📄 slaed8.c
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#include "blaswrap.h"
/* -- translated by f2c (version 19990503).
You must link the resulting object file with the libraries:
-lf2c -lm (in that order)
*/
#include "f2c.h"
/* Common Block Declarations */
struct {
real ops, itcnt;
} latime_;
#define latime_1 latime_
/* Table of constant values */
static real c_b3 = -1.f;
static integer c__1 = 1;
/* Subroutine */ int slaed8_(integer *icompq, integer *k, integer *n, integer
*qsiz, real *d__, real *q, integer *ldq, integer *indxq, real *rho,
integer *cutpnt, real *z__, real *dlamda, real *q2, integer *ldq2,
real *w, integer *perm, integer *givptr, integer *givcol, real *
givnum, integer *indxp, integer *indx, integer *info)
{
/* System generated locals */
integer q_dim1, q_offset, q2_dim1, q2_offset, i__1;
real r__1;
/* Builtin functions */
double sqrt(doublereal);
/* Local variables */
static integer jlam, imax, jmax;
extern /* Subroutine */ int srot_(integer *, real *, integer *, real *,
integer *, real *, real *);
static real c__;
static integer i__, j;
static real s, t;
extern /* Subroutine */ int sscal_(integer *, real *, real *, integer *);
static integer k2;
extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *,
integer *);
static integer n1, n2;
extern doublereal slapy2_(real *, real *);
static integer jp;
extern doublereal slamch_(char *);
extern /* Subroutine */ int xerbla_(char *, integer *);
extern integer isamax_(integer *, real *, integer *);
extern /* Subroutine */ int slamrg_(integer *, integer *, real *, integer
*, integer *, integer *), slacpy_(char *, integer *, integer *,
real *, integer *, real *, integer *);
static integer n1p1;
static real eps, tau, tol;
#define q_ref(a_1,a_2) q[(a_2)*q_dim1 + a_1]
#define q2_ref(a_1,a_2) q2[(a_2)*q2_dim1 + a_1]
#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, Oak Ridge National Lab, Argonne National Lab,
Courant Institute, NAG Ltd., and Rice University
September 30, 1994
Common block to return operation count and iteration count
ITCNT is unchanged, OPS is only incremented
Purpose
=======
SLAED8 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
=========
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.
K (output) INTEGER
The number of non-deflated eigenvalues, and 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 orthogonal matrix used to reduce
the full matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1.
D (input/output) REAL array, dimension (N)
On entry, the eigenvalues of the two submatrices to be
combined. On exit, the trailing (N-K) updated eigenvalues
(those which were deflated) sorted into increasing order.
Q (input/output) REAL array, dimension (LDQ,N)
If ICOMPQ = 0, Q is not referenced. Otherwise,
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).
INDXQ (input) INTEGER array, dimension (N)
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.
RHO (input/output) REAL
On entry, the off-diagonal element associated with the rank-1
cut which originally split the two submatrices which are now
being recombined.
On exit, RHO has been modified to the value required by
SLAED3.
CUTPNT (input) INTEGER
The location of the last eigenvalue in the leading
sub-matrix. min(1,N) <= CUTPNT <= N.
Z (input) REAL array, dimension (N)
On entry, Z contains the updating vector (the last row of
the first sub-eigenvector matrix and the first row of the
second sub-eigenvector matrix).
On exit, the contents of Z are destroyed by the updating
process.
DLAMDA (output) REAL array, dimension (N)
A copy of the first K eigenvalues which will be used by
SLAED3 to form the secular equation.
Q2 (output) REAL array, dimension (LDQ2,N)
If ICOMPQ = 0, Q2 is not referenced. Otherwise,
a copy of the first K eigenvectors which will be used by
SLAED7 in a matrix multiply (SGEMM) to update the new
eigenvectors.
LDQ2 (input) INTEGER
The leading dimension of the array Q2. LDQ2 >= max(1,N).
W (output) REAL array, dimension (N)
The first k values of the final deflation-altered z-vector and
will be passed to SLAED3.
PERM (output) INTEGER array, dimension (N)
The permutations (from deflation and sorting) to be applied
to each eigenblock.
GIVPTR (output) INTEGER
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) REAL array, dimension (2, N)
Each number indicates the S value to be used in the
corresponding Givens rotation.
INDXP (workspace) INTEGER array, dimension (N)
The permutation used to place deflated values of D at the end
of the array. 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)
The permutation used to sort the contents of D into ascending
order.
