/* Subroutine */ int clalsa_(integer *icompq, integer *smlsiz, integer *n,
integer *nrhs, complex *b, integer *ldb, complex *bx, integer *ldbx,
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 *rwork,
integer *iwork, integer *info)
{
/* -- LAPACK routine (version 3.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
June 30, 1999
Purpose
=======
CLALSA is an itermediate step in solving the least squares problem
by computing the SVD of the coefficient matrix in compact form (The
singular vectors are computed as products of simple orthorgonal
matrices.).
If ICOMPQ = 0, CLALSA applies the inverse of the left singular vector
matrix of an upper bidiagonal matrix to the right hand side; and if
ICOMPQ = 1, CLALSA applies the right singular vector matrix to the
right hand side. The singular vector matrices were generated in
compact form by CLALSA.
Arguments
=========
ICOMPQ (input) INTEGER
Specifies whether the left or the right singular vector
matrix is involved.
= 0: Left singular vector matrix
= 1: Right singular vector matrix
SMLSIZ (input) INTEGER
The maximum size of the subproblems at the bottom of the
computation tree.
N (input) INTEGER
The row and column dimensions of the upper bidiagonal matrix.
NRHS (input) INTEGER
The number of columns of B and BX. NRHS must be at least 1.
B (input) COMPLEX array, dimension ( LDB, NRHS )
On input, B contains the right hand sides of the least
squares problem in rows 1 through M. On output, B contains
the solution X in rows 1 through N.
LDB (input) INTEGER
The leading dimension of B in the calling subprogram.
LDB must be at least max(1,MAX( M, N ) ).
BX (output) COMPLEX array, dimension ( LDBX, NRHS )
On exit, the result of applying the left or right singular
vector matrix to B.
LDBX (input) INTEGER
The leading dimension of BX.
U (input) REAL array, dimension ( LDU, SMLSIZ ).
On entry, 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 (input) REAL array, dimension ( LDU, SMLSIZ+1 ).
On entry, VT' contains the right singular vector matrices of
all subproblems at the bottom level.
K (input) INTEGER array, dimension ( N ).
DIFL (input) REAL array, dimension ( LDU, NLVL ).
where NLVL = INT(log_2 (N/(SMLSIZ+1))) + 1.
DIFR (input) REAL array, dimension ( LDU, 2 * NLVL ).
On entry, DIFL(*, I) and DIFR(*, 2 * I -1) record
distances between singular values on the I-th level and
singular values on the (I -1)-th level, and DIFR(*, 2 * I)
record the normalizing factors of the right singular vectors
matrices of subproblems on I-th level.
Z (input) REAL array, dimension ( LDU, NLVL ).
On entry, Z(1, I) contains the components of the deflation-
adjusted updating row vector for subproblems on the I-th
level.
POLES (input) REAL array, dimension ( LDU, 2 * NLVL ).
On entry, POLES(*, 2 * I -1: 2 * I) contains the new and old
singular values involved in the secular equations on the I-th
level.
GIVPTR (input) INTEGER array, dimension ( N ).
On entry, GIVPTR( I ) records the number of Givens
rotations performed on the I-th problem on the computation
tree.
GIVCOL (input) INTEGER array, dimension ( LDGCOL, 2 * NLVL ).
On entry, for each I, GIVCOL(*, 2 * I - 1: 2 * I) records 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 (input) INTEGER array, dimension ( LDGCOL, NLVL ).
On entry, PERM(*, I) records permutations done on the I-th
level of the computation tree.
GIVNUM (input) REAL array, dimension ( LDU, 2 * NLVL ).
On entry, GIVNUM(*, 2 *I -1 : 2 * I) records the C- and S-
values of Givens rotations performed on the I-th level on the
computation tree.
C (input) REAL array, dimension ( N ).
On entry, if the I-th subproblem is not square,
C( I ) contains the C-value of a Givens rotation related to
the right null space of the I-th subproblem.
S (input) REAL array, dimension ( N ).
On entry, if the I-th subproblem is not square,
S( I ) contains the S-value of a Givens rotation related to
the right null space of the I-th subproblem.
RWORK (workspace) REAL array, dimension at least
max ( N, (SMLSZ+1)*NRHS*3 ).
IWORK (workspace) INTEGER array.
The dimension must be at least 3 * N
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
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 parameters.
Parameter adjustments */
/* Table of constant values */
static real c_b9 = 1.f;
static real c_b10 = 0.f;
static integer c__2 = 2;
/* 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, b_dim1, b_offset, bx_dim1, bx_offset, i__1,
i__2, i__3, i__4, i__5, i__6;
complex q__1;
/* Builtin functions */
double r_imag(complex *);
integer pow_ii(integer *, integer *);
/* Local variables */
static integer jcol, nlvl, sqre, jrow, i__, j, jimag, jreal, inode, ndiml;
extern /* Subroutine */ int sgemm_(char *, char *, integer *, integer *,
integer *, real *, real *, integer *, real *, integer *, real *,
real *, integer *);
static integer ndimr;
extern /* Subroutine */ int ccopy_(integer *, complex *, integer *,
complex *, integer *);
static integer i1;
extern /* Subroutine */ int clals0_(integer *, integer *, integer *,
integer *, integer *, complex *, integer *, complex *, integer *,
integer *, integer *, integer *, integer *, real *, integer *,
real *, real *, real *, real *, integer *, real *, real *, real *,
integer *);
static integer ic, lf, nd, ll, nl, nr;
extern /* Subroutine */ int xerbla_(char *, integer *), slasdt_(
integer *, integer *, integer *, integer *, integer *, integer *,
integer *);
static integer im1, nlf, nrf, lvl, ndb1, nlp1, lvl2, nrp1;
#define difl_ref(a_1,a_2) difl[(a_2)*difl_dim1 + a_1]
#define difr_ref(a_1,a_2) difr[(a_2)*difr_dim1 + a_1]
#define perm_ref(a_1,a_2) perm[(a_2)*perm_dim1 + a_1]
#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)]
#define u_ref(a_1,a_2) u[(a_2)*u_dim1 + a_1]
#define z___ref(a_1,a_2) z__[(a_2)*z_dim1 + a_1]
#define poles_ref(a_1,a_2) poles[(a_2)*poles_dim1 + a_1]
#define bx_subscr(a_1,a_2) (a_2)*bx_dim1 + a_1
#define bx_ref(a_1,a_2) bx[bx_subscr(a_1,a_2)]
#define vt_ref(a_1,a_2) vt[(a_2)*vt_dim1 + a_1]
#define givcol_ref(a_1,a_2) givcol[(a_2)*givcol_dim1 + a_1]
#define givnum_ref(a_1,a_2) givnum[(a_2)*givnum_dim1 + a_1]
b_dim1 = *ldb;
b_offset = 1 + b_dim1 * 1;
b -= b_offset;
bx_dim1 = *ldbx;
bx_offset = 1 + bx_dim1 * 1;
bx -= bx_offset;
givnum_dim1 = *ldu;
givnum_offset = 1 + givnum_dim1 * 1;
givnum -= givnum_offset;
poles_dim1 = *ldu;
poles_offset = 1 + poles_dim1 * 1;
poles -= poles_offset;
z_dim1 = *ldu;
z_offset = 1 + z_dim1 * 1;
z__ -= z_offset;
difr_dim1 = *ldu;
difr_offset = 1 + difr_dim1 * 1;
difr -= difr_offset;
difl_dim1 = *ldu;
difl_offset = 1 + difl_dim1 * 1;
difl -= difl_offset;
vt_dim1 = *ldu;
vt_offset = 1 + vt_dim1 * 1;
vt -= vt_offset;
u_dim1 = *ldu;
u_offset = 1 + u_dim1 * 1;
u -= u_offset;
--k;
--givptr;
perm_dim1 = *ldgcol;
perm_offset = 1 + perm_dim1 * 1;
perm -= perm_offset;
givcol_dim1 = *ldgcol;
givcol_offset = 1 + givcol_dim1 * 1;
givcol -= givcol_offset;
--c__;
--s;
--rwork;
--iwork;
/* Function Body */
*info = 0;
if (*icompq < 0 || *icompq > 1) {
*info = -1;
} else if (*smlsiz < 3) {
*info = -2;
} else if (*n < *smlsiz) {
*info = -3;
} else if (*nrhs < 1) {
*info = -4;
} else if (*ldb < *n) {
*info = -6;
} else if (*ldbx < *n) {
*info = -8;
} else if (*ldu < *n) {
*info = -10;
} else if (*ldgcol < *n) {
*info = -19;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CLALSA", &i__1);
return 0;
}
/* Book-keeping and setting up the computation tree. */
inode = 1;
ndiml = inode + *n;
ndimr = ndiml + *n;
slasdt_(n, &nlvl, &nd, &iwork[inode], &iwork[ndiml], &iwork[ndimr],
smlsiz);
/* The following code applies back the left singular vector factors.
For applying back the right singular vector factors, go to 170. */
if (*icompq == 1) {
goto L170;
}
/* The nodes on the bottom level of the tree were solved
by SLASDQ. The corresponding left and right singular vector
matrices are in explicit form. First apply back the left
singular vector matrices. */
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];
nr = iwork[ndimr + i1];
nlf = ic - nl;
nrf = ic + 1;
/* Since B and BX are complex, the following call to SGEMM
is performed in two steps (real and imaginary parts).
CALL SGEMM( 'T', 'N', NL, NRHS, NL, ONE, U( NLF, 1 ), LDU,
$ B( NLF, 1 ), LDB, ZERO, BX( NLF, 1 ), LDBX ) */
j = nl * *nrhs << 1;
i__2 = *nrhs;
for (jcol = 1; jcol <= i__2; ++jcol) {
i__3 = nlf + nl - 1;
for (jrow = nlf; jrow <= i__3; ++jrow) {
++j;
i__4 = b_subscr(jrow, jcol);
rwork[j] = b[i__4].r;
/* L10: */
}
/* L20: */
}
sgemm_("T", "N", &nl, nrhs, &nl, &c_b9, &u_ref(nlf, 1), ldu, &rwork[(
nl * *nrhs << 1) + 1], &nl, &c_b10, &rwork[1], &nl);
j = nl * *nrhs << 1;
i__2 = *nrhs;
for (jcol = 1; jcol <= i__2; ++jcol) {
i__3 = nlf + nl - 1;
for (jrow = nlf; jrow <= i__3; ++jrow) {
++j;
rwork[j] = r_imag(&b_ref(jrow, jcol));
/* L30: */
}
/* L40: */
}
sgemm_("T", "N", &nl, nrhs, &nl, &c_b9, &u_ref(nlf, 1), ldu, &rwork[(
nl * *nrhs << 1) + 1], &nl, &c_b10, &rwork[nl * *nrhs + 1], &
nl);
jreal = 0;
jimag = nl * *nrhs;
i__2 = *nrhs;
for (jcol = 1; jcol <= i__2; ++jcol) {
i__3 = nlf + nl - 1;
for (jrow = nlf; jrow <= i__3; ++jrow) {
++jreal;
++jimag;
i__4 = bx_subscr(jrow, jcol);
i__5 = jreal;
i__6 = jimag;
q__1.r = rwork[i__5], q__1.i = rwork[i__6];
bx[i__4].r = q__1.r, bx[i__4].i = q__1.i;
/* L50: */
}
/* L60: */
}
/* Since B and BX are complex, the following call to SGEMM
is performed in two steps (real and imaginary parts).
