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/* dsyequb.f -- translated by f2c (version 20061008).
You must link the resulting object file with libf2c:
on Microsoft Windows system, link with libf2c.lib;
on Linux or Unix systems, link with .../path/to/libf2c.a -lm
or, if you install libf2c.a in a standard place, with -lf2c -lm
-- in that order, at the end of the command line, as in
cc *.o -lf2c -lm
Source for libf2c is in /netlib/f2c/libf2c.zip, e.g.,
http://www.netlib.org/f2c/libf2c.zip
*/
#include "f2c.h"
#include "blaswrap.h"
/* Table of constant values */
static integer c__1 = 1;
/* Subroutine */ int dsyequb_(char *uplo, integer *n, doublereal *a, integer *
lda, doublereal *s, doublereal *scond, doublereal *amax, doublereal *
work, integer *info)
{
/* System generated locals */
integer a_dim1, a_offset, i__1, i__2;
doublereal d__1, d__2, d__3;
/* Builtin functions */
double sqrt(doublereal), log(doublereal), pow_di(doublereal *, integer *);
/* Local variables */
doublereal d__;
integer i__, j;
doublereal t, u, c0, c1, c2, si;
logical up;
doublereal avg, std, tol, base;
integer iter;
doublereal smin, smax, scale;
extern logical lsame_(char *, char *);
doublereal sumsq;
extern doublereal dlamch_(char *);
extern /* Subroutine */ int xerbla_(char *, integer *);
doublereal bignum;
extern /* Subroutine */ int dlassq_(integer *, doublereal *, integer *,
doublereal *, doublereal *);
doublereal smlnum;
/* -- LAPACK routine (version 3.2) -- */
/* -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and -- */
/* -- Jason Riedy of Univ. of California Berkeley. -- */
/* -- November 2008 -- */
/* -- LAPACK is a software package provided by Univ. of Tennessee, -- */
/* -- Univ. of California Berkeley and NAG Ltd. -- */
/* .. */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* DSYEQUB computes row and column scalings intended to equilibrate a */
/* symmetric matrix A and reduce its condition number */
/* (with respect to the two-norm). S contains the scale factors, */
/* S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with */
/* elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This */
/* choice of S puts the condition number of B within a factor N of the */
/* smallest possible condition number over all possible diagonal */
/* scalings. */
/* Arguments */
/* ========= */
/* N (input) INTEGER */
/* The order of the matrix A. N >= 0. */
/* A (input) DOUBLE PRECISION array, dimension (LDA,N) */
/* The N-by-N symmetric matrix whose scaling */
/* factors are to be computed. Only the diagonal elements of A */
/* are referenced. */
/* LDA (input) INTEGER */
/* The leading dimension of the array A. LDA >= max(1,N). */
/* S (output) DOUBLE PRECISION array, dimension (N) */
/* If INFO = 0, S contains the scale factors for A. */
/* SCOND (output) DOUBLE PRECISION */
/* If INFO = 0, S contains the ratio of the smallest S(i) to */
/* the largest S(i). If SCOND >= 0.1 and AMAX is neither too */
/* large nor too small, it is not worth scaling by S. */
/* AMAX (output) DOUBLE PRECISION */
/* Absolute value of largest matrix element. If AMAX is very */
/* close to overflow or very close to underflow, the matrix */
/* should be scaled. */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* > 0: if INFO = i, the i-th diagonal element is nonpositive. */
/* Further Details */
/* ======= ======= */
/* Reference: Livne, O.E. and Golub, G.H., "Scaling by Binormalization", */
/* Numerical Algorithms, vol. 35, no. 1, pp. 97-120, January 2004. */
/* DOI 10.1023/B:NUMA.0000016606.32820.69 */
/* Tech report version: http://ruready.utah.edu/archive/papers/bin.pdf */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Executable Statements .. */
/* Test input parameters. */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
--s;
--work;
/* Function Body */
*info = 0;
if (! (lsame_(uplo, "U") || lsame_(uplo, "L"))) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*lda < max(1,*n)) {
*info = -4;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("DSYEQUB", &i__1);
return 0;
}
up = lsame_(uplo, "U");
*amax = 0.;
/* Quick return if possible. */
if (*n == 0) {
*scond = 1.;
