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