CALL SGEMM( 'T', 'N', NR, NRHS, NR, ONE, U( NRF, 1 ), LDU,
$ B( NRF, 1 ), LDB, ZERO, BX( NRF, 1 ), LDBX ) */
j = nr * *nrhs << 1;
i__2 = *nrhs;
for (jcol = 1; jcol <= i__2; ++jcol) {
i__3 = nrf + nr - 1;
for (jrow = nrf; jrow <= i__3; ++jrow) {
++j;
i__4 = b_subscr(jrow, jcol);
rwork[j] = b[i__4].r;
/* L70: */
}
/* L80: */
}
sgemm_("T", "N", &nr, nrhs, &nr, &c_b9, &u_ref(nrf, 1), ldu, &rwork[(
nr * *nrhs << 1) + 1], &nr, &c_b10, &rwork[1], &nr);
j = nr * *nrhs << 1;
i__2 = *nrhs;
for (jcol = 1; jcol <= i__2; ++jcol) {
i__3 = nrf + nr - 1;
for (jrow = nrf; jrow <= i__3; ++jrow) {
++j;
rwork[j] = r_imag(&b_ref(jrow, jcol));
/* L90: */
}
/* L100: */
}
sgemm_("T", "N", &nr, nrhs, &nr, &c_b9, &u_ref(nrf, 1), ldu, &rwork[(
nr * *nrhs << 1) + 1], &nr, &c_b10, &rwork[nr * *nrhs + 1], &
nr);
jreal = 0;
jimag = nr * *nrhs;
i__2 = *nrhs;
for (jcol = 1; jcol <= i__2; ++jcol) {
i__3 = nrf + nr - 1;
for (jrow = nrf; jrow <= i__3; ++jrow) {
++jreal;
++jimag;
i__4 = bx_subscr(jrow, jcol);
i__5 = jreal;
i__6 = jimag;
q__1.r = rwork[i__5], q__1.i = rwork[i__6];
bx[i__4].r = q__1.r, bx[i__4].i = q__1.i;
/* L110: */
}
/* L120: */
}
/* L130: */
}
/* Next copy the rows of B that correspond to unchanged rows
in the bidiagonal matrix to BX. */
i__1 = nd;
for (i__ = 1; i__ <= i__1; ++i__) {
ic = iwork[inode + i__ - 1];
ccopy_(nrhs, &b_ref(ic, 1), ldb, &bx_ref(ic, 1), ldbx);
/* L140: */
}
/* Finally go through the left singular vector matrices of all
the other subproblems bottom-up on the tree. */
j = pow_ii(&c__2, &nlvl);
sqre = 0;
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;
--j;
clals0_(icompq, &nl, &nr, &sqre, nrhs, &bx_ref(nlf, 1), ldbx, &
b_ref(nlf, 1), ldb, &perm_ref(nlf, lvl), &givptr[j], &
givcol_ref(nlf, lvl2), ldgcol, &givnum_ref(nlf, lvl2),
ldu, &poles_ref(nlf, lvl2), &difl_ref(nlf, lvl), &
difr_ref(nlf, lvl2), &z___ref(nlf, lvl), &k[j], &c__[j], &
s[j], &rwork[1], info);
/* L150: */
}
/* L160: */
}
goto L330;
/* ICOMPQ = 1: applying back the right singular vector factors. */
L170:
/* First now go through the right singular vector matrices of all
the tree nodes top-down. */
j = 0;
i__1 = nlvl;
for (lvl = 1; lvl <= i__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__2 = lvl - 1;
lf = pow_ii(&c__2, &i__2);
ll = (lf << 1) - 1;
}
i__2 = lf;
for (i__ = ll; i__ >= i__2; --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) {
sqre = 0;
} else {
sqre = 1;
}
++j;
clals0_(icompq, &nl, &nr, &sqre, nrhs, &b_ref(nlf, 1), ldb, &
bx_ref(nlf, 1), ldbx, &perm_ref(nlf, lvl), &givptr[j], &
givcol_ref(nlf, lvl2), ldgcol, &givnum_ref(nlf, lvl2),
ldu, &poles_ref(nlf, lvl2), &difl_ref(nlf, lvl), &
difr_ref(nlf, lvl2), &z___ref(nlf, lvl), &k[j], &c__[j], &
s[j], &rwork[1], info);
/* L180: */
}
/* L190: */
}
/* The nodes on the bottom level of the tree were solved
by SLASDQ. The corresponding right singular vector
matrices are in explicit form. Apply them back. */
ndb1 = (nd + 1) / 2;
i__1 = nd;
for (i__ = ndb1; i__ <= i__1; ++i__) {
i1 = i__ - 1;
ic = iwork[inode + i1];
nl = iwork[ndiml + i1];
nr = iwork[ndimr + i1];
nlp1 = nl + 1;
if (i__ == nd) {
nrp1 = nr;
} else {
nrp1 = nr + 1;
}
nlf = ic - nl;
nrf = ic + 1;
/* Since B and BX are complex, the following call to SGEMM is
performed in two steps (real and imaginary parts).
CALL SGEMM( 'T', 'N', NLP1, NRHS, NLP1, ONE, VT( NLF, 1 ), LDU,
$ B( NLF, 1 ), LDB, ZERO, BX( NLF, 1 ), LDBX ) */
j = nlp1 * *nrhs << 1;
i__2 = *nrhs;
for (jcol = 1; jcol <= i__2; ++jcol) {
i__3 = nlf + nlp1 - 1;
for (jrow = nlf; jrow <= i__3; ++jrow) {
++j;
i__4 = b_subscr(jrow, jcol);
rwork[j] = b[i__4].r;
/* L200: */
}
/* L210: */
}
sgemm_("T", "N", &nlp1, nrhs, &nlp1, &c_b9, &vt_ref(nlf, 1), ldu, &
rwork[(nlp1 * *nrhs << 1) + 1], &nlp1, &c_b10, &rwork[1], &
nlp1);
j = nlp1 * *nrhs << 1;
i__2 = *nrhs;
for (jcol = 1; jcol <= i__2; ++jcol) {
i__3 = nlf + nlp1 - 1;
for (jrow = nlf; jrow <= i__3; ++jrow) {
++j;
rwork[j] = r_imag(&b_ref(jrow, jcol));
/* L220: */
}
/* L230: */
}
sgemm_("T", "N", &nlp1, nrhs, &nlp1, &c_b9, &vt_ref(nlf, 1), ldu, &
rwork[(nlp1 * *nrhs << 1) + 1], &nlp1, &c_b10, &rwork[nlp1 * *
nrhs + 1], &nlp1);
jreal = 0;
jimag = nlp1 * *nrhs;
i__2 = *nrhs;
for (jcol = 1; jcol <= i__2; ++jcol) {
i__3 = nlf + nlp1 - 1;
for (jrow = nlf; jrow <= i__3; ++jrow) {
++jreal;
++jimag;
i__4 = bx_subscr(jrow, jcol);
i__5 = jreal;
i__6 = jimag;
q__1.r = rwork[i__5], q__1.i = rwork[i__6];
bx[i__4].r = q__1.r, bx[i__4].i = q__1.i;
/* L240: */
}
/* L250: */
}
/* Since B and BX are complex, the following call to SGEMM is
performed in two steps (real and imaginary parts).
CALL SGEMM( 'T', 'N', NRP1, NRHS, NRP1, ONE, VT( NRF, 1 ), LDU,
$ B( NRF, 1 ), LDB, ZERO, BX( NRF, 1 ), LDBX ) */
j = nrp1 * *nrhs << 1;
i__2 = *nrhs;
for (jcol = 1; jcol <= i__2; ++jcol) {
i__3 = nrf + nrp1 - 1;
for (jrow = nrf; jrow <= i__3; ++jrow) {
++j;
i__4 = b_subscr(jrow, jcol);
rwork[j] = b[i__4].r;
/* L260: */
}
/* L270: */
}
sgemm_("T", "N", &nrp1, nrhs, &nrp1, &c_b9, &vt_ref(nrf, 1), ldu, &
rwork[(nrp1 * *nrhs << 1) + 1], &nrp1, &c_b10, &rwork[1], &
nrp1);
j = nrp1 * *nrhs << 1;
i__2 = *nrhs;
for (jcol = 1; jcol <= i__2; ++jcol) {
i__3 = nrf + nrp1 - 1;
for (jrow = nrf; jrow <= i__3; ++jrow) {
++j;
rwork[j] = r_imag(&b_ref(jrow, jcol));
/* L280: */
}
/* L290: */
}
sgemm_("T", "N", &nrp1, nrhs, &nrp1, &c_b9, &vt_ref(nrf, 1), ldu, &
rwork[(nrp1 * *nrhs << 1) + 1], &nrp1, &c_b10, &rwork[nrp1 * *
nrhs + 1], &nrp1);
jreal = 0;
jimag = nrp1 * *nrhs;
i__2 = *nrhs;
for (jcol = 1; jcol <= i__2; ++jcol) {
i__3 = nrf + nrp1 - 1;
for (jrow = nrf; jrow <= i__3; ++jrow) {
++jreal;
++jimag;
i__4 = bx_subscr(jrow, jcol);
i__5 = jreal;
i__6 = jimag;
q__1.r = rwork[i__5], q__1.i = rwork[i__6];
bx[i__4].r = q__1.r, bx[i__4].i = q__1.i;
/* L300: */
}
/* L310: */
}
/* L320: */
}
L330:
return 0;
/* End of CLALSA */
} /* clalsa_ */
/* Subroutine */ int dlaed7_(integer *icompq, integer *n, integer *qsiz,
integer *tlvls, integer *curlvl, integer *curpbm, doublereal *d__,
doublereal *q, integer *ldq, integer *indxq, doublereal *rho, integer
*cutpnt, doublereal *qstore, integer *qptr, integer *prmptr, integer *
perm, integer *givptr, integer *givcol, doublereal *givnum,
doublereal *work, integer *iwork, integer *info)
{
/* -- LAPACK routine (version 3.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
September 30, 1994
Purpose
=======
DLAED7 computes the updated eigensystem of a diagonal
matrix after modification by a rank-one symmetric matrix. This
routine is used only for the eigenproblem which requires all
eigenvalues and optionally eigenvectors of a dense symmetric matrix
that has been reduced to tridiagonal form. DLAED1 handles
the case in which all eigenvalues and eigenvectors of a symmetric
tridiagonal matrix are desired.
T = Q(in) ( D(in) + RHO * Z*Z' ) Q'(in) = Q(out) * D(out) * Q'(out)
where Z = Q'u, u is a vector of length N with ones in the
CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.
The eigenvectors of the original matrix are stored in Q, and the
eigenvalues are in D. The algorithm consists of three stages:
The first stage consists of deflating the size of the problem
when there are multiple eigenvalues or if there is a zero in
the Z vector. For each such occurence the dimension of the
secular equation problem is reduced by one. This stage is
performed by the routine DLAED8.
