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/* poly/qr.c
*
* Copyright (C) 1996, 1997, 1998, 1999, 2000 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 2 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.
*/
static int qr_companion (double *h, size_t nc, gsl_complex_packed_ptr z);
static int
qr_companion (double *h, size_t nc, gsl_complex_packed_ptr zroot)
{
double t = 0.0;
size_t iterations, e, i, j, k, m;
double w, x, y, s, z;
double p = 0, q = 0, r = 0;
/* FIXME: if p,q,r, are not set to zero then the compiler complains
that they ``might be used uninitialized in this
function''. Looking at the code this does seem possible, so this
should be checked. */
int notlast;
size_t n = nc;
next_root:
if (n == 0)
return GSL_SUCCESS ;
iterations = 0;
next_iteration:
for (e = n; e >= 2; e--)
{
double a1 = fabs (FMAT (h, e, e - 1, nc));
double a2 = fabs (FMAT (h, e - 1, e - 1, nc));
double a3 = fabs (FMAT (h, e, e, nc));
if (a1 <= GSL_DBL_EPSILON * (a2 + a3))
break;
}
x = FMAT (h, n, n, nc);
if (e == n)
{
GSL_SET_COMPLEX_PACKED (zroot, n-1, x + t, 0); /* one real root */
n--;
goto next_root;
/*continue;*/
}
y = FMAT (h, n - 1, n - 1, nc);
w = FMAT (h, n - 1, n, nc) * FMAT (h, n, n - 1, nc);
if (e == n - 1)
{
p = (y - x) / 2;
q = p * p + w;
y = sqrt (fabs (q));
x += t;
if (q > 0) /* two real roots */
{
if (p < 0)
y = -y;
y += p;
GSL_SET_COMPLEX_PACKED (zroot, n-1, x - w / y, 0);
GSL_SET_COMPLEX_PACKED (zroot, n-2, x + y, 0);
}
else
{
GSL_SET_COMPLEX_PACKED (zroot, n-1, x + p, -y);
GSL_SET_COMPLEX_PACKED (zroot, n-2, x + p, y);
}
n -= 2;
goto next_root;
/*continue;*/
}
/* No more roots found yet, do another iteration */
if (iterations == 60) /* increased from 30 to 60 */
{
/* too many iterations - give up! */
return GSL_FAILURE ;
}
if (iterations % 10 == 0 && iterations > 0)
{
/* use an exceptional shift */
t += x;
for (i = 1; i <= n; i++)
{
FMAT (h, i, i, nc) -= x;
}
s = fabs (FMAT (h, n, n - 1, nc)) + fabs (FMAT (h, n - 1, n - 2, nc));
y = 0.75 * s;
x = y;
w = -0.4375 * s * s;
}
iterations++;
for (m = n - 2; m >= e; m--)
{
double a1, a2, a3;
z = FMAT (h, m, m, nc);
r = x - z;
s = y - z;
p = FMAT (h, m, m + 1, nc) + (r * s - w) / FMAT (h, m + 1, m, nc);
q = FMAT (h, m + 1, m + 1, nc) - z - r - s;
r = FMAT (h, m + 2, m + 1, nc);
s = fabs (p) + fabs (q) + fabs (r);
p /= s;
q /= s;
r /= s;
if (m == e)
break;
a1 = fabs (FMAT (h, m, m - 1, nc));
a2 = fabs (FMAT (h, m - 1, m - 1, nc));
a3 = fabs (FMAT (h, m + 1, m + 1, nc));
if (a1 * (fabs (q) + fabs (r)) <= GSL_DBL_EPSILON * fabs (p) * (a2 + a3))
break;
}
for (i = m + 2; i <= n; i++)
{
FMAT (h, i, i - 2, nc) = 0;
}
for (i = m + 3; i <= n; i++)
{
FMAT (h, i, i - 3, nc) = 0;
}
/* double QR step */
for (k = m; k <= n - 1; k++)
{
notlast = (k != n - 1);
if (k != m)
{
p = FMAT (h, k, k - 1, nc);
q = FMAT (h, k + 1, k - 1, nc);
r = notlast ? FMAT (h, k + 2, k - 1, nc) : 0.0;
x = fabs (p) + fabs (q) + fabs (r);
if (x == 0)
continue; /* FIXME????? */
p /= x;
q /= x;
r /= x;
}
s = sqrt (p * p + q * q + r * r);
if (p < 0)
s = -s;
if (k != m)
{
FMAT (h, k, k - 1, nc) = -s * x;
}
else if (e != m)
{
FMAT (h, k, k - 1, nc) *= -1;
}
p += s;
x = p / s;
y = q / s;
z = r / s;
q /= p;
r /= p;
/* do row modifications */
for (j = k; j <= n; j++)
{
p = FMAT (h, k, j, nc) + q * FMAT (h, k + 1, j, nc);
if (notlast)
{
p += r * FMAT (h, k + 2, j, nc);
FMAT (h, k + 2, j, nc) -= p * z;
}
FMAT (h, k + 1, j, nc) -= p * y;
FMAT (h, k, j, nc) -= p * x;
}
j = (k + 3 < n) ? (k + 3) : n;
/* do column modifications */
for (i = e; i <= j; i++)
{
p = x * FMAT (h, i, k, nc) + y * FMAT (h, i, k + 1, nc);
if (notlast)
{
p += z * FMAT (h, i, k + 2, nc);
FMAT (h, i, k + 2, nc) -= p * r;
}
FMAT (h, i, k + 1, nc) -= p * q;
FMAT (h, i, k, nc) -= p;
}
}
goto next_iteration;
}
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