/* csyequb.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 csyequb_(char *uplo, integer *n, complex *a, integer *
lda, real *s, real *scond, real *amax, complex *work, integer *info)
{
/* System generated locals */
integer a_dim1, a_offset, i__1, i__2, i__3, i__4, i__5;
real r__1, r__2, r__3, r__4;
doublereal d__1;
complex q__1, q__2, q__3, q__4;
/* Builtin functions */
double r_imag(complex *), sqrt(doublereal), log(doublereal), pow_ri(real *
, integer *);
/* Local variables */
real d__;
integer i__, j;
real t, u, c0, c1, c2, si;
logical up;
real avg, std, tol, base;
integer iter;
real smin, smax, scale;
extern logical lsame_(char *, char *);
real sumsq;
extern doublereal slamch_(char *);
extern /* Subroutine */ int xerbla_(char *, integer *);
real bignum;
extern /* Subroutine */ int classq_(integer *, complex *, integer *, real
*, real *);
real 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 */
/* ======= */
/* CSYEQUB 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) COMPLEX 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) REAL array, dimension (N) */
/* If INFO = 0, S contains the scale factors for A. */
/* SCOND (output) REAL */
/* 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) REAL */
/* 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 .. */
/* .. */
/* .. Statement Functions .. */
/* .. */
/* Statement Function Definitions */
/* .. */
/* .. Executable Statements .. */
/* Test the 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_("CSYEQUB", &i__1);
return 0;
}
up = lsame_(uplo, "U");
*amax = 0.f;
/* Quick return if possible. */
if (*n == 0) {
*scond = 1.f;
return 0;
}
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
s[i__] = 0.f;
}
*amax = 0.f;
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 */
i__3 = i__ + j * a_dim1;
r__3 = s[i__], r__4 = (r__1 = a[i__3].r, dabs(r__1)) + (r__2 =
r_imag(&a[i__ + j * a_dim1]), dabs(r__2));
s[i__] = dmax(r__3,r__4);
/* Computing MAX */
i__3 = i__ + j * a_dim1;
r__3 = s[j], r__4 = (r__1 = a[i__3].r, dabs(r__1)) + (r__2 =
r_imag(&a[i__ + j * a_dim1]), dabs(r__2));
s[j] = dmax(r__3,r__4);
/* Computing MAX */
i__3 = i__ + j * a_dim1;
r__3 = *amax, r__4 = (r__1 = a[i__3].r, dabs(r__1)) + (r__2 =
r_imag(&a[i__ + j * a_dim1]), dabs(r__2));
*amax = dmax(r__3,r__4);
}
/* Computing MAX */
i__2 = j + j * a_dim1;
r__3 = s[j], r__4 = (r__1 = a[i__2].r, dabs(r__1)) + (r__2 =
r_imag(&a[j + j * a_dim1]), dabs(r__2));
s[j] = dmax(r__3,r__4);
/* Computing MAX */
i__2 = j + j * a_dim1;
r__3 = *amax, r__4 = (r__1 = a[i__2].r, dabs(r__1)) + (r__2 =
r_imag(&a[j + j * a_dim1]), dabs(r__2));
*amax = dmax(r__3,r__4);
}
} else {
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
/* Computing MAX */
i__2 = j + j * a_dim1;
r__3 = s[j], r__4 = (r__1 = a[i__2].r, dabs(r__1)) + (r__2 =
r_imag(&a[j + j * a_dim1]), dabs(r__2));
s[j] = dmax(r__3,r__4);
/* Computing MAX */
i__2 = j + j * a_dim1;
r__3 = *amax, r__4 = (r__1 = a[i__2].r, dabs(r__1)) + (r__2 =
r_imag(&a[j + j * a_dim1]), dabs(r__2));
*amax = dmax(r__3,r__4);
i__2 = *n;
