *, integer *, integer *, integer
*, integer *, integer *, real *, real *, real *, integer *, real *
, integer *, real *, integer *, real *, integer *);
static integer icompq;
extern /* Subroutine */ int slaset_(char *, integer *, integer *, real *,
real *, real *, integer *), slartg_(real *, real *, real *
, real *, real *);
static real orgnrm;
static integer givnum;
extern doublereal slanst_(char *, integer *, real *, real *);
static integer givptr, nm1, qstart, smlsiz, wstart, smlszp;
static real eps;
static integer ivt;
#define u_ref(a_1,a_2) u[(a_2)*u_dim1 + a_1]
#define vt_ref(a_1,a_2) vt[(a_2)*vt_dim1 + a_1]
/* -- LAPACK routine (instrumented to count ops, version 3.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
October 31, 1999
Purpose
=======
SBDSDC computes the singular value decomposition (SVD) of a real
N-by-N (upper or lower) bidiagonal matrix B: B = U * S * VT,
using a divide and conquer method, where S is a diagonal matrix
with non-negative diagonal elements (the singular values of B), and
U and VT are orthogonal matrices of left and right singular vectors,
respectively. SBDSDC can be used to compute all singular values,
and optionally, singular vectors or singular vectors in compact form.
This code 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. See SLASD3 for details.
The code currently call SLASDQ if singular values only are desired.
However, it can be slightly modified to compute singular values
using the divide and conquer method.
Arguments
=========
UPLO (input) CHARACTER*1
= 'U': B is upper bidiagonal.
= 'L': B is lower bidiagonal.
COMPQ (input) CHARACTER*1
Specifies whether singular vectors are to be computed
as follows:
= 'N': Compute singular values only;
= 'P': Compute singular values and compute singular
vectors in compact form;
= 'I': Compute singular values and singular vectors.
N (input) INTEGER
The order of the matrix B. N >= 0.
D (input/output) REAL array, dimension (N)
On entry, the n diagonal elements of the bidiagonal matrix B.
On exit, if INFO=0, the singular values of B.
E (input/output) REAL array, dimension (N)
On entry, the elements of E contain the offdiagonal
elements of the bidiagonal matrix whose SVD is desired.
On exit, E has been destroyed.
U (output) REAL array, dimension (LDU,N)
If COMPQ = 'I', then:
On exit, if INFO = 0, U contains the left singular vectors
of the bidiagonal matrix.
For other values of COMPQ, U is not referenced.
LDU (input) INTEGER
The leading dimension of the array U. LDU >= 1.
If singular vectors are desired, then LDU >= max( 1, N ).
VT (output) REAL array, dimension (LDVT,N)
If COMPQ = 'I', then:
On exit, if INFO = 0, VT' contains the right singular
vectors of the bidiagonal matrix.
For other values of COMPQ, VT is not referenced.
LDVT (input) INTEGER
The leading dimension of the array VT. LDVT >= 1.
If singular vectors are desired, then LDVT >= max( 1, N ).
Q (output) REAL array, dimension (LDQ)
If COMPQ = 'P', then:
On exit, if INFO = 0, Q and IQ contain the left
and right singular vectors in a compact form,
requiring O(N log N) space instead of 2*N**2.
In particular, Q contains all the REAL data in
LDQ >= N*(11 + 2*SMLSIZ + 8*INT(LOG_2(N/(SMLSIZ+1))))
words of memory, where 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).
For other values of COMPQ, Q is not referenced.
IQ (output) INTEGER array, dimension (LDIQ)
If COMPQ = 'P', then:
On exit, if INFO = 0, Q and IQ contain the left
and right singular vectors in a compact form,
requiring O(N log N) space instead of 2*N**2.
In particular, IQ contains all INTEGER data in
LDIQ >= N*(3 + 3*INT(LOG_2(N/(SMLSIZ+1))))
words of memory, where 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).
For other values of COMPQ, IQ is not referenced.
WORK (workspace) REAL array, dimension (LWORK)
If COMPQ = 'N' then LWORK >= (4 * N).
If COMPQ = 'P' then LWORK >= (6 * N).
If COMPQ = 'I' then LWORK >= (3 * N**2 + 4 * N).
IWORK (workspace) INTEGER array, dimension (7*N)
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: The algorithm failed to compute an singular value.
The update process of divide and conquer failed.
Further Details
===============
Based on contributions by
Ming Gu and Huan Ren, Computer Science Division, University of
California at Berkeley, USA
=====================================================================
Test the input parameters.