The second stage consists of calculating the updated
eigenvalues. This is done by finding the roots of the secular
equation via the routine DLAED4 (as called by DLAED9).
This routine also calculates the eigenvectors of the current
problem.
The final stage consists of computing the updated eigenvectors
directly using the updated eigenvalues. The eigenvectors for
the current problem are multiplied with the eigenvectors from
the overall problem.
Arguments
=========
ICOMPQ (input) INTEGER
= 0: Compute eigenvalues only.
= 1: Compute eigenvectors of original dense symmetric matrix
also. On entry, Q contains the orthogonal matrix used
to reduce the original matrix to tridiagonal form.
N (input) INTEGER
The dimension of the symmetric tridiagonal matrix. N >= 0.
QSIZ (input) INTEGER
The dimension of the orthogonal matrix used to reduce
the full matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1.
TLVLS (input) INTEGER
The total number of merging levels in the overall divide and
conquer tree.
CURLVL (input) INTEGER
The current level in the overall merge routine,
0 <= CURLVL <= TLVLS.
CURPBM (input) INTEGER
The current problem in the current level in the overall
merge routine (counting from upper left to lower right).
D (input/output) DOUBLE PRECISION array, dimension (N)
On entry, the eigenvalues of the rank-1-perturbed matrix.
On exit, the eigenvalues of the repaired matrix.
Q (input/output) DOUBLE PRECISION array, dimension (LDQ, N)
On entry, the eigenvectors of the rank-1-perturbed matrix.
On exit, the eigenvectors of the repaired tridiagonal matrix.
LDQ (input) INTEGER
The leading dimension of the array Q. LDQ >= max(1,N).
INDXQ (output) INTEGER array, dimension (N)
The permutation which will reintegrate the subproblem just
solved back into sorted order, i.e., D( INDXQ( I = 1, N ) )
will be in ascending order.
RHO (input) DOUBLE PRECISION
The subdiagonal element used to create the rank-1
modification.
CUTPNT (input) INTEGER
Contains the location of the last eigenvalue in the leading
sub-matrix. min(1,N) <= CUTPNT <= N.
QSTORE (input/output) DOUBLE PRECISION array, dimension (N**2+1)
Stores eigenvectors of submatrices encountered during
divide and conquer, packed together. QPTR points to
beginning of the submatrices.
QPTR (input/output) INTEGER array, dimension (N+2)
List of indices pointing to beginning of submatrices stored
in QSTORE. The submatrices are numbered starting at the
bottom left of the divide and conquer tree, from left to
right and bottom to top.
PRMPTR (input) INTEGER array, dimension (N lg N)
Contains a list of pointers which indicate where in PERM a
level's permutation is stored. PRMPTR(i+1) - PRMPTR(i)
indicates the size of the permutation and also the size of
the full, non-deflated problem.
PERM (input) INTEGER array, dimension (N lg N)
Contains the permutations (from deflation and sorting) to be
applied to each eigenblock.
GIVPTR (input) INTEGER array, dimension (N lg N)
Contains a list of pointers which indicate where in GIVCOL a
level's Givens rotations are stored. GIVPTR(i+1) - GIVPTR(i)
indicates the number of Givens rotations.
GIVCOL (input) INTEGER array, dimension (2, N lg N)
Each pair of numbers indicates a pair of columns to take place
in a Givens rotation.
GIVNUM (input) DOUBLE PRECISION array, dimension (2, N lg N)
Each number indicates the S value to be used in the
corresponding Givens rotation.
WORK (workspace) DOUBLE PRECISION array, dimension (3*N+QSIZ*N)
IWORK (workspace) INTEGER array, dimension (4*N)
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if INFO = 1, an eigenvalue did not converge
Further Details
===============
Based on contributions by
Jeff Rutter, Computer Science Division, University of California
at Berkeley, USA
=====================================================================
Test the input parameters.
Parameter adjustments */
/* Table of constant values */
static integer c__2 = 2;
static integer c__1 = 1;
static doublereal c_b10 = 1.;
static doublereal c_b11 = 0.;
static integer c_n1 = -1;
/* System generated locals */
integer q_dim1, q_offset, i__1, i__2;
/* Builtin functions */
integer pow_ii(integer *, integer *);
/* Local variables */
static integer indx, curr, i__, k;
extern /* Subroutine */ int dgemm_(char *, char *, integer *, integer *,
integer *, doublereal *, doublereal *, integer *, doublereal *,
integer *, doublereal *, doublereal *, integer *);
static integer indxc, indxp, n1, n2;
extern /* Subroutine */ int dlaed8_(integer *, integer *, integer *,
integer *, doublereal *, doublereal *, integer *, integer *,
doublereal *, integer *, doublereal *, doublereal *, doublereal *,
integer *, doublereal *, integer *, integer *, integer *,
doublereal *, integer *, integer *, integer *), dlaed9_(integer *,
integer *, integer *, integer *, doublereal *, doublereal *,
integer *, doublereal *, doublereal *, doublereal *, doublereal *,
integer *, integer *), dlaeda_(integer *, integer *, integer *,
integer *, integer *, integer *, integer *, integer *, doublereal
*, doublereal *, integer *, doublereal *, doublereal *, integer *)
;
static integer idlmda, is, iw, iz;
extern /* Subroutine */ int dlamrg_(integer *, integer *, doublereal *,
integer *, integer *, integer *), xerbla_(char *, integer *);
static integer coltyp, iq2, ptr, ldq2;
#define givcol_ref(a_1,a_2) givcol[(a_2)*2 + a_1]
#define givnum_ref(a_1,a_2) givnum[(a_2)*2 + a_1]
--d__;
q_dim1 = *ldq;
q_offset = 1 + q_dim1 * 1;
q -= q_offset;
--indxq;
--qstore;
--qptr;
--prmptr;
--perm;
--givptr;
givcol -= 3;
givnum -= 3;
--work;
--iwork;
/* Function Body */
*info = 0;
if (*icompq < 0 || *icompq > 1) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*icompq == 1 && *qsiz < *n) {
*info = -4;
} else if (*ldq < max(1,*n)) {
*info = -9;
} else if (min(1,*n) > *cutpnt || *n < *cutpnt) {
*info = -12;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("DLAED7", &i__1);
return 0;
}
/* Quick return if possible */
if (*n == 0) {
return 0;
}
/* The following values are for bookkeeping purposes only. They are
integer pointers which indicate the portion of the workspace
used by a particular array in DLAED8 and DLAED9. */
if (*icompq == 1) {
ldq2 = *qsiz;
} else {
ldq2 = *n;
}
iz = 1;
idlmda = iz + *n;
iw = idlmda + *n;
iq2 = iw + *n;
is = iq2 + *n * ldq2;
indx = 1;
indxc = indx + *n;
coltyp = indxc + *n;
indxp = coltyp + *n;
/* Form the z-vector which consists of the last row of Q_1 and the
first row of Q_2. */
ptr = pow_ii(&c__2, tlvls) + 1;
i__1 = *curlvl - 1;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = *tlvls - i__;
ptr += pow_ii(&c__2, &i__2);
/* L10: */
}
curr = ptr + *curpbm;
dlaeda_(n, tlvls, curlvl, curpbm, &prmptr[1], &perm[1], &givptr[1], &
givcol[3], &givnum[3], &qstore[1], &qptr[1], &work[iz], &work[iz
+ *n], info);
/* When solving the final problem, we no longer need the stored data,
so we will overwrite the data from this level onto the previously
used storage space. */
if (*curlvl == *tlvls) {
qptr[curr] = 1;
prmptr[curr] = 1;
givptr[curr] = 1;
}
/* Sort and Deflate eigenvalues. */
dlaed8_(icompq, &k, n, qsiz, &d__[1], &q[q_offset], ldq, &indxq[1], rho,
cutpnt, &work[iz], &work[idlmda], &work[iq2], &ldq2, &work[iw], &
perm[prmptr[curr]], &givptr[curr + 1], &givcol_ref(1, givptr[curr]
), &givnum_ref(1, givptr[curr]), &iwork[indxp], &iwork[indx],
info);
prmptr[curr + 1] = prmptr[curr] + *n;
givptr[curr + 1] += givptr[curr];
/* Solve Secular Equation. */
if (k != 0) {
dlaed9_(&k, &c__1, &k, n, &d__[1], &work[is], &k, rho, &work[idlmda],
&work[iw], &qstore[qptr[curr]], &k, info);
if (*info != 0) {
goto L30;
}
if (*icompq == 1) {
dgemm_("N", "N", qsiz, &k, &k, &c_b10, &work[iq2], &ldq2, &qstore[
qptr[curr]], &k, &c_b11, &q[q_offset], ldq);
}
/* Computing 2nd power */
i__1 = k;
qptr[curr + 1] = qptr[curr] + i__1 * i__1;
/* Prepare the INDXQ sorting permutation. */
n1 = k;
n2 = *n - k;
dlamrg_(&n1, &n2, &d__[1], &c__1, &c_n1, &indxq[1]);
} else {
qptr[curr + 1] = qptr[curr];
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
indxq[i__] = i__;
/* L20: */
}
}
L30:
return 0;
/* End of DLAED7 */
} /* dlaed7_ */
/* Subroutine */ int dlasda_(integer *icompq, integer *smlsiz, integer *n,
integer *sqre, doublereal *d__, doublereal *e, doublereal *u, integer
*ldu, doublereal *vt, integer *k, doublereal *difl, doublereal *difr,
doublereal *z__, doublereal *poles, integer *givptr, integer *givcol,
integer *ldgcol, integer *perm, doublereal *givnum, doublereal *c__,
doublereal *s, doublereal *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 */
static doublereal beta;
static integer idxq, nlvl, i__, j, m;
static doublereal alpha;
static integer inode, ndiml, ndimr, idxqi, itemp;
extern /* Subroutine */ int dcopy_(integer *, doublereal *, integer *,
doublereal *, integer *);
static integer sqrei, i1;
extern /* Subroutine */ int dlasd6_(integer *, integer *, integer *,
integer *, doublereal *, doublereal *, doublereal *, doublereal *,
doublereal *, integer *, integer *, integer *, integer *,
integer *, doublereal *, integer *, doublereal *, doublereal *,
doublereal *, doublereal *, integer *, doublereal *, doublereal *,
doublereal *, integer *, integer *);
static integer ic, nwork1, lf, nd, nwork2, ll, nl, vf, nr, vl;
extern /* Subroutine */ int dlasdq_(char *, integer *, integer *, integer
*, integer *, integer *, doublereal *, doublereal *, doublereal *,
integer *, doublereal *, integer *, doublereal *, integer *,
doublereal *, integer *), dlasdt_(integer *, integer *,
integer *, integer *, integer *, integer *, integer *), dlaset_(
char *, integer *, integer *, doublereal *, doublereal *,
doublereal *, integer *), xerbla_(char *, integer *);
static integer im1, smlszp, ncc, nlf, nrf, vfi, iwk, vli, lvl, nru, ndb1,
nlp1, lvl2, nrp1;
#define difl_ref(a_1,a_2) difl[(a_2)*difl_dim1 + a_1]
#define difr_ref(a_1,a_2) difr[(a_2)*difr_dim1 + a_1]
#define perm_ref(a_1,a_2) perm[(a_2)*perm_dim1 + a_1]
#define u_ref(a_1,a_2) u[(a_2)*u_dim1 + a_1]
#define z___ref(a_1,a_2) z__[(a_2)*z_dim1 + a_1]
#define poles_ref(a_1,a_2) poles[(a_2)*poles_dim1 + a_1]
#define vt_ref(a_1,a_2) vt[(a_2)*vt_dim1 + a_1]
#define givcol_ref(a_1,a_2) givcol[(a_2)*givcol_dim1 + a_1]
#define givnum_ref(a_1,a_2) givnum[(a_2)*givnum_dim1 + a_1]
/* -- LAPACK auxiliary 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
=======
Using a divide and conquer approach, DLASDA 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, DLASD0, 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension ( M-1 )
Contains the subdiagonal entries of the bidiagonal matrix.
