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// Copyright (C) 2003--2004 Zbigniew Leyk (zbigniew.leyk@anu.edu.au)
// and David E. Stewart (david.stewart@anu.edu.au)
// and Ronan Collobert (collober@idiap.ch)
//
// This file is part of Torch 3.1.
//
// All rights reserved.
//
// Redistribution and use in source and binary forms, with or without
// modification, are permitted provided that the following conditions
// are met:
// 1. Redistributions of source code must retain the above copyright
// notice, this list of conditions and the following disclaimer.
// 2. Redistributions in binary form must reproduce the above copyright
// notice, this list of conditions and the following disclaimer in the
// documentation and/or other materials provided with the distribution.
// 3. The name of the author may not be used to endorse or promote products
// derived from this software without specific prior written permission.
//
// THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR
// IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
// OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
// IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
// INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
// NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
// DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
// THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
// (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
// THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
/*
File containing routines for symmetric eigenvalue problems
*/
#include "mx_sym_eig.h"
namespace Torch {
#define SQRT2 1.4142135623730949
#define sgn(x) ( (x) >= 0 ? 1 : -1 )
/* trieig -- finds eigenvalues of symmetric tridiagonal matrices
-- matrix represented by a pair of vectors a (diag entries)
and b (sub- & super-diag entries)
-- eigenvalues in a on return */
void mxTriEig(Vec * a, Vec * b, Mat * mat_q)
{
int i_min, i_max;
real b_sqr, bk, ak1, bk1, ak2, bk2, z;
real c, c2, cs, s, s2, d, mu;
int n = a->n;
real *a_ptr = a->ptr;
real *b_ptr = b->ptr;
i_min = 0;
while (i_min < n) /* outer while loop */
{
/* find i_max to suit;
submatrix i_min..i_max should be irreducible */
i_max = n - 1;
for (int i = i_min; i < n - 1; i++)
{
if (b_ptr[i] == 0.0)
{
i_max = i;
break;
}
}
if (i_max <= i_min)
{
i_min = i_max + 1;
continue; /* outer while loop */
}
/* repeatedly perform QR method until matrix splits */
bool split = false;
while (!split) /* inner while loop */
{
/* find Wilkinson shift */
d = (a_ptr[i_max - 1] - a_ptr[i_max]) / 2;
b_sqr = b_ptr[i_max - 1] * b_ptr[i_max - 1];
mu = a_ptr[i_max] - b_sqr / (d + sgn(d) * sqrt(d * d + b_sqr));
/* initial Givens' rotation */
mx_givens(a_ptr[i_min] - mu, b_ptr[i_min], &c, &s);
s = -s;
if (fabs(c) < SQRT2)
{
c2 = c * c;
s2 = 1 - c2;
}
else
{
s2 = s * s;
c2 = 1 - s2;
}
cs = c * s;
ak1 =
c2 * a_ptr[i_min] + s2 * a_ptr[i_min + 1] -
2 * cs * b_ptr[i_min];
bk1 =
cs * (a_ptr[i_min] - a_ptr[i_min + 1]) + (c2 -
s2) * b_ptr[i_min];
ak2 =
s2 * a_ptr[i_min] + c2 * a_ptr[i_min + 1] +
2 * cs * b_ptr[i_min];
bk2 = (i_min < i_max - 1) ? c * b_ptr[i_min + 1] : 0.0;
z = (i_min < i_max - 1) ? -s * b_ptr[i_min + 1] : 0.0;
a_ptr[i_min] = ak1;
a_ptr[i_min + 1] = ak2;
b_ptr[i_min] = bk1;
if (i_min < i_max - 1)
b_ptr[i_min + 1] = bk2;
if (mat_q)
mx_rot_cols(mat_q, i_min, i_min + 1, c, -s, mat_q);
for (int i = i_min + 1; i < i_max; i++)
{
/* get Givens' rotation for sub-block -- k == i-1 */
mx_givens(b_ptr[i - 1], z, &c, &s);
s = -s;
/* perform Givens' rotation on sub-block */
if (fabs(c) < SQRT2)
{
c2 = c * c;
s2 = 1 - c2;
}
else
{
s2 = s * s;
c2 = 1 - s2;
}
cs = c * s;
bk = c * b_ptr[i - 1] - s * z;
ak1 = c2 * a_ptr[i] + s2 * a_ptr[i + 1] - 2 * cs * b_ptr[i];
bk1 = cs * (a_ptr[i] - a_ptr[i + 1]) + (c2 - s2) * b_ptr[i];
ak2 = s2 * a_ptr[i] + c2 * a_ptr[i + 1] + 2 * cs * b_ptr[i];
bk2 = (i + 1 < i_max) ? c * b_ptr[i + 1] : 0.0;
z = (i + 1 < i_max) ? -s * b_ptr[i + 1] : 0.0;
a_ptr[i] = ak1;
a_ptr[i + 1] = ak2;
b_ptr[i] = bk1;
if (i < i_max - 1)
b_ptr[i + 1] = bk2;
if (i > i_min)
b_ptr[i - 1] = bk;
if (mat_q)
mx_rot_cols(mat_q, i, i + 1, c, -s, mat_q);
}
/* test to see if matrix should be split */
for (int i = i_min; i < i_max; i++)
{
if (fabs(b_ptr[i]) <
REAL_EPSILON * (fabs(a_ptr[i]) + fabs(a_ptr[i + 1])))
{
b_ptr[i] = 0.0;
split = true;
}
}
}
}
}
/* symmeig -- computes eigenvalues of a dense symmetric matrix
-- mat_a **must** be symmetric on entry
-- eigenvalues stored in out
-- mat_q contains orthogonal matrix of eigenvectors
-- returns vector of eigenvalues
-- je pense: if mat_q is NULL, eigenvectors won't be computed
*/
void mxSymEig(Mat * mat_a, Mat * mat_q, Vec * out)
{
Mat *tmp = new Mat(mat_a->m, mat_a->n);
tmp->copy(mat_a);
Vec *b = new Vec(mat_a->m - 1);
Vec *diag = new Vec(mat_a->m);
Vec *beta = new Vec(mat_a->m);
mxHFactor(tmp, diag, beta);
if (mat_q)
mxMakeHQ(tmp, diag, beta, mat_q);
int i;
for (i = 0; i < mat_a->m - 1; i++)
{
out->ptr[i] = tmp->ptr[i][i];
b->ptr[i] = tmp->ptr[i][i + 1];
}
out->ptr[i] = tmp->ptr[i][i];
mxTriEig(out, b, mat_q);
delete beta;
delete diag;
delete b;
delete tmp;
}
}
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