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/* eigen/qrstep.c
*
* Copyright (C) 2007, 2010 Brian Gough
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation; either version 3 of the License, or (at
* your option) any later version.
*
* This program is distributed in the hope that it will be useful, but
* WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
* General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with this program; if not, write to the Free Software
* Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
*/
/* remove off-diagonal elements which are neglegible compared with the
neighboring diagonal elements */
static void
chop_small_elements (const size_t N, const double d[], double sd[])
{
double d_i = d[0];
size_t i;
for (i = 0; i < N - 1; i++)
{
double sd_i = sd[i];
double d_ip1 = d[i + 1];
if (fabs (sd_i) < GSL_DBL_EPSILON * (fabs (d_i) + fabs (d_ip1)))
{
sd[i] = 0.0;
}
d_i = d_ip1;
}
}
/* Generate a Givens rotation (cos,sin) which takes v=(x,y) to (|v|,0)
From Golub and Van Loan, "Matrix Computations", Section 5.1.8 */
inline static void
create_givens (const double a, const double b, double *c, double *s)
{
if (b == 0)
{
*c = 1;
*s = 0;
}
else if (fabs (b) > fabs (a))
{
double t = -a / b;
double s1 = 1.0 / sqrt (1 + t * t);
*s = s1;
*c = s1 * t;
}
else
{
double t = -b / a;
double c1 = 1.0 / sqrt (1 + t * t);
*c = c1;
*s = c1 * t;
}
}
inline static double
trailing_eigenvalue (const size_t n, const double d[], const double sd[])
{
double ta = d[n - 2];
double tb = d[n - 1];
double tab = sd[n - 2];
double dt = (ta - tb) / 2.0;
double mu;
if (dt > 0)
{
mu = tb - tab * (tab / (dt + hypot (dt, tab)));
}
else if (dt == 0)
{
mu = tb - fabs(tab);
}
else
{
mu = tb + tab * (tab / ((-dt) + hypot (dt, tab)));
}
return mu;
}
static void
qrstep (const size_t n, double d[], double sd[], double gc[], double gs[])
{
double x, z;
double ak, bk, zk, ap, bp, aq, bq;
size_t k;
double mu = trailing_eigenvalue (n, d, sd);
/* If mu is large relative to d_0 and sd_0 then the Givens rotation
will have no effect, leading to an infinite loop.
We set mu to zero in this case, which at least diagonalises the
submatrix [d_0, sd_0 ; sd_0, d_0] and allows further progress. */
if (GSL_DBL_EPSILON * fabs(mu) > (fabs(d[0]) + fabs(sd[0]))) {
mu = 0;
}
x = d[0] - mu;
z = sd[0];
ak = 0;
bk = 0;
zk = 0;
ap = d[0];
bp = sd[0];
aq = d[1];
if (n == 2)
{
double c, s;
create_givens (x, z, &c, &s);
if (gc != NULL)
gc[0] = c;
if (gs != NULL)
gs[0] = s;
{
double ap1 = c * (c * ap - s * bp) + s * (s * aq - c * bp);
double bp1 = c * (s * ap + c * bp) - s * (s * bp + c * aq);
double aq1 = s * (s * ap + c * bp) + c * (s * bp + c * aq);
ak = ap1;
bk = bp1;
ap = aq1;
}
d[0] = ak;
sd[0] = bk;
d[1] = ap;
return;
}
bq = sd[1];
for (k = 0; k < n - 1; k++)
{
double c, s;
create_givens (x, z, &c, &s);
/* store Givens rotation */
if (gc != NULL)
gc[k] = c;
if (gs != NULL)
gs[k] = s;
/* compute G' T G */
{
double bk1 = c * bk - s * zk;
double ap1 = c * (c * ap - s * bp) + s * (s * aq - c * bp);
double bp1 = c * (s * ap + c * bp) - s * (s * bp + c * aq);
double zp1 = -s * bq;
double aq1 = s * (s * ap + c * bp) + c * (s * bp + c * aq);
double bq1 = c * bq;
ak = ap1;
bk = bp1;
zk = zp1;
ap = aq1;
bp = bq1;
if (k < n - 2)
aq = d[k + 2];
if (k < n - 3)
bq = sd[k + 2];
d[k] = ak;
if (k > 0)
sd[k - 1] = bk1;
if (k < n - 2)
sd[k + 1] = bp;
x = bk;
z = zk;
}
}
/* k = n - 1 */
d[k] = ap;
sd[k - 1] = bk;
}
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