On exit, E has been destroyed.
U (output) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension ( LDU, NLVL ),
where NLVL = floor(log_2 (N/SMLSIZ))).
DIFR (output) DOUBLE PRECISION 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 DLASD8 for details.
Z (output) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension
(6 * N + (SMLSIZ + 1)*(SMLSIZ + 1)).
IWORK (workspace) INTEGER array.
Dimension must be at least (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
=====================================================================
Test the input parameters.
Parameter adjustments */
--d__;
--e;
givnum_dim1 = *ldu;
givnum_offset = 1 + givnum_dim1 * 1;
givnum -= givnum_offset;
poles_dim1 = *ldu;
poles_offset = 1 + poles_dim1 * 1;
poles -= poles_offset;
z_dim1 = *ldu;
z_offset = 1 + z_dim1 * 1;
z__ -= z_offset;
difr_dim1 = *ldu;
difr_offset = 1 + difr_dim1 * 1;
difr -= difr_offset;
difl_dim1 = *ldu;
difl_offset = 1 + difl_dim1 * 1;
difl -= difl_offset;
vt_dim1 = *ldu;
vt_offset = 1 + vt_dim1 * 1;
vt -= vt_offset;
u_dim1 = *ldu;
u_offset = 1 + u_dim1 * 1;
u -= u_offset;
--k;
--givptr;
perm_dim1 = *ldgcol;
perm_offset = 1 + perm_dim1 * 1;
perm -= perm_offset;
givcol_dim1 = *ldgcol;
givcol_offset = 1 + givcol_dim1 * 1;
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_("DLASDA", &i__1);
return 0;
}
m = *n + *sqre;
/* If the input matrix is too small, call DLASDQ to find the SVD. */
if (*n <= *smlsiz) {
if (*icompq == 0) {
dlasdq_("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 {
dlasdq_("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;
dlasdt_(n, &nlvl, &nd, &iwork[inode], &iwork[ndiml], &iwork[ndimr],
smlsiz);
/* for the nodes on bottom level of the tree, solve
their subproblems by DLASDQ. */
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) {
dlaset_("A", &nlp1, &nlp1, &c_b11, &c_b12, &work[nwork1], &smlszp);
dlasdq_("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;
dcopy_(&nlp1, &work[nwork1], &c__1, &work[vfi], &c__1);
dcopy_(&nlp1, &work[itemp], &c__1, &work[vli], &c__1);
} else {
dlaset_("A", &nl, &nl, &c_b11, &c_b12, &u_ref(nlf, 1), ldu);
dlaset_("A", &nlp1, &nlp1, &c_b11, &c_b12, &vt_ref(nlf, 1), ldu);
dlasdq_("U", &sqrei, &nl, &nlp1, &nl, &ncc, &d__[nlf], &e[nlf], &
vt_ref(nlf, 1), ldu, &u_ref(nlf, 1), ldu, &u_ref(nlf, 1),
ldu, &work[nwork1], info);
dcopy_(&nlp1, &vt_ref(nlf, 1), &c__1, &work[vfi], &c__1);
dcopy_(&nlp1, &vt_ref(nlf, nlp1), &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) {
dlaset_("A", &nrp1, &nrp1, &c_b11, &c_b12, &work[nwork1], &smlszp);
dlasdq_("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;
dcopy_(&nrp1, &work[nwork1], &c__1, &work[vfi], &c__1);
dcopy_(&nrp1, &work[itemp], &c__1, &work[vli], &c__1);
} else {
dlaset_("A", &nr, &nr, &c_b11, &c_b12, &u_ref(nrf, 1), ldu);
dlaset_("A", &nrp1, &nrp1, &c_b11, &c_b12, &vt_ref(nrf, 1), ldu);
dlasdq_("U", &sqrei, &nr, &nrp1, &nr, &ncc, &d__[nrf], &e[nrf], &
vt_ref(nrf, 1), ldu, &u_ref(nrf, 1), ldu, &u_ref(nrf, 1),
ldu, &work[nwork1], info);
dcopy_(&nrp1, &vt_ref(nrf, 1), &c__1, &work[vfi], &c__1);
dcopy_(&nrp1, &vt_ref(nrf, nrp1), &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) {
dlasd6_(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;
dlasd6_(icompq, &nl, &nr, &sqrei, &d__[nlf], &work[vfi], &
work[vli], &alpha, &beta, &iwork[idxqi], &perm_ref(
nlf, lvl), &givptr[j], &givcol_ref(nlf, lvl2), ldgcol,
&givnum_ref(nlf, lvl2), ldu, &poles_ref(nlf, lvl2), &
difl_ref(nlf, lvl), &difr_ref(nlf, lvl2), &z___ref(
nlf, lvl), &k[j], &c__[j], &s[j], &work[nwork1], &
iwork[iwk], info);
}
if (*info != 0) {
return 0;
}
/* L40: */
}
/* L50: */
}
return 0;
/* End of DLASDA */
} /* dlasda_ */
/* Subroutine */ int zlaed8_(integer *k, integer *n, integer *qsiz,
doublecomplex *q, integer *ldq, doublereal *d__, doublereal *rho,
integer *cutpnt, doublereal *z__, doublereal *dlamda, doublecomplex *
q2, integer *ldq2, doublereal *w, integer *indxp, integer *indx,
integer *indxq, integer *perm, integer *givptr, integer *givcol,
doublereal *givnum, integer *info)
{
/* -- LAPACK routine (version 3.0) --
Univ. of Tennessee, Oak Ridge National Lab, Argonne National Lab,
Courant Institute, NAG Ltd., and Rice University
September 30, 1994
Purpose
=======
ZLAED8 merges the two sets of eigenvalues together into a single
sorted set. Then it tries to deflate the size of the problem.
There are two ways in which deflation can occur: when two or more
eigenvalues are close together or if there is a tiny element in the
Z vector. For each such occurrence the order of the related secular
equation problem is reduced by one.
Arguments
=========
K (output) INTEGER
Contains the number of non-deflated eigenvalues.
This is the order of the related secular equation.
N (input) INTEGER
The dimension of the symmetric tridiagonal matrix. N >= 0.
QSIZ (input) INTEGER
The dimension of the unitary matrix used to reduce
the dense or band matrix to tridiagonal form.
QSIZ >= N if ICOMPQ = 1.
Q (input/output) COMPLEX*16 array, dimension (LDQ,N)
On entry, Q contains the eigenvectors of the partially solved
system which has been previously updated in matrix
multiplies with other partially solved eigensystems.
On exit, Q contains the trailing (N-K) updated eigenvectors
(those which were deflated) in its last N-K columns.
LDQ (input) INTEGER
The leading dimension of the array Q. LDQ >= max( 1, N ).
D (input/output) DOUBLE PRECISION array, dimension (N)
On entry, D contains the eigenvalues of the two submatrices to
be combined. On exit, D contains the trailing (N-K) updated
eigenvalues (those which were deflated) sorted into increasing
order.
RHO (input/output) DOUBLE PRECISION
Contains the off diagonal element associated with the rank-1
cut which originally split the two submatrices which are now
being recombined. RHO is modified during the computation to
the value required by DLAED3.
CUTPNT (input) INTEGER
Contains the location of the last eigenvalue in the leading
sub-matrix. MIN(1,N) <= CUTPNT <= N.
Z (input) DOUBLE PRECISION array, dimension (N)
On input this vector contains the updating vector (the last
row of the first sub-eigenvector matrix and the first row of
the second sub-eigenvector matrix). The contents of Z are
destroyed during the updating process.
DLAMDA (output) DOUBLE PRECISION array, dimension (N)
Contains a copy of the first K eigenvalues which will be used
by DLAED3 to form the secular equation.
Q2 (output) COMPLEX*16 array, dimension (LDQ2,N)
If ICOMPQ = 0, Q2 is not referenced. Otherwise,
Contains a copy of the first K eigenvectors which will be used
by DLAED7 in a matrix multiply (DGEMM) to update the new
eigenvectors.
LDQ2 (input) INTEGER
The leading dimension of the array Q2. LDQ2 >= max( 1, N ).
W (output) DOUBLE PRECISION array, dimension (N)
This will hold the first k values of the final
deflation-altered z-vector and will be passed to DLAED3.
INDXP (workspace) INTEGER array, dimension (N)
This will contain the permutation used to place deflated
values of D at the end of the array. On output INDXP(1:K)
points to the nondeflated D-values and INDXP(K+1:N)
points to the deflated eigenvalues.
INDX (workspace) INTEGER array, dimension (N)
This will contain the permutation used to sort the contents of
D into ascending order.
INDXQ (input) INTEGER array, dimension (N)
This contains the permutation which separately sorts the two
sub-problems in D into ascending order. Note that elements in
the second half of this permutation must first have CUTPNT
added to their values in order to be accurate.
PERM (output) INTEGER array, dimension (N)
Contains the permutations (from deflation and sorting) to be
applied to each eigenblock.
GIVPTR (output) INTEGER
Contains the number of Givens rotations which took place in
this subproblem.
GIVCOL (output) INTEGER array, dimension (2, N)
Each pair of numbers indicates a pair of columns to take place
in a Givens rotation.
GIVNUM (output) DOUBLE PRECISION array, dimension (2, N)
Each number indicates the S value to be used in the
corresponding Givens rotation.
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
=====================================================================
Test the input parameters.