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 */
--d__;
q_dim1 = *ldq;
q_offset = 1 + q_dim1 * 1;
q -= q_offset;
--indxq;
--z__;
--dlamda;
q2_dim1 = *ldq2;
q2_offset = 1 + q2_dim1 * 1;
q2 -= q2_offset;
--w;
--perm;
givcol -= 3;
givnum -= 3;
--indxp;
--indx;
/* Function Body */
*info = 0;
if (*icompq < 0 || *icompq > 1) {
*info = -1;
} else if (*n < 0) {
*info = -3;
} else if (*icompq == 1 && *qsiz < *n) {
*info = -4;
} else if (*ldq < max(1,*n)) {
*info = -7;
} else if (*cutpnt < min(1,*n) || *cutpnt > *n) {
*info = -10;
} else if (*ldq2 < max(1,*n)) {
*info = -14;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SLAED8", &i__1);
return 0;
}
/* Quick return if possible */
if (*n == 0) {
return 0;
}
n1 = *cutpnt;
n2 = *n - n1;
n1p1 = n1 + 1;
if (*rho < 0.f) {
latime_1.ops += n2;
sscal_(&n2, &c_b3, &z__[n1p1], &c__1);
}
/* Normalize z so that norm(z) = 1 */
latime_1.ops = latime_1.ops + *n + 6;
t = 1.f / sqrt(2.f);
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
indx[j] = j;
/* L10: */
}
sscal_(n, &t, &z__[1], &c__1);
*rho = (r__1 = *rho * 2.f, dabs(r__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;
slamrg_(&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 tolerence */
imax = isamax_(n, &z__[1], &c__1);
jmax = isamax_(n, &d__[1], &c__1);
eps = slamch_("Epsilon");
tol = eps * 8.f * (r__1 = d__[jmax], dabs(r__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 * (r__1 = z__[imax], dabs(r__1)) <= tol) {
*k = 0;
if (*icompq == 0) {
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
perm[j] = indxq[indx[j]];
/* L50: */
}
} else {
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
perm[j] = indxq[indx[j]];
scopy_(qsiz, &q_ref(1, perm[j]), &c__1, &q2_ref(1, j), &c__1);
/* L60: */
}
slacpy_("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) {
latime_1.ops += 1;
if (*rho * (r__1 = z__[j], dabs(r__1)) <= tol) {
/* Deflate due to small z component. */
--k2;
indxp[k2] = j;
if (j == *n) {
goto L110;
}
} else {
jlam = j;
goto L80;
}
/* L70: */
}
L80:
++j;
if (j > *n) {
goto L100;
}
latime_1.ops += 1;
if (*rho * (r__1 = z__[j], dabs(r__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. */
latime_1.ops += 10;
tau = slapy2_(&c__, &s);
t = d__[j] - d__[jlam];
c__ /= tau;
s = -s / tau;
if ((r__1 = t * c__ * s, dabs(r__1)) <= tol) {
/* Deflation is possible. */
z__[j] = tau;
z__[jlam] = 0.f;
/* 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;
if (*icompq == 1) {
latime_1.ops += *qsiz * 6;
srot_(qsiz, &q_ref(1, indxq[indx[jlam]]), &c__1, &q_ref(1,
indxq[indx[j]]), &c__1, &c__, &s);
}
latime_1.ops += 10;
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;
L90:
if (k2 + i__ <= *n) {
if (d__[jlam] < d__[indxp[k2 + i__]]) {
indxp[k2 + i__ - 1] = indxp[k2 + i__];
indxp[k2 + i__] = jlam;
++i__;
goto L90;
} 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 L80;
L100:
/* Record the last eigenvalue. */
++(*k);
w[*k] = z__[jlam];
dlamda[*k] = d__[jlam];
indxp[*k] = jlam;
L110:
/* 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. */
if (*icompq == 0) {
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
jp = indxp[j];
dlamda[j] = d__[jp];
perm[j] = indxq[indx[jp]];
/* L120: */
}
} else {
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
jp = indxp[j];
dlamda[j] = d__[jp];
perm[j] = indxq[indx[jp]];
scopy_(qsiz, &q_ref(1, perm[j]), &c__1, &q2_ref(1, j), &c__1);
/* L130: */
}
}
/* The deflated eigenvalues and their corresponding vectors go back
into the last N - K slots of D and Q respectively. */
if (*k < *n) {
if (*icompq == 0) {
i__1 = *n - *k;
scopy_(&i__1, &dlamda[*k + 1], &c__1, &d__[*k + 1], &c__1);
} else {
i__1 = *n - *k;
scopy_(&i__1, &dlamda[*k + 1], &c__1, &d__[*k + 1], &c__1);
i__1 = *n - *k;
slacpy_("A", qsiz, &i__1, &q2_ref(1, *k + 1), ldq2, &q_ref(1, *k
+ 1), ldq);
}
}
return 0;
/* End of SLAED8 */
} /* slaed8_ */
#undef givnum_ref
#undef givcol_ref
#undef q2_ref
#undef q_ref
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