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/**************************************************************************
**
** Copyright (C) 1993 David E. Steward & Zbigniew Leyk, all rights reserved.
**
** Meschach Library
**
** This Meschach Library is provided "as is" without any express
** or implied warranty of any kind with respect to this software.
** In particular the authors shall not be liable for any direct,
** indirect, special, incidental or consequential damages arising
** in any way from use of the software.
**
** Everyone is granted permission to copy, modify and redistribute this
** Meschach Library, provided:
** 1. All copies contain this copyright notice.
** 2. All modified copies shall carry a notice stating who
** made the last modification and the date of such modification.
** 3. No charge is made for this software or works derived from it.
** This clause shall not be construed as constraining other software
** distributed on the same medium as this software, nor is a
** distribution fee considered a charge.
**
***************************************************************************/
/*
File containing routines for symmetric eigenvalue problems
*/
#include <stdio.h>
#include "matrix.h"
#include "matrix2.h"
#include <math.h>
static char rcsid[] = "$Id: symmeig.c,v 1.6 1995/03/27 15:45:55 des Exp $";
#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 */
VEC *trieig(a,b,Q)
VEC *a, *b;
MAT *Q;
{
int i, i_min, i_max, n, split;
Real *a_ve, *b_ve;
Real b_sqr, bk, ak1, bk1, ak2, bk2, z;
Real c, c2, cs, s, s2, d, mu;
if ( ! a || ! b )
error(E_NULL,"trieig");
if ( a->dim != b->dim + 1 || ( Q && Q->m != a->dim ) )
error(E_SIZES,"trieig");
if ( Q && Q->m != Q->n )
error(E_SQUARE,"trieig");
n = a->dim;
a_ve = a->ve; b_ve = b->ve;
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 ( i = i_min; i < n-1; i++ )
if ( b_ve[i] == 0.0 )
{ i_max = i; break; }
if ( i_max <= i_min )
{
/* printf("# i_min = %d, i_max = %d\n",i_min,i_max); */
i_min = i_max + 1;
continue; /* outer while loop */
}
/* printf("# i_min = %d, i_max = %d\n",i_min,i_max); */
/* repeatedly perform QR method until matrix splits */
split = FALSE;
while ( ! split ) /* inner while loop */
{
/* find Wilkinson shift */
d = (a_ve[i_max-1] - a_ve[i_max])/2;
b_sqr = b_ve[i_max-1]*b_ve[i_max-1];
mu = a_ve[i_max] - b_sqr/(d + sgn(d)*sqrt(d*d+b_sqr));
/* printf("# Wilkinson shift = %g\n",mu); */
/* initial Givens' rotation */
givens(a_ve[i_min]-mu,b_ve[i_min],&c,&s);
s = -s;
/* printf("# c = %g, s = %g\n",c,s); */
if ( fabs(c) < SQRT2 )
{ c2 = c*c; s2 = 1-c2; }
else
{ s2 = s*s; c2 = 1-s2; }
cs = c*s;
ak1 = c2*a_ve[i_min]+s2*a_ve[i_min+1]-2*cs*b_ve[i_min];
bk1 = cs*(a_ve[i_min]-a_ve[i_min+1]) +
(c2-s2)*b_ve[i_min];
ak2 = s2*a_ve[i_min]+c2*a_ve[i_min+1]+2*cs*b_ve[i_min];
bk2 = ( i_min < i_max-1 ) ? c*b_ve[i_min+1] : 0.0;
z = ( i_min < i_max-1 ) ? -s*b_ve[i_min+1] : 0.0;
a_ve[i_min] = ak1;
a_ve[i_min+1] = ak2;
b_ve[i_min] = bk1;
if ( i_min < i_max-1 )
b_ve[i_min+1] = bk2;
if ( Q )
rot_cols(Q,i_min,i_min+1,c,-s,Q);
/* printf("# z = %g\n",z); */
/* printf("# a [temp1] =\n"); v_output(a); */
/* printf("# b [temp1] =\n"); v_output(b); */
for ( i = i_min+1; i < i_max; i++ )
{
/* get Givens' rotation for sub-block -- k == i-1 */
givens(b_ve[i-1],z,&c,&s);
s = -s;
/* printf("# c = %g, s = %g\n",c,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_ve[i-1] - s*z;
ak1 = c2*a_ve[i]+s2*a_ve[i+1]-2*cs*b_ve[i];
bk1 = cs*(a_ve[i]-a_ve[i+1]) +
(c2-s2)*b_ve[i];
ak2 = s2*a_ve[i]+c2*a_ve[i+1]+2*cs*b_ve[i];
bk2 = ( i+1 < i_max ) ? c*b_ve[i+1] : 0.0;
z = ( i+1 < i_max ) ? -s*b_ve[i+1] : 0.0;
a_ve[i] = ak1; a_ve[i+1] = ak2;
b_ve[i] = bk1;
if ( i < i_max-1 )
b_ve[i+1] = bk2;
if ( i > i_min )
b_ve[i-1] = bk;
if ( Q )
rot_cols(Q,i,i+1,c,-s,Q);
/* printf("# a [temp2] =\n"); v_output(a); */
/* printf("# b [temp2] =\n"); v_output(b); */
}
/* test to see if matrix should be split */
for ( i = i_min; i < i_max; i++ )
if ( fabs(b_ve[i]) < MACHEPS*
(fabs(a_ve[i])+fabs(a_ve[i+1])) )
{ b_ve[i] = 0.0; split = TRUE; }
/* printf("# a =\n"); v_output(a); */
/* printf("# b =\n"); v_output(b); */
}
}
return a;
}
/* symmeig -- computes eigenvalues of a dense symmetric matrix
-- A **must** be symmetric on entry
-- eigenvalues stored in out
-- Q contains orthogonal matrix of eigenvectors
-- returns vector of eigenvalues */
VEC *symmeig(A,Q,out)
MAT *A, *Q;
VEC *out;
{
int i;
static MAT *tmp = MNULL;
static VEC *b = VNULL, *diag = VNULL, *beta = VNULL;
if ( ! A )
error(E_NULL,"symmeig");
if ( A->m != A->n )
error(E_SQUARE,"symmeig");
if ( ! out || out->dim != A->m )
out = v_resize(out,A->m);
tmp = m_resize(tmp,A->m,A->n);
tmp = m_copy(A,tmp);
b = v_resize(b,A->m - 1);
diag = v_resize(diag,(u_int)A->m);
beta = v_resize(beta,(u_int)A->m);
MEM_STAT_REG(tmp,TYPE_MAT);
MEM_STAT_REG(b,TYPE_VEC);
MEM_STAT_REG(diag,TYPE_VEC);
MEM_STAT_REG(beta,TYPE_VEC);
Hfactor(tmp,diag,beta);
if ( Q )
makeHQ(tmp,diag,beta,Q);
for ( i = 0; i < A->m - 1; i++ )
{
out->ve[i] = tmp->me[i][i];
b->ve[i] = tmp->me[i][i+1];
}
out->ve[i] = tmp->me[i][i];
trieig(out,b,Q);
return out;
}
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