return 0;
}
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
s[i__] = 0.;
}
*amax = 0.;
if (up) {
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
i__2 = j - 1;
for (i__ = 1; i__ <= i__2; ++i__) {
/* Computing MAX */
d__2 = s[i__], d__3 = (d__1 = a[i__ + j * a_dim1], abs(d__1));
s[i__] = max(d__2,d__3);
/* Computing MAX */
d__2 = s[j], d__3 = (d__1 = a[i__ + j * a_dim1], abs(d__1));
s[j] = max(d__2,d__3);
/* Computing MAX */
d__2 = *amax, d__3 = (d__1 = a[i__ + j * a_dim1], abs(d__1));
*amax = max(d__2,d__3);
}
/* Computing MAX */
d__2 = s[j], d__3 = (d__1 = a[j + j * a_dim1], abs(d__1));
s[j] = max(d__2,d__3);
/* Computing MAX */
d__2 = *amax, d__3 = (d__1 = a[j + j * a_dim1], abs(d__1));
*amax = max(d__2,d__3);
}
} else {
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
/* Computing MAX */
d__2 = s[j], d__3 = (d__1 = a[j + j * a_dim1], abs(d__1));
s[j] = max(d__2,d__3);
/* Computing MAX */
d__2 = *amax, d__3 = (d__1 = a[j + j * a_dim1], abs(d__1));
*amax = max(d__2,d__3);
i__2 = *n;
for (i__ = j + 1; i__ <= i__2; ++i__) {
/* Computing MAX */
d__2 = s[i__], d__3 = (d__1 = a[i__ + j * a_dim1], abs(d__1));
s[i__] = max(d__2,d__3);
/* Computing MAX */
d__2 = s[j], d__3 = (d__1 = a[i__ + j * a_dim1], abs(d__1));
s[j] = max(d__2,d__3);
/* Computing MAX */
d__2 = *amax, d__3 = (d__1 = a[i__ + j * a_dim1], abs(d__1));
*amax = max(d__2,d__3);
}
}
}
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
s[j] = 1. / s[j];
}
tol = 1. / sqrt(*n * 2.);
for (iter = 1; iter <= 100; ++iter) {
scale = 0.;
sumsq = 0.;
/* BETA = |A|S */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
work[i__] = 0.;
}
if (up) {
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
i__2 = j - 1;
for (i__ = 1; i__ <= i__2; ++i__) {
t = (d__1 = a[i__ + j * a_dim1], abs(d__1));
work[i__] += (d__1 = a[i__ + j * a_dim1], abs(d__1)) * s[
j];
work[j] += (d__1 = a[i__ + j * a_dim1], abs(d__1)) * s[
i__];
}
work[j] += (d__1 = a[j + j * a_dim1], abs(d__1)) * s[j];
}
} else {
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
work[j] += (d__1 = a[j + j * a_dim1], abs(d__1)) * s[j];
i__2 = *n;
for (i__ = j + 1; i__ <= i__2; ++i__) {
t = (d__1 = a[i__ + j * a_dim1], abs(d__1));
work[i__] += (d__1 = a[i__ + j * a_dim1], abs(d__1)) * s[
j];
work[j] += (d__1 = a[i__ + j * a_dim1], abs(d__1)) * s[
i__];
}
}
}
/* avg = s^T beta / n */
avg = 0.;
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
avg += s[i__] * work[i__];
}
avg /= *n;
std = 0.;
i__1 = *n * 3;
for (i__ = (*n << 1) + 1; i__ <= i__1; ++i__) {
work[i__] = s[i__ - (*n << 1)] * work[i__ - (*n << 1)] - avg;
}
dlassq_(n, &work[(*n << 1) + 1], &c__1, &scale, &sumsq);
std = scale * sqrt(sumsq / *n);
if (std < tol * avg) {
goto L999;
}
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
t = (d__1 = a[i__ + i__ * a_dim1], abs(d__1));
si = s[i__];
c2 = (*n - 1) * t;
c1 = (*n - 2) * (work[i__] - t * si);
c0 = -(t * si) * si + work[i__] * 2 * si - *n * avg;
d__ = c1 * c1 - c0 * 4 * c2;
if (d__ <= 0.) {
*info = -1;
return 0;
}
si = c0 * -2 / (c1 + sqrt(d__));
d__ = si - s[i__];
u = 0.;
if (up) {
i__2 = i__;
for (j = 1; j <= i__2; ++j) {
t = (d__1 = a[j + i__ * a_dim1], abs(d__1));
u += s[j] * t;
work[j] += d__ * t;
}
i__2 = *n;
for (j = i__ + 1; j <= i__2; ++j) {
t = (d__1 = a[i__ + j * a_dim1], abs(d__1));
u += s[j] * t;
work[j] += d__ * t;
}
} else {
i__2 = i__;
for (j = 1; j <= i__2; ++j) {
t = (d__1 = a[i__ + j * a_dim1], abs(d__1));
u += s[j] * t;
work[j] += d__ * t;
}
i__2 = *n;
for (j = i__ + 1; j <= i__2; ++j) {
t = (d__1 = a[j + i__ * a_dim1], abs(d__1));
u += s[j] * t;
work[j] += d__ * t;
}
}
avg += (u + work[i__]) * d__ / *n;
s[i__] = si;
}
}
L999:
smlnum = dlamch_("SAFEMIN");
bignum = 1. / smlnum;
smin = bignum;
smax = 0.;
t = 1. / sqrt(avg);
base = dlamch_("B");
u = 1. / log(base);
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = (integer) (u * log(s[i__] * t));
s[i__] = pow_di(&base, &i__2);
/* Computing MIN */
d__1 = smin, d__2 = s[i__];
smin = min(d__1,d__2);
/* Computing MAX */
d__1 = smax, d__2 = s[i__];
smax = max(d__1,d__2);
}
*scond = max(smin,smlnum) / min(smax,bignum);
return 0;
} /* dsyequb_ */
|