Parameter adjustments */
/* Table of constant values */
static doublereal c_b3 = -1.;
static integer c__1 = 1;
/* System generated locals */
integer q_dim1, q_offset, q2_dim1, q2_offset, i__1;
doublereal d__1;
/* Builtin functions */
double sqrt(doublereal);
/* Local variables */
static integer jlam, imax, jmax;
static doublereal c__;
static integer i__, j;
static doublereal s, t;
extern /* Subroutine */ int dscal_(integer *, doublereal *, doublereal *,
integer *), dcopy_(integer *, doublereal *, integer *, doublereal
*, integer *);
static integer k2, n1, n2;
extern /* Subroutine */ int zdrot_(integer *, doublecomplex *, integer *,
doublecomplex *, integer *, doublereal *, doublereal *), zcopy_(
integer *, doublecomplex *, integer *, doublecomplex *, integer *)
;
extern doublereal dlapy2_(doublereal *, doublereal *), dlamch_(char *);
static integer jp;
extern integer idamax_(integer *, doublereal *, integer *);
extern /* Subroutine */ int dlamrg_(integer *, integer *, doublereal *,
integer *, integer *, integer *), xerbla_(char *, integer *), zlacpy_(char *, integer *, integer *, doublecomplex *,
integer *, doublecomplex *, integer *);
static integer n1p1;
static doublereal eps, tau, tol;
#define q_subscr(a_1,a_2) (a_2)*q_dim1 + a_1
#define q_ref(a_1,a_2) q[q_subscr(a_1,a_2)]
#define q2_subscr(a_1,a_2) (a_2)*q2_dim1 + a_1
#define q2_ref(a_1,a_2) q2[q2_subscr(a_1,a_2)]
#define givcol_ref(a_1,a_2) givcol[(a_2)*2 + a_1]
#define givnum_ref(a_1,a_2) givnum[(a_2)*2 + a_1]
q_dim1 = *ldq;
q_offset = 1 + q_dim1 * 1;
q -= q_offset;
--d__;
--z__;
--dlamda;
q2_dim1 = *ldq2;
q2_offset = 1 + q2_dim1 * 1;
q2 -= q2_offset;
--w;
--indxp;
--indx;
--indxq;
--perm;
givcol -= 3;
givnum -= 3;
/* Function Body */
*info = 0;
if (*n < 0) {
*info = -2;
} else if (*qsiz < *n) {
*info = -3;
} else if (*ldq < max(1,*n)) {
*info = -5;
} else if (*cutpnt < min(1,*n) || *cutpnt > *n) {
*info = -8;
} else if (*ldq2 < max(1,*n)) {
*info = -12;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("ZLAED8", &i__1);
return 0;
}
/* Quick return if possible */
if (*n == 0) {
return 0;
}
n1 = *cutpnt;
n2 = *n - n1;
n1p1 = n1 + 1;
if (*rho < 0.) {
dscal_(&n2, &c_b3, &z__[n1p1], &c__1);
}
/* Normalize z so that norm(z) = 1 */
t = 1. / sqrt(2.);
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
indx[j] = j;
/* L10: */
}
dscal_(n, &t, &z__[1], &c__1);
*rho = (d__1 = *rho * 2., abs(d__1));
/* Sort the eigenvalues into increasing order */
i__1 = *n;
for (i__ = *cutpnt + 1; i__ <= i__1; ++i__) {
indxq[i__] += *cutpnt;
/* L20: */
}
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
dlamda[i__] = d__[indxq[i__]];
w[i__] = z__[indxq[i__]];
/* L30: */
}
i__ = 1;
j = *cutpnt + 1;
dlamrg_(&n1, &n2, &dlamda[1], &c__1, &c__1, &indx[1]);
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
d__[i__] = dlamda[indx[i__]];
z__[i__] = w[indx[i__]];
/* L40: */
}
/* Calculate the allowable deflation tolerance */
imax = idamax_(n, &z__[1], &c__1);
jmax = idamax_(n, &d__[1], &c__1);
eps = dlamch_("Epsilon");
tol = eps * 8. * (d__1 = d__[jmax], abs(d__1));
/* If the rank-1 modifier is small enough, no more needs to be done
-- except to reorganize Q so that its columns correspond with the
elements in D. */
if (*rho * (d__1 = z__[imax], abs(d__1)) <= tol) {
*k = 0;
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
perm[j] = indxq[indx[j]];
zcopy_(qsiz, &q_ref(1, perm[j]), &c__1, &q2_ref(1, j), &c__1);
/* L50: */
}
zlacpy_("A", qsiz, n, &q2_ref(1, 1), ldq2, &q_ref(1, 1), ldq);
return 0;
}
/* If there are multiple eigenvalues then the problem deflates. Here
the number of equal eigenvalues are found. As each equal
eigenvalue is found, an elementary reflector is computed to rotate
the corresponding eigensubspace so that the corresponding
components of Z are zero in this new basis. */
*k = 0;
*givptr = 0;
k2 = *n + 1;
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
if (*rho * (d__1 = z__[j], abs(d__1)) <= tol) {
/* Deflate due to small z component. */
--k2;
indxp[k2] = j;
if (j == *n) {
goto L100;
}
} else {
jlam = j;
goto L70;
}
/* L60: */
}
L70:
++j;
if (j > *n) {
goto L90;
}
if (*rho * (d__1 = z__[j], abs(d__1)) <= tol) {
/* Deflate due to small z component. */
--k2;
indxp[k2] = j;
} else {
/* Check if eigenvalues are close enough to allow deflation. */
s = z__[jlam];
c__ = z__[j];
/* Find sqrt(a**2+b**2) without overflow or
destructive underflow. */
tau = dlapy2_(&c__, &s);
t = d__[j] - d__[jlam];
c__ /= tau;
s = -s / tau;
if ((d__1 = t * c__ * s, abs(d__1)) <= tol) {
/* Deflation is possible. */
z__[j] = tau;
z__[jlam] = 0.;
/* Record the appropriate Givens rotation */
++(*givptr);
givcol_ref(1, *givptr) = indxq[indx[jlam]];
givcol_ref(2, *givptr) = indxq[indx[j]];
givnum_ref(1, *givptr) = c__;
givnum_ref(2, *givptr) = s;
zdrot_(qsiz, &q_ref(1, indxq[indx[jlam]]), &c__1, &q_ref(1, indxq[
indx[j]]), &c__1, &c__, &s);
t = d__[jlam] * c__ * c__ + d__[j] * s * s;
d__[j] = d__[jlam] * s * s + d__[j] * c__ * c__;
d__[jlam] = t;
--k2;
i__ = 1;
L80:
if (k2 + i__ <= *n) {
if (d__[jlam] < d__[indxp[k2 + i__]]) {
indxp[k2 + i__ - 1] = indxp[k2 + i__];
indxp[k2 + i__] = jlam;
++i__;
goto L80;
} else {
indxp[k2 + i__ - 1] = jlam;
}
} else {
indxp[k2 + i__ - 1] = jlam;
}
jlam = j;
} else {
++(*k);
w[*k] = z__[jlam];
dlamda[*k] = d__[jlam];
indxp[*k] = jlam;
jlam = j;
}
}
goto L70;
L90:
/* Record the last eigenvalue. */
++(*k);
w[*k] = z__[jlam];
dlamda[*k] = d__[jlam];
indxp[*k] = jlam;
L100:
/* Sort the eigenvalues and corresponding eigenvectors into DLAMDA
and Q2 respectively. The eigenvalues/vectors which were not
deflated go into the first K slots of DLAMDA and Q2 respectively,
while those which were deflated go into the last N - K slots. */
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
jp = indxp[j];
dlamda[j] = d__[jp];
perm[j] = indxq[indx[jp]];
zcopy_(qsiz, &q_ref(1, perm[j]), &c__1, &q2_ref(1, j), &c__1);
/* L110: */
}
/* The deflated eigenvalues and their corresponding vectors go back
into the last N - K slots of D and Q respectively. */
if (*k < *n) {
i__1 = *n - *k;
dcopy_(&i__1, &dlamda[*k + 1], &c__1, &d__[*k + 1], &c__1);
i__1 = *n - *k;
zlacpy_("A", qsiz, &i__1, &q2_ref(1, *k + 1), ldq2, &q_ref(1, *k + 1),
ldq);
}
return 0;
/* End of ZLAED8 */
} /* zlaed8_ */
/* Subroutine */ int slals0_(integer *icompq, integer *nl, integer *nr,
integer *sqre, integer *nrhs, real *b, integer *ldb, real *bx,
integer *ldbx, integer *perm, integer *givptr, integer *givcol,
integer *ldgcol, real *givnum, integer *ldgnum, real *poles, real *
difl, real *difr, real *z__, integer *k, real *c__, real *s, real *
work, integer *info)
{
/* -- LAPACK routine (version 3.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
December 1, 1999
Purpose
=======
SLALS0 applies back the multiplying factors of either the left or the
right singular vector matrix of a diagonal matrix appended by a row
to the right hand side matrix B in solving the least squares problem
using the divide-and-conquer SVD approach.
For the left singular vector matrix, three types of orthogonal
matrices are involved:
(1L) Givens rotations: the number of such rotations is GIVPTR; the
pairs of columns/rows they were applied to are stored in GIVCOL;
and the C- and S-values of these rotations are stored in GIVNUM.
(2L) Permutation. The (NL+1)-st row of B is to be moved to the first
row, and for J=2:N, PERM(J)-th row of B is to be moved to the
J-th row.
(3L) The left singular vector matrix of the remaining matrix.
For the right singular vector matrix, four types of orthogonal
matrices are involved:
(1R) The right singular vector matrix of the remaining matrix.
(2R) If SQRE = 1, one extra Givens rotation to generate the right
null space.
(3R) The inverse transformation of (2L).
(4R) The inverse transformation of (1L).
Arguments
=========
ICOMPQ (input) INTEGER
Specifies whether singular vectors are to be computed in
factored form:
= 0: Left singular vector matrix.
= 1: Right singular vector matrix.
NL (input) INTEGER
The row dimension of the upper block. NL >= 1.
NR (input) INTEGER
The row dimension of the lower block. NR >= 1.
SQRE (input) INTEGER
= 0: the lower block is an NR-by-NR square matrix.
= 1: the lower block is an NR-by-(NR+1) rectangular matrix.
The bidiagonal matrix has row dimension N = NL + NR + 1,
and column dimension M = N + SQRE.
NRHS (input) INTEGER
The number of columns of B and BX. NRHS must be at least 1.
B (input/output) REAL array, dimension ( LDB, NRHS )
On input, B contains the right hand sides of the least
squares problem in rows 1 through M. On output, B contains
the solution X in rows 1 through N.
LDB (input) INTEGER
The leading dimension of B. LDB must be at least
max(1,MAX( M, N ) ).
BX (workspace) REAL array, dimension ( LDBX, NRHS )
LDBX (input) INTEGER
The leading dimension of BX.
PERM (input) INTEGER array, dimension ( N )
The permutations (from deflation and sorting) applied
to the two blocks.
GIVPTR (input) INTEGER
The number of Givens rotations which took place in this
subproblem.
GIVCOL (input) INTEGER array, dimension ( LDGCOL, 2 )
Each pair of numbers indicates a pair of rows/columns
involved in a Givens rotation.
LDGCOL (input) INTEGER
The leading dimension of GIVCOL, must be at least N.
GIVNUM (input) REAL array, dimension ( LDGNUM, 2 )
Each number indicates the C or S value used in the
corresponding Givens rotation.
LDGNUM (input) INTEGER
The leading dimension of arrays DIFR, POLES and
GIVNUM, must be at least K.
POLES (input) REAL array, dimension ( LDGNUM, 2 )
On entry, POLES(1:K, 1) contains the new singular
values obtained from solving the secular equation, and
POLES(1:K, 2) is an array containing the poles in the secular
equation.
DIFL (input) REAL array, dimension ( K ).