Parameter adjustments */
--d__;
--e;
u_dim1 = *ldu;
u_offset = 1 + u_dim1 * 1;
u -= u_offset;
vt_dim1 = *ldvt;
vt_offset = 1 + vt_dim1 * 1;
vt -= vt_offset;
--q;
--iq;
--work;
--iwork;
/* Function Body */
*info = 0;
iuplo = 0;
if (lsame_(uplo, "U")) {
iuplo = 1;
}
if (lsame_(uplo, "L")) {
iuplo = 2;
}
if (lsame_(compq, "N")) {
icompq = 0;
} else if (lsame_(compq, "P")) {
icompq = 1;
} else if (lsame_(compq, "I")) {
icompq = 2;
} else {
icompq = -1;
}
if (iuplo == 0) {
*info = -1;
} else if (icompq < 0) {
*info = -2;
} else if (*n < 0) {
*info = -3;
} else if (*ldu < 1 || icompq == 2 && *ldu < *n) {
*info = -7;
} else if (*ldvt < 1 || icompq == 2 && *ldvt < *n) {
*info = -9;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SBDSDC", &i__1);
return 0;
}
/* Quick return if possible */
if (*n == 0) {
return 0;
}
smlsiz = ilaenv_(&c__9, "SBDSDC", " ", &c__0, &c__0, &c__0, &c__0, (
ftnlen)6, (ftnlen)1);
if (*n == 1) {
if (icompq == 1) {
q[1] = r_sign(&c_b15, &d__[1]);
q[smlsiz * *n + 1] = 1.f;
} else if (icompq == 2) {
u_ref(1, 1) = r_sign(&c_b15, &d__[1]);
vt_ref(1, 1) = 1.f;
}
d__[1] = dabs(d__[1]);
return 0;
}
nm1 = *n - 1;
/* If matrix lower bidiagonal, rotate to be upper bidiagonal
by applying Givens rotations on the left */
wstart = 1;
qstart = 3;
if (icompq == 1) {
scopy_(n, &d__[1], &c__1, &q[1], &c__1);
i__1 = *n - 1;
scopy_(&i__1, &e[1], &c__1, &q[*n + 1], &c__1);
}
if (iuplo == 2) {
qstart = 5;
wstart = (*n << 1) - 1;
latime_1.ops += (real) (*n - 1 << 3);
i__1 = *n - 1;
for (i__ = 1; i__ <= i__1; ++i__) {
slartg_(&d__[i__], &e[i__], &cs, &sn, &r__);
d__[i__] = r__;
e[i__] = sn * d__[i__ + 1];
d__[i__ + 1] = cs * d__[i__ + 1];
if (icompq == 1) {
q[i__ + (*n << 1)] = cs;
q[i__ + *n * 3] = sn;
} else if (icompq == 2) {
work[i__] = cs;
work[nm1 + i__] = -sn;
}
/* L10: */
}
}
/* If ICOMPQ = 0, use SLASDQ to compute the singular values. */
if (icompq == 0) {
slasdq_("U", &c__0, n, &c__0, &c__0, &c__0, &d__[1], &e[1], &vt[
vt_offset], ldvt, &u[u_offset], ldu, &u[u_offset], ldu, &work[
wstart], info);
goto L40;
}
/* If N is smaller than the minimum divide size SMLSIZ, then solve
the problem with another solver. */
if (*n <= smlsiz) {
if (icompq == 2) {
slaset_("A", n, n, &c_b29, &c_b15, &u[u_offset], ldu);
slaset_("A", n, n, &c_b29, &c_b15, &vt[vt_offset], ldvt);
slasdq_("U", &c__0, n, n, n, &c__0, &d__[1], &e[1], &vt[vt_offset]
, ldvt, &u[u_offset], ldu, &u[u_offset], ldu, &work[
wstart], info);
} else if (icompq == 1) {
iu = 1;
ivt = iu + *n;
slaset_("A", n, n, &c_b29, &c_b15, &q[iu + (qstart - 1) * *n], n);
slaset_("A", n, n, &c_b29, &c_b15, &q[ivt + (qstart - 1) * *n], n);
slasdq_("U", &c__0, n, n, n, &c__0, &d__[1], &e[1], &q[ivt + (
qstart - 1) * *n], n, &q[iu + (qstart - 1) * *n], n, &q[
iu + (qstart - 1) * *n], n, &work[wstart], info);
}
goto L40;
}
if (icompq == 2) {
slaset_("A", n, n, &c_b29, &c_b15, &u[u_offset], ldu);
slaset_("A", n, n, &c_b29, &c_b15, &vt[vt_offset], ldvt);
}
/* Scale. */
orgnrm = slanst_("M", n, &d__[1], &e[1]);
if (orgnrm == 0.f) {
return 0;
}
latime_1.ops += (real) (*n + nm1);
slascl_("G", &c__0, &c__0, &orgnrm, &c_b15, n, &c__1, &d__[1], n, &ierr);
slascl_("G", &c__0, &c__0, &orgnrm, &c_b15, &nm1, &c__1, &e[1], &nm1, &