for (i__ = j + 1; i__ <= i__2; ++i__) {
/* Computing MAX */
i__3 = i__ + j * a_dim1;
r__3 = s[i__], r__4 = (r__1 = a[i__3].r, dabs(r__1)) + (r__2 =
r_imag(&a[i__ + j * a_dim1]), dabs(r__2));
s[i__] = dmax(r__3,r__4);
/* Computing MAX */
i__3 = i__ + j * a_dim1;
r__3 = s[j], r__4 = (r__1 = a[i__3].r, dabs(r__1)) + (r__2 =
r_imag(&a[i__ + j * a_dim1]), dabs(r__2));
s[j] = dmax(r__3,r__4);
/* Computing MAX */
i__3 = i__ + j * a_dim1;
r__3 = *amax, r__4 = (r__1 = a[i__3].r, dabs(r__1)) + (r__2 =
r_imag(&a[i__ + j * a_dim1]), dabs(r__2));
*amax = dmax(r__3,r__4);
}
}
}
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
s[j] = 1.f / s[j];
}
tol = 1.f / sqrt(*n * 2.f);
for (iter = 1; iter <= 100; ++iter) {
scale = 0.f;
sumsq = 0.f;
/* beta = |A|s */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = i__;
work[i__2].r = 0.f, work[i__2].i = 0.f;
}
if (up) {
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
i__2 = j - 1;
for (i__ = 1; i__ <= i__2; ++i__) {
i__3 = i__ + j * a_dim1;
t = (r__1 = a[i__3].r, dabs(r__1)) + (r__2 = r_imag(&a[
i__ + j * a_dim1]), dabs(r__2));
i__3 = i__;
i__4 = i__;
i__5 = i__ + j * a_dim1;
r__3 = ((r__1 = a[i__5].r, dabs(r__1)) + (r__2 = r_imag(&
a[i__ + j * a_dim1]), dabs(r__2))) * s[j];
q__1.r = work[i__4].r + r__3, q__1.i = work[i__4].i;
work[i__3].r = q__1.r, work[i__3].i = q__1.i;
i__3 = j;
i__4 = j;
i__5 = i__ + j * a_dim1;
r__3 = ((r__1 = a[i__5].r, dabs(r__1)) + (r__2 = r_imag(&
a[i__ + j * a_dim1]), dabs(r__2))) * s[i__];
q__1.r = work[i__4].r + r__3, q__1.i = work[i__4].i;
work[i__3].r = q__1.r, work[i__3].i = q__1.i;
}
i__2 = j;
i__3 = j;
i__4 = j + j * a_dim1;
r__3 = ((r__1 = a[i__4].r, dabs(r__1)) + (r__2 = r_imag(&a[j
+ j * a_dim1]), dabs(r__2))) * s[j];
q__1.r = work[i__3].r + r__3, q__1.i = work[i__3].i;
work[i__2].r = q__1.r, work[i__2].i = q__1.i;
}
} else {
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
i__2 = j;
i__3 = j;
i__4 = j + j * a_dim1;
r__3 = ((r__1 = a[i__4].r, dabs(r__1)) + (r__2 = r_imag(&a[j
+ j * a_dim1]), dabs(r__2))) * s[j];
q__1.r = work[i__3].r + r__3, q__1.i = work[i__3].i;
work[i__2].r = q__1.r, work[i__2].i = q__1.i;
i__2 = *n;
for (i__ = j + 1; i__ <= i__2; ++i__) {
i__3 = i__ + j * a_dim1;
t = (r__1 = a[i__3].r, dabs(r__1)) + (r__2 = r_imag(&a[
i__ + j * a_dim1]), dabs(r__2));
i__3 = i__;
i__4 = i__;
i__5 = i__ + j * a_dim1;
r__3 = ((r__1 = a[i__5].r, dabs(r__1)) + (r__2 = r_imag(&
a[i__ + j * a_dim1]), dabs(r__2))) * s[j];
q__1.r = work[i__4].r + r__3, q__1.i = work[i__4].i;
work[i__3].r = q__1.r, work[i__3].i = q__1.i;
i__3 = j;
i__4 = j;
i__5 = i__ + j * a_dim1;
r__3 = ((r__1 = a[i__5].r, dabs(r__1)) + (r__2 = r_imag(&
a[i__ + j * a_dim1]), dabs(r__2))) * s[i__];
q__1.r = work[i__4].r + r__3, q__1.i = work[i__4].i;
work[i__3].r = q__1.r, work[i__3].i = q__1.i;
}
}
}
/* avg = s^T beta / n */
avg = 0.f;
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = i__;
i__3 = i__;
q__2.r = s[i__2] * work[i__3].r, q__2.i = s[i__2] * work[i__3].i;
q__1.r = avg + q__2.r, q__1.i = q__2.i;
avg = q__1.r;
}
avg /= *n;
std = 0.f;
i__1 = *n << 1;