On entry, DIFL(I) is the distance between I-th updated
(undeflated) singular value and the I-th (undeflated) old
singular value.
DIFR (input) REAL array, dimension ( LDGNUM, 2 ).
On entry, DIFR(I, 1) contains the distances between I-th
updated (undeflated) singular value and the I+1-th
(undeflated) old singular value. And DIFR(I, 2) is the
normalizing factor for the I-th right singular vector.
Z (input) REAL array, dimension ( K )
Contain the components of the deflation-adjusted updating row
vector.
K (input) INTEGER
Contains the dimension of the non-deflated matrix,
This is the order of the related secular equation. 1 <= K <=N.
C (input) REAL
C contains garbage if SQRE =0 and the C-value of a Givens
rotation related to the right null space if SQRE = 1.
S (input) REAL
S contains garbage if SQRE =0 and the S-value of a Givens
rotation related to the right null space if SQRE = 1.
WORK (workspace) REAL array, dimension ( K )
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
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 parameters.
Parameter adjustments */
/* Table of constant values */
static real c_b5 = -1.f;
static integer c__1 = 1;
static real c_b11 = 1.f;
static real c_b13 = 0.f;
static integer c__0 = 0;
/* System generated locals */
integer givcol_dim1, givcol_offset, b_dim1, b_offset, bx_dim1, bx_offset,
difr_dim1, difr_offset, givnum_dim1, givnum_offset, poles_dim1,
poles_offset, i__1, i__2;
real r__1;
/* Local variables */
static real temp;
extern /* Subroutine */ int srot_(integer *, real *, integer *, real *,
integer *, real *, real *);
extern doublereal snrm2_(integer *, real *, integer *);
static integer i__, j, m, n;
static real diflj, difrj, dsigj;
extern /* Subroutine */ int sscal_(integer *, real *, real *, integer *),
sgemv_(char *, integer *, integer *, real *, real *, integer *,
real *, integer *, real *, real *, integer *), scopy_(
integer *, real *, integer *, real *, integer *);
extern doublereal slamc3_(real *, real *);
static real dj;
extern /* Subroutine */ int xerbla_(char *, integer *);
static real dsigjp;
extern /* Subroutine */ int slascl_(char *, integer *, integer *, real *,
real *, integer *, integer *, real *, integer *, integer *), slacpy_(char *, integer *, integer *, real *, integer *,
real *, integer *);
static integer nlp1;
#define difr_ref(a_1,a_2) difr[(a_2)*difr_dim1 + a_1]
#define b_ref(a_1,a_2) b[(a_2)*b_dim1 + a_1]
#define poles_ref(a_1,a_2) poles[(a_2)*poles_dim1 + a_1]
#define bx_ref(a_1,a_2) bx[(a_2)*bx_dim1 + a_1]
#define givcol_ref(a_1,a_2) givcol[(a_2)*givcol_dim1 + a_1]
#define givnum_ref(a_1,a_2) givnum[(a_2)*givnum_dim1 + a_1]
b_dim1 = *ldb;
b_offset = 1 + b_dim1 * 1;
b -= b_offset;
bx_dim1 = *ldbx;
bx_offset = 1 + bx_dim1 * 1;
bx -= bx_offset;
--perm;
givcol_dim1 = *ldgcol;
givcol_offset = 1 + givcol_dim1 * 1;
givcol -= givcol_offset;
difr_dim1 = *ldgnum;
difr_offset = 1 + difr_dim1 * 1;
difr -= difr_offset;
poles_dim1 = *ldgnum;
poles_offset = 1 + poles_dim1 * 1;
poles -= poles_offset;
givnum_dim1 = *ldgnum;
givnum_offset = 1 + givnum_dim1 * 1;
givnum -= givnum_offset;
--difl;
--z__;
--work;
/* Function Body */
*info = 0;
if (*icompq < 0 || *icompq > 1) {
*info = -1;
} else if (*nl < 1) {
*info = -2;
} else if (*nr < 1) {
*info = -3;
} else if (*sqre < 0 || *sqre > 1) {
*info = -4;
}
n = *nl + *nr + 1;
if (*nrhs < 1) {
*info = -5;
} else if (*ldb < n) {
*info = -7;
} else if (*ldbx < n) {
*info = -9;
} else if (*givptr < 0) {
*info = -11;
} else if (*ldgcol < n) {
*info = -13;
} else if (*ldgnum < n) {
*info = -15;
} else if (*k < 1) {
*info = -20;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SLALS0", &i__1);
return 0;
}
m = n + *sqre;
nlp1 = *nl + 1;
if (*icompq == 0) {
/* Apply back orthogonal transformations from the left.
Step (1L): apply back the Givens rotations performed. */
i__1 = *givptr;
for (i__ = 1; i__ <= i__1; ++i__) {
srot_(nrhs, &b_ref(givcol_ref(i__, 2), 1), ldb, &b_ref(givcol_ref(
i__, 1), 1), ldb, &givnum_ref(i__, 2), &givnum_ref(i__, 1)
);
/* L10: */
}
/* Step (2L): permute rows of B. */
scopy_(nrhs, &b_ref(nlp1, 1), ldb, &bx_ref(1, 1), ldbx);
i__1 = n;
for (i__ = 2; i__ <= i__1; ++i__) {
scopy_(nrhs, &b_ref(perm[i__], 1), ldb, &bx_ref(i__, 1), ldbx);
/* L20: */
}
/* Step (3L): apply the inverse of the left singular vector
matrix to BX. */
if (*k == 1) {
scopy_(nrhs, &bx[bx_offset], ldbx, &b[b_offset], ldb);
if (z__[1] < 0.f) {
sscal_(nrhs, &c_b5, &b[b_offset], ldb);
}
} else {
i__1 = *k;
for (j = 1; j <= i__1; ++j) {
diflj = difl[j];
dj = poles_ref(j, 1);
dsigj = -poles_ref(j, 2);
if (j < *k) {
difrj = -difr_ref(j, 1);
dsigjp = -poles_ref(j + 1, 2);
}
if (z__[j] == 0.f || poles_ref(j, 2) == 0.f) {
work[j] = 0.f;
} else {
work[j] = -poles_ref(j, 2) * z__[j] / diflj / (poles_ref(
j, 2) + dj);
}
i__2 = j - 1;
for (i__ = 1; i__ <= i__2; ++i__) {
if (z__[i__] == 0.f || poles_ref(i__, 2) == 0.f) {
work[i__] = 0.f;
} else {
work[i__] = poles_ref(i__, 2) * z__[i__] / (slamc3_(&
poles_ref(i__, 2), &dsigj) - diflj) / (
poles_ref(i__, 2) + dj);
}
/* L30: */
}
i__2 = *k;
for (i__ = j + 1; i__ <= i__2; ++i__) {
if (z__[i__] == 0.f || poles_ref(i__, 2) == 0.f) {
work[i__] = 0.f;
} else {
work[i__] = poles_ref(i__, 2) * z__[i__] / (slamc3_(&
poles_ref(i__, 2), &dsigjp) + difrj) / (
poles_ref(i__, 2) + dj);
}
/* L40: */
}
work[1] = -1.f;
temp = snrm2_(k, &work[1], &c__1);
sgemv_("T", k, nrhs, &c_b11, &bx[bx_offset], ldbx, &work[1], &
c__1, &c_b13, &b_ref(j, 1), ldb);
slascl_("G", &c__0, &c__0, &temp, &c_b11, &c__1, nrhs, &b_ref(
j, 1), ldb, info);
/* L50: */
}
}
/* Move the deflated rows of BX to B also. */
if (*k < max(m,n)) {
i__1 = n - *k;
slacpy_("A", &i__1, nrhs, &bx_ref(*k + 1, 1), ldbx, &b_ref(*k + 1,
1), ldb);
}
} else {
/* Apply back the right orthogonal transformations.
Step (1R): apply back the new right singular vector matrix
to B. */
if (*k == 1) {
scopy_(nrhs, &b[b_offset], ldb, &bx[bx_offset], ldbx);
} else {
i__1 = *k;
for (j = 1; j <= i__1; ++j) {
dsigj = poles_ref(j, 2);
if (z__[j] == 0.f) {
work[j] = 0.f;
} else {
work[j] = -z__[j] / difl[j] / (dsigj + poles_ref(j, 1)) /
difr_ref(j, 2);
}
i__2 = j - 1;
for (i__ = 1; i__ <= i__2; ++i__) {
if (z__[j] == 0.f) {
work[i__] = 0.f;
} else {
r__1 = -poles_ref(i__ + 1, 2);
work[i__] = z__[j] / (slamc3_(&dsigj, &r__1) -
difr_ref(i__, 1)) / (dsigj + poles_ref(i__, 1)
) / difr_ref(i__, 2);
}
/* L60: */
}
i__2 = *k;
for (i__ = j + 1; i__ <= i__2; ++i__) {
if (z__[j] == 0.f) {
work[i__] = 0.f;
} else {
r__1 = -poles_ref(i__, 2);
work[i__] = z__[j] / (slamc3_(&dsigj, &r__1) - difl[
i__]) / (dsigj + poles_ref(i__, 1)) /
difr_ref(i__, 2);
}
/* L70: */
}
sgemv_("T", k, nrhs, &c_b11, &b[b_offset], ldb, &work[1], &
c__1, &c_b13, &bx_ref(j, 1), ldbx);
/* L80: */
}
}
/* Step (2R): if SQRE = 1, apply back the rotation that is
related to the right null space of the subproblem. */
if (*sqre == 1) {
scopy_(nrhs, &b_ref(m, 1), ldb, &bx_ref(m, 1), ldbx);
srot_(nrhs, &bx_ref(1, 1), ldbx, &bx_ref(m, 1), ldbx, c__, s);
}
if (*k < max(m,n)) {
i__1 = n - *k;
slacpy_("A", &i__1, nrhs, &b_ref(*k + 1, 1), ldb, &bx_ref(*k + 1,
1), ldbx);
}
/* Step (3R): permute rows of B. */
scopy_(nrhs, &bx_ref(1, 1), ldbx, &b_ref(nlp1, 1), ldb);
if (*sqre == 1) {
scopy_(nrhs, &bx_ref(m, 1), ldbx, &b_ref(m, 1), ldb);
}
i__1 = n;
for (i__ = 2; i__ <= i__1; ++i__) {
scopy_(nrhs, &bx_ref(i__, 1), ldbx, &b_ref(perm[i__], 1), ldb);
/* L90: */
}
/* Step (4R): apply back the Givens rotations performed. */
for (i__ = *givptr; i__ >= 1; --i__) {
r__1 = -givnum_ref(i__, 1);
srot_(nrhs, &b_ref(givcol_ref(i__, 2), 1), ldb, &b_ref(givcol_ref(
i__, 1), 1), ldb, &givnum_ref(i__, 2), &r__1);
/* L100: */
}
}
return 0;
/* End of SLALS0 */
} /* slals0_ */
/* Subroutine */ int slaeda_(integer *n, integer *tlvls, integer *curlvl,
integer *curpbm, integer *prmptr, integer *perm, integer *givptr,
integer *givcol, real *givnum, real *q, integer *qptr, real *z__,
real *ztemp, integer *info)
{
/* System generated locals */
integer i__1, i__2, i__3;
/* Builtin functions */
integer pow_ii(integer *, integer *);
double sqrt(doublereal);
/* Local variables */
static integer curr;
extern /* Subroutine */ int srot_(integer *, real *, integer *, real *,
integer *, real *, real *);
static integer bsiz1, bsiz2, psiz1, psiz2, i__, k, zptr1;
extern /* Subroutine */ int sgemv_(char *, integer *, integer *, real *,
real *, integer *, real *, integer *, real *, real *, integer *), scopy_(integer *, real *, integer *, real *, integer *),
xerbla_(char *, integer *);
static integer mid, ptr;
#define givcol_ref(a_1,a_2) givcol[(a_2)*2 + a_1]
#define givnum_ref(a_1,a_2) givnum[(a_2)*2 + a_1]
/* -- LAPACK routine (instrumented to count operations, version 3.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
September 30, 1994
Common block to return operation count and iteration count
ITCNT is unchanged, OPS is only incremented
Purpose
=======
SLAEDA computes the Z vector corresponding to the merge step in the
CURLVLth step of the merge process with TLVLS steps for the CURPBMth
problem.