ierr);
eps = slamch_("Epsilon");
mlvl = (integer) (log((real) (*n) / (real) (smlsiz + 1)) / log(2.f)) + 1;
smlszp = smlsiz + 1;
if (icompq == 1) {
iu = 1;
ivt = smlsiz + 1;
difl = ivt + smlszp;
difr = difl + mlvl;
z__ = difr + (mlvl << 1);
ic = z__ + mlvl;
is = ic + 1;
poles = is + 1;
givnum = poles + (mlvl << 1);
k = 1;
givptr = 2;
perm = 3;
givcol = perm + mlvl;
}
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
if ((r__1 = d__[i__], dabs(r__1)) < eps) {
d__[i__] = r_sign(&eps, &d__[i__]);
}
/* L20: */
}
start = 1;
sqre = 0;
i__1 = nm1;
for (i__ = 1; i__ <= i__1; ++i__) {
if ((r__1 = e[i__], dabs(r__1)) < eps || i__ == nm1) {
/* Subproblem found. First determine its size and then
apply divide and conquer on it. */
if (i__ < nm1) {
/* A subproblem with E(I) small for I < NM1. */
nsize = i__ - start + 1;
} else if ((r__1 = e[i__], dabs(r__1)) >= eps) {
/* A subproblem with E(NM1) not too small but I = NM1. */
nsize = *n - start + 1;
} else {
/* A subproblem with E(NM1) small. This implies an
1-by-1 subproblem at D(N). Solve this 1-by-1 problem
first. */
nsize = i__ - start + 1;
if (icompq == 2) {
u_ref(*n, *n) = r_sign(&c_b15, &d__[*n]);
vt_ref(*n, *n) = 1.f;
} else if (icompq == 1) {
q[*n + (qstart - 1) * *n] = r_sign(&c_b15, &d__[*n]);
q[*n + (smlsiz + qstart - 1) * *n] = 1.f;
}
d__[*n] = (r__1 = d__[*n], dabs(r__1));
}
if (icompq == 2) {
slasd0_(&nsize, &sqre, &d__[start], &e[start], &u_ref(start,
start), ldu, &vt_ref(start, start), ldvt, &smlsiz, &
iwork[1], &work[wstart], info);
} else {
slasda_(&icompq, &smlsiz, &nsize, &sqre, &d__[start], &e[
start], &q[start + (iu + qstart - 2) * *n], n, &q[
start + (ivt + qstart - 2) * *n], &iq[start + k * *n],
&q[start + (difl + qstart - 2) * *n], &q[start + (
difr + qstart - 2) * *n], &q[start + (z__ + qstart -
2) * *n], &q[start + (poles + qstart - 2) * *n], &iq[
start + givptr * *n], &iq[start + givcol * *n], n, &
iq[start + perm * *n], &q[start + (givnum + qstart -
2) * *n], &q[start + (ic + qstart - 2) * *n], &q[
start + (is + qstart - 2) * *n], &work[wstart], &
iwork[1], info);
if (*info != 0) {
return 0;
}
}
start = i__ + 1;
}
/* L30: */
}
/* Unscale */
latime_1.ops += (real) (*n);
slascl_("G", &c__0, &c__0, &c_b15, &orgnrm, n, &c__1, &d__[1], n, &ierr);
L40:
/* Use Selection Sort to minimize swaps of singular vectors */
i__1 = *n;
for (ii = 2; ii <= i__1; ++ii) {
i__ = ii - 1;
kk = i__;
p = d__[i__];
i__2 = *n;
for (j = ii; j <= i__2; ++j) {
if (d__[j] > p) {
kk = j;
p = d__[j];
}
/* L50: */
}
if (kk != i__) {
d__[kk] = d__[i__];
d__[i__] = p;
if (icompq == 1) {
iq[i__] = kk;
} else if (icompq == 2) {
sswap_(n, &u_ref(1, i__), &c__1, &u_ref(1, kk), &c__1);
sswap_(n, &vt_ref(i__, 1), ldvt, &vt_ref(kk, 1), ldvt);
}
} else if (icompq == 1) {
iq[i__] = i__;
}
/* L60: */
}
/* If ICOMPQ = 1, use IQ(N,1) as the indicator for UPLO */
if (icompq == 1) {
if (iuplo == 1) {
iq[*n] = 1;
} else {
iq[*n] = 0;
}
}
/* If B is lower bidiagonal, update U by those Givens rotations
which rotated B to be upper bidiagonal */
if (iuplo == 2 && icompq == 2) {
latime_1.ops += (real) ((*n - 1) * 6 * *n);
slasr_("L", "V", "B", n, n, &work[1], &work[*n], &u[u_offset], ldu);
}
return 0;
/* End of SBDSDC */
} /* sbdsdc_ */