for (i__ = *n + 1; i__ <= i__1; ++i__) {
i__2 = i__;
i__3 = i__ - *n;
i__4 = i__ - *n;
q__2.r = s[i__3] * work[i__4].r, q__2.i = s[i__3] * work[i__4].i;
q__1.r = q__2.r - avg, q__1.i = q__2.i;
work[i__2].r = q__1.r, work[i__2].i = q__1.i;
}
classq_(n, &work[*n + 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__) {
i__2 = i__ + i__ * a_dim1;
t = (r__1 = a[i__2].r, dabs(r__1)) + (r__2 = r_imag(&a[i__ + i__ *
a_dim1]), dabs(r__2));
si = s[i__];
c2 = (*n - 1) * t;
i__2 = *n - 2;
i__3 = i__;
r__1 = t * si;
q__2.r = work[i__3].r - r__1, q__2.i = work[i__3].i;
d__1 = (doublereal) i__2;
q__1.r = d__1 * q__2.r, q__1.i = d__1 * q__2.i;
c1 = q__1.r;
r__1 = -(t * si) * si;
i__2 = i__;
d__1 = 2.;
q__4.r = d__1 * work[i__2].r, q__4.i = d__1 * work[i__2].i;
q__3.r = si * q__4.r, q__3.i = si * q__4.i;
q__2.r = r__1 + q__3.r, q__2.i = q__3.i;
r__2 = *n * avg;
q__1.r = q__2.r - r__2, q__1.i = q__2.i;
c0 = q__1.r;
d__ = c1 * c1 - c0 * 4 * c2;
if (d__ <= 0.f) {
*info = -1;
return 0;
}
si = c0 * -2 / (c1 + sqrt(d__));
d__ = si - s[i__];
u = 0.f;
if (up) {
i__2 = i__;
for (j = 1; j <= i__2; ++j) {
i__3 = j + i__ * a_dim1;
t = (r__1 = a[i__3].r, dabs(r__1)) + (r__2 = r_imag(&a[j
+ i__ * a_dim1]), dabs(r__2));
u += s[j] * t;
i__3 = j;
i__4 = j;
r__1 = d__ * t;
q__1.r = work[i__4].r + r__1, q__1.i = work[i__4].i;
work[i__3].r = q__1.r, work[i__3].i = q__1.i;
}
i__2 = *n;
for (j = i__ + 1; j <= i__2; ++j) {
i__3 = i__ + j * a_dim1;
t = (r__1 = a[i__3].r, dabs(r__1)) + (r__2 = r_imag(&a[
i__ + j * a_dim1]), dabs(r__2));
u += s[j] * t;
i__3 = j;
i__4 = j;
r__1 = d__ * t;
q__1.r = work[i__4].r + r__1, q__1.i = work[i__4].i;
work[i__3].r = q__1.r, work[i__3].i = q__1.i;
}
} else {
i__2 = i__;
for (j = 1; j <= i__2; ++j) {
i__3 = i__ + j * a_dim1;
t = (r__1 = a[i__3].r, dabs(r__1)) + (r__2 = r_imag(&a[
i__ + j * a_dim1]), dabs(r__2));
u += s[j] * t;
i__3 = j;
i__4 = j;
r__1 = d__ * t;
q__1.r = work[i__4].r + r__1, q__1.i = work[i__4].i;
work[i__3].r = q__1.r, work[i__3].i = q__1.i;
}
i__2 = *n;
for (j = i__ + 1; j <= i__2; ++j) {
i__3 = j + i__ * a_dim1;
t = (r__1 = a[i__3].r, dabs(r__1)) + (r__2 = r_imag(&a[j
+ i__ * a_dim1]), dabs(r__2));
u += s[j] * t;
i__3 = j;
i__4 = j;
r__1 = d__ * t;
q__1.r = work[i__4].r + r__1, q__1.i = work[i__4].i;
work[i__3].r = q__1.r, work[i__3].i = q__1.i;
}
}
i__2 = i__;
q__4.r = u + work[i__2].r, q__4.i = work[i__2].i;
q__3.r = d__ * q__4.r, q__3.i = d__ * q__4.i;
d__1 = (doublereal) (*n);
q__2.r = q__3.r / d__1, q__2.i = q__3.i / d__1;
q__1.r = avg + q__2.r, q__1.i = q__2.i;
avg = q__1.r;
s[i__] = si;
}
}
L999:
smlnum = slamch_("SAFEMIN");
bignum = 1.f / smlnum;
smin = bignum;
smax = 0.f;
t = 1.f / sqrt(avg);
base = slamch_("B");
u = 1.f / log(base);
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = (integer) (u * log(s[i__] * t));
s[i__] = pow_ri(&base, &i__2);
/* Computing MIN */
r__1 = smin, r__2 = s[i__];
smin = dmin(r__1,r__2);
/* Computing MAX */
r__1 = smax, r__2 = s[i__];
smax = dmax(r__1,r__2);
}
*scond = dmax(smin,smlnum) / dmin(smax,bignum);
return 0;
} /* csyequb_ */