Arguments
=========
N (input) INTEGER
The dimension of the symmetric tridiagonal matrix. N >= 0.
TLVLS (input) INTEGER
The total number of merging levels in the overall divide and
conquer tree.
CURLVL (input) INTEGER
The current level in the overall merge routine,
0 <= curlvl <= tlvls.
CURPBM (input) INTEGER
The current problem in the current level in the overall
merge routine (counting from upper left to lower right).
PRMPTR (input) INTEGER array, dimension (N lg N)
Contains a list of pointers which indicate where in PERM a
level's permutation is stored. PRMPTR(i+1) - PRMPTR(i)
indicates the size of the permutation and incidentally the
size of the full, non-deflated problem.
PERM (input) INTEGER array, dimension (N lg N)
Contains the permutations (from deflation and sorting) to be
applied to each eigenblock.
GIVPTR (input) INTEGER array, dimension (N lg N)
Contains a list of pointers which indicate where in GIVCOL a
level's Givens rotations are stored. GIVPTR(i+1) - GIVPTR(i)
indicates the number of Givens rotations.
GIVCOL (input) INTEGER array, dimension (2, N lg N)
Each pair of numbers indicates a pair of columns to take place
in a Givens rotation.
GIVNUM (input) REAL array, dimension (2, N lg N)
Each number indicates the S value to be used in the
corresponding Givens rotation.
Q (input) REAL array, dimension (N**2)
Contains the square eigenblocks from previous levels, the
starting positions for blocks are given by QPTR.
QPTR (input) INTEGER array, dimension (N+2)
Contains a list of pointers which indicate where in Q an
eigenblock is stored. SQRT( QPTR(i+1) - QPTR(i) ) indicates
the size of the block.
Z (output) REAL array, dimension (N)
On output this vector contains the updating vector (the last
row of the first sub-eigenvector matrix and the first row of
the second sub-eigenvector matrix).
ZTEMP (workspace) REAL array, dimension (N)
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
Further Details
===============
Based on contributions by
Jeff Rutter, Computer Science Division, University of California
at Berkeley, USA
=====================================================================
Test the input parameters.
Parameter adjustments */
--ztemp;
--z__;
--qptr;
--q;
givnum -= 3;
givcol -= 3;
--givptr;
--perm;
--prmptr;
/* Function Body */
*info = 0;
if (*n < 0) {
*info = -1;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SLAEDA", &i__1);
return 0;
}
/* Quick return if possible */
if (*n == 0) {
return 0;
}
/* Determine location of first number in second half. */
mid = *n / 2 + 1;
/* Gather last/first rows of appropriate eigenblocks into center of Z */
ptr = 1;
/* Determine location of lowest level subproblem in the full storage
scheme */
i__1 = *curlvl - 1;
curr = ptr + *curpbm * pow_ii(&c__2, curlvl) + pow_ii(&c__2, &i__1) - 1;
/* Determine size of these matrices. We add HALF to the value of
the SQRT in case the machine underestimates one of these square
roots. */
latime_1.ops += 8;
bsiz1 = (integer) (sqrt((real) (qptr[curr + 1] - qptr[curr])) + .5f);
bsiz2 = (integer) (sqrt((real) (qptr[curr + 2] - qptr[curr + 1])) + .5f);
i__1 = mid - bsiz1 - 1;
for (k = 1; k <= i__1; ++k) {
z__[k] = 0.f;
/* L10: */
}
scopy_(&bsiz1, &q[qptr[curr] + bsiz1 - 1], &bsiz1, &z__[mid - bsiz1], &
c__1);
scopy_(&bsiz2, &q[qptr[curr + 1]], &bsiz2, &z__[mid], &c__1);
i__1 = *n;
for (k = mid + bsiz2; k <= i__1; ++k) {
z__[k] = 0.f;
/* L20: */
}
/* Loop thru remaining levels 1 -> CURLVL applying the Givens
rotations and permutation and then multiplying the center matrices
against the current Z. */
ptr = pow_ii(&c__2, tlvls) + 1;
i__1 = *curlvl - 1;
for (k = 1; k <= i__1; ++k) {
i__2 = *curlvl - k;
i__3 = *curlvl - k - 1;
curr = ptr + *curpbm * pow_ii(&c__2, &i__2) + pow_ii(&c__2, &i__3) -
1;
psiz1 = prmptr[curr + 1] - prmptr[curr];
psiz2 = prmptr[curr + 2] - prmptr[curr + 1];
zptr1 = mid - psiz1;
/* Apply Givens at CURR and CURR+1 */
latime_1.ops += (givptr[curr + 2] - givptr[curr]) * 6;
i__2 = givptr[curr + 1] - 1;
for (i__ = givptr[curr]; i__ <= i__2; ++i__) {
srot_(&c__1, &z__[zptr1 + givcol_ref(1, i__) - 1], &c__1, &z__[
zptr1 + givcol_ref(2, i__) - 1], &c__1, &givnum_ref(1,
i__), &givnum_ref(2, i__));
/* L30: */
}
i__2 = givptr[curr + 2] - 1;
for (i__ = givptr[curr + 1]; i__ <= i__2; ++i__) {
srot_(&c__1, &z__[mid - 1 + givcol_ref(1, i__)], &c__1, &z__[mid
- 1 + givcol_ref(2, i__)], &c__1, &givnum_ref(1, i__), &
givnum_ref(2, i__));
/* L40: */
}
psiz1 = prmptr[curr + 1] - prmptr[curr];
psiz2 = prmptr[curr + 2] - prmptr[curr + 1];
i__2 = psiz1 - 1;
for (i__ = 0; i__ <= i__2; ++i__) {
ztemp[i__ + 1] = z__[zptr1 + perm[prmptr[curr] + i__] - 1];
/* L50: */
}
i__2 = psiz2 - 1;
for (i__ = 0; i__ <= i__2; ++i__) {
ztemp[psiz1 + i__ + 1] = z__[mid + perm[prmptr[curr + 1] + i__] -
1];
/* L60: */
}
/* Multiply Blocks at CURR and CURR+1
Determine size of these matrices. We add HALF to the value of
the SQRT in case the machine underestimates one of these
square roots. */
latime_1.ops += 8;
bsiz1 = (integer) (sqrt((real) (qptr[curr + 1] - qptr[curr])) + .5f);
bsiz2 = (integer) (sqrt((real) (qptr[curr + 2] - qptr[curr + 1])) +
.5f);
if (bsiz1 > 0) {
latime_1.ops += (bsiz1 << 1) * bsiz1;
sgemv_("T", &bsiz1, &bsiz1, &c_b24, &q[qptr[curr]], &bsiz1, &
ztemp[1], &c__1, &c_b26, &z__[zptr1], &c__1);
}
i__2 = psiz1 - bsiz1;
scopy_(&i__2, &ztemp[bsiz1 + 1], &c__1, &z__[zptr1 + bsiz1], &c__1);
if (bsiz2 > 0) {
latime_1.ops += (bsiz2 << 1) * bsiz2;
sgemv_("T", &bsiz2, &bsiz2, &c_b24, &q[qptr[curr + 1]], &bsiz2, &
ztemp[psiz1 + 1], &c__1, &c_b26, &z__[mid], &c__1);
}
i__2 = psiz2 - bsiz2;
scopy_(&i__2, &ztemp[psiz1 + bsiz2 + 1], &c__1, &z__[mid + bsiz2], &
c__1);
i__2 = *tlvls - k;
ptr += pow_ii(&c__2, &i__2);
/* L70: */
}
return 0;
/* End of SLAEDA */
} /* slaeda_ */
/* Subroutine */ int dlalsa_(integer *icompq, integer *smlsiz, integer *n,
integer *nrhs, doublereal *b, integer *ldb, doublereal *bx, integer *
ldbx, doublereal *u, integer *ldu, doublereal *vt, integer *k,
doublereal *difl, doublereal *difr, doublereal *z__, doublereal *
poles, integer *givptr, integer *givcol, integer *ldgcol, integer *
perm, doublereal *givnum, doublereal *c__, doublereal *s, doublereal *
work, integer *iwork, integer *info)
{
/* System generated locals */
integer givcol_dim1, givcol_offset, perm_dim1, perm_offset, b_dim1,
b_offset, bx_dim1, bx_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 */
static integer nlvl, sqre, i__, j;
extern /* Subroutine */ int dgemm_(char *, char *, integer *, integer *,
integer *, doublereal *, doublereal *, integer *, doublereal *,
integer *, doublereal *, doublereal *, integer *);
static integer inode, ndiml, ndimr;
extern /* Subroutine */ int dcopy_(integer *, doublereal *, integer *,
doublereal *, integer *);
static integer i1;
extern /* Subroutine */ int dlals0_(integer *, integer *, integer *,
integer *, integer *, doublereal *, integer *, doublereal *,
integer *, integer *, integer *, integer *, integer *, doublereal
*, integer *, doublereal *, doublereal *, doublereal *,
doublereal *, integer *, doublereal *, doublereal *, doublereal *,
integer *);
extern doublereal dopbl3_(char *, integer *, integer *, integer *)
;
static integer ic, lf, nd, ll, nl, nr;
extern /* Subroutine */ int dlasdt_(integer *, integer *, integer *,
integer *, integer *, integer *, integer *), xerbla_(char *,
integer *);
static integer im1, nlf, nrf, lvl, ndb1, nlp1, lvl2, nrp1;
#define difl_ref(a_1,a_2) difl[(a_2)*difl_dim1 + a_1]
#define difr_ref(a_1,a_2) difr[(a_2)*difr_dim1 + a_1]
#define perm_ref(a_1,a_2) perm[(a_2)*perm_dim1 + a_1]
#define b_ref(a_1,a_2) b[(a_2)*b_dim1 + a_1]
#define u_ref(a_1,a_2) u[(a_2)*u_dim1 + a_1]
#define z___ref(a_1,a_2) z__[(a_2)*z_dim1 + a_1]
#define poles_ref(a_1,a_2) poles[(a_2)*poles_dim1 + a_1]
#define bx_ref(a_1,a_2) bx[(a_2)*bx_dim1 + a_1]
#define vt_ref(a_1,a_2) vt[(a_2)*vt_dim1 + a_1]
#define givcol_ref(a_1,a_2) givcol[(a_2)*givcol_dim1 + a_1]
#define givnum_ref(a_1,a_2) givnum[(a_2)*givnum_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
June 30, 1999
Purpose
=======
DLALSA is an itermediate step in solving the least squares problem
by computing the SVD of the coefficient matrix in compact form (The
singular vectors are computed as products of simple orthorgonal
matrices.).
If ICOMPQ = 0, DLALSA applies the inverse of the left singular vector
matrix of an upper bidiagonal matrix to the right hand side; and if
ICOMPQ = 1, DLALSA applies the right singular vector matrix to the
right hand side. The singular vector matrices were generated in
compact form by DLALSA.
Arguments
=========
ICOMPQ (input) INTEGER
Specifies whether the left or the right singular vector
matrix is involved.
= 0: Left singular vector matrix
= 1: Right singular vector matrix
SMLSIZ (input) INTEGER
The maximum size of the subproblems at the bottom of the
computation tree.
N (input) INTEGER
The row and column dimensions of the upper bidiagonal matrix.
NRHS (input) INTEGER
The number of columns of B and BX. NRHS must be at least 1.
B (input) DOUBLE PRECISION array, dimension ( LDB, NRHS )
On input, B contains the right hand sides of the least
squares problem in rows 1 through M. On output, B contains
the solution X in rows 1 through N.
LDB (input) INTEGER
The leading dimension of B in the calling subprogram.
LDB must be at least max(1,MAX( M, N ) ).
BX (output) DOUBLE PRECISION array, dimension ( LDBX, NRHS )
On exit, the result of applying the left or right singular
vector matrix to B.
LDBX (input) INTEGER
The leading dimension of BX.
U (input) DOUBLE PRECISION array, dimension ( LDU, SMLSIZ ).
On entry, 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 (input) DOUBLE PRECISION array, dimension ( LDU, SMLSIZ+1 ).
On entry, VT' contains the right singular vector matrices of
all subproblems at the bottom level.
K (input) INTEGER array, dimension ( N ).
DIFL (input) DOUBLE PRECISION array, dimension ( LDU, NLVL ).
where NLVL = INT(log_2 (N/(SMLSIZ+1))) + 1.
DIFR (input) DOUBLE PRECISION array, dimension ( LDU, 2 * NLVL ).
On entry, DIFL(*, I) and DIFR(*, 2 * I -1) record
distances between singular values on the I-th level and
singular values on the (I -1)-th level, and DIFR(*, 2 * I)
record the normalizing factors of the right singular vectors
matrices of subproblems on I-th level.
Z (input) DOUBLE PRECISION array, dimension ( LDU, NLVL ).
On entry, Z(1, I) contains the components of the deflation-
adjusted updating row vector for subproblems on the I-th
level.
POLES (input) DOUBLE PRECISION array, dimension ( LDU, 2 * NLVL ).
On entry, POLES(*, 2 * I -1: 2 * I) contains the new and old
singular values involved in the secular equations on the I-th
level.
GIVPTR (input) INTEGER array, dimension ( N ).
On entry, GIVPTR( I ) records the number of Givens
rotations performed on the I-th problem on the computation
tree.
GIVCOL (input) INTEGER array, dimension ( LDGCOL, 2 * NLVL ).
On entry, for each I, GIVCOL(*, 2 * I - 1: 2 * I) records 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 (input) INTEGER array, dimension ( LDGCOL, NLVL ).
On entry, PERM(*, I) records permutations done on the I-th
level of the computation tree.
GIVNUM (input) DOUBLE PRECISION array, dimension ( LDU, 2 * NLVL ).
On entry, GIVNUM(*, 2 *I -1 : 2 * I) records the C- and S-
values of Givens rotations performed on the I-th level on the
computation tree.
C (input) DOUBLE PRECISION array, dimension ( N ).
On entry, if the I-th subproblem is not square,
C( I ) contains the C-value of a Givens rotation related to
the right null space of the I-th subproblem.
S (input) DOUBLE PRECISION array, dimension ( N ).
On entry, if the I-th subproblem is not square,
S( I ) contains the S-value of a Givens rotation related to
the right null space of the I-th subproblem.
WORK (workspace) DOUBLE PRECISION array.
The dimension must be at least N.
IWORK (workspace) INTEGER array.
The dimension must be at least 3 * N
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
=====================================================================
Test the input parameters.
Parameter adjustments */
b_dim1 = *ldb;
b_offset = 1 + b_dim1 * 1;
b -= b_offset;
bx_dim1 = *ldbx;
bx_offset = 1 + bx_dim1 * 1;
bx -= bx_offset;
givnum_dim1 = *ldu;
givnum_offset = 1 + givnum_dim1 * 1;
givnum -= givnum_offset;
poles_dim1 = *ldu;
poles_offset = 1 + poles_dim1 * 1;
poles -= poles_offset;
z_dim1 = *ldu;
z_offset = 1 + z_dim1 * 1;
z__ -= z_offset;
difr_dim1 = *ldu;
difr_offset = 1 + difr_dim1 * 1;
difr -= difr_offset;
difl_dim1 = *ldu;
difl_offset = 1 + difl_dim1 * 1;
difl -= difl_offset;
vt_dim1 = *ldu;
vt_offset = 1 + vt_dim1 * 1;
vt -= vt_offset;
u_dim1 = *ldu;
u_offset = 1 + u_dim1 * 1;
u -= u_offset;
--k;
--givptr;
perm_dim1 = *ldgcol;
perm_offset = 1 + perm_dim1 * 1;
perm -= perm_offset;
givcol_dim1 = *ldgcol;
givcol_offset = 1 + givcol_dim1 * 1;
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 < *smlsiz) {
*info = -3;
} else if (*nrhs < 1) {
*info = -4;
} else if (*ldb < *n) {
*info = -6;
} else if (*ldbx < *n) {
*info = -8;
} else if (*ldu < *n) {
*info = -10;
} else if (*ldgcol < *n) {
*info = -19;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("DLALSA", &i__1);
return 0;
}
/* Book-keeping and setting up the computation tree. */
inode = 1;
ndiml = inode + *n;
ndimr = ndiml + *n;
dlasdt_(n, &nlvl, &nd, &iwork[inode], &iwork[ndiml], &iwork[ndimr],
smlsiz);
/* The following code applies back the left singular vector factors.
For applying back the right singular vector factors, go to 50. */
if (*icompq == 1) {
goto L50;
}
/* The nodes on the bottom level of the tree were solved by DLASDQ.
The corresponding left and right singular vector matrices are in
explicit form. First apply back the left singular vector matrices. */
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];
nr = iwork[ndimr + i1];
nlf = ic - nl;
nrf = ic + 1;
latime_1.ops += dopbl3_("DGEMM ", &nl, nrhs, &nl);
latime_1.ops += dopbl3_("DGEMM ", &nr, nrhs, &nr);
dgemm_("T", "N", &nl, nrhs, &nl, &c_b9, &u_ref(nlf, 1), ldu, &b_ref(
nlf, 1), ldb, &c_b10, &bx_ref(nlf, 1), ldbx);
dgemm_("T", "N", &nr, nrhs, &nr, &c_b9, &u_ref(nrf, 1), ldu, &b_ref(
nrf, 1), ldb, &c_b10, &bx_ref(nrf, 1), ldbx);
/* L10: */
}
/* Next copy the rows of B that correspond to unchanged rows
in the bidiagonal matrix to BX. */
i__1 = nd;
for (i__ = 1; i__ <= i__1; ++i__) {
ic = iwork[inode + i__ - 1];
dcopy_(nrhs, &b_ref(ic, 1), ldb, &bx_ref(ic, 1), ldbx);
/* L20: */
}
/* Finally go through the left singular vector matrices of all
the other subproblems bottom-up on the tree. */
j = pow_ii(&c__2, &nlvl);
sqre = 0;
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;
--j;
dlals0_(icompq, &nl, &nr, &sqre, nrhs, &bx_ref(nlf, 1), ldbx, &
b_ref(nlf, 1), ldb, &perm_ref(nlf, lvl), &givptr[j], &
givcol_ref(nlf, lvl2), ldgcol, &givnum_ref(nlf, lvl2),
ldu, &poles_ref(nlf, lvl2), &difl_ref(nlf, lvl), &
difr_ref(nlf, lvl2), &z___ref(nlf, lvl), &k[j], &c__[j], &
s[j], &work[1], info);
/* L30: */
}
/* L40: */
}
goto L90;
/* ICOMPQ = 1: applying back the right singular vector factors. */
L50:
/* First now go through the right singular vector matrices of all
the tree nodes top-down. */
j = 0;
i__1 = nlvl;
for (lvl = 1; lvl <= i__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__2 = lvl - 1;
lf = pow_ii(&c__2, &i__2);
ll = (lf << 1) - 1;
}
i__2 = lf;
for (i__ = ll; i__ >= i__2; --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) {
sqre = 0;
} else {
sqre = 1;
}
++j;
dlals0_(icompq, &nl, &nr, &sqre, nrhs, &b_ref(nlf, 1), ldb, &
bx_ref(nlf, 1), ldbx, &perm_ref(nlf, lvl), &givptr[j], &
givcol_ref(nlf, lvl2), ldgcol, &givnum_ref(nlf, lvl2),
ldu, &poles_ref(nlf, lvl2), &difl_ref(nlf, lvl), &
difr_ref(nlf, lvl2), &z___ref(nlf, lvl), &k[j], &c__[j], &
s[j], &work[1], info);
/* L60: */
}
/* L70: */
}
/* The nodes on the bottom level of the tree were solved by DLASDQ.
The corresponding right singular vector matrices are in explicit
form. Apply them back. */
ndb1 = (nd + 1) / 2;
i__1 = nd;
for (i__ = ndb1; i__ <= i__1; ++i__) {
i1 = i__ - 1;
ic = iwork[inode + i1];
nl = iwork[ndiml + i1];
nr = iwork[ndimr + i1];
nlp1 = nl + 1;
if (i__ == nd) {
nrp1 = nr;
} else {
nrp1 = nr + 1;
}
nlf = ic - nl;
nrf = ic + 1;
latime_1.ops += dopbl3_("DGEMM ", &nlp1, nrhs, &nlp1);
latime_1.ops += dopbl3_("DGEMM ", &nrp1, nrhs, &nrp1);
dgemm_("T", "N", &nlp1, nrhs, &nlp1, &c_b9, &vt_ref(nlf, 1), ldu, &
b_ref(nlf, 1), ldb, &c_b10, &bx_ref(nlf, 1), ldbx);
dgemm_("T", "N", &nrp1, nrhs, &nrp1, &c_b9, &vt_ref(nrf, 1), ldu, &
b_ref(nrf, 1), ldb, &c_b10, &bx_ref(nrf, 1), ldbx);
/* L80: */
}
L90:
return 0;
/* End of DLALSA */
} /* dlalsa_ */