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mesch12a.zip
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bkpfacto.c
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1994-01-13
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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.
**
***************************************************************************/
/*
Matrix factorisation routines to work with the other matrix files.
*/
static char rcsid[] = "$Id: bkpfacto.c,v 1.7 1994/01/13 05:45:50 des Exp $";
#include <stdio.h>
#include <math.h>
#include "matrix.h"
#include "matrix2.h"
#define btos(x) ((x) ? "TRUE" : "FALSE")
/* Most matrix factorisation routines are in-situ unless otherwise specified */
#define alpha 0.6403882032022076 /* = (1+sqrt(17))/8 */
/* sqr -- returns square of x -- utility function */
double sqr(x)
double x;
{ return x*x; }
/* interchange -- a row/column swap routine */
static void interchange(A,i,j)
MAT *A; /* assumed != NULL & also SQUARE */
int i, j; /* assumed in range */
{
Real **A_me, tmp;
int k, n;
A_me = A->me; n = A->n;
if ( i == j )
return;
if ( i > j )
{ k = i; i = j; j = k; }
for ( k = 0; k < i; k++ )
{
/* tmp = A_me[k][i]; */
tmp = m_entry(A,k,i);
/* A_me[k][i] = A_me[k][j]; */
m_set_val(A,k,i,m_entry(A,k,j));
/* A_me[k][j] = tmp; */
m_set_val(A,k,j,tmp);
}
for ( k = j+1; k < n; k++ )
{
/* tmp = A_me[j][k]; */
tmp = m_entry(A,j,k);
/* A_me[j][k] = A_me[i][k]; */
m_set_val(A,j,k,m_entry(A,i,k));
/* A_me[i][k] = tmp; */
m_set_val(A,i,k,tmp);
}
for ( k = i+1; k < j; k++ )
{
/* tmp = A_me[k][j]; */
tmp = m_entry(A,k,j);
/* A_me[k][j] = A_me[i][k]; */
m_set_val(A,k,j,m_entry(A,i,k));
/* A_me[i][k] = tmp; */
m_set_val(A,i,k,tmp);
}
/* tmp = A_me[i][i]; */
tmp = m_entry(A,i,i);
/* A_me[i][i] = A_me[j][j]; */
m_set_val(A,i,i,m_entry(A,j,j));
/* A_me[j][j] = tmp; */
m_set_val(A,j,j,tmp);
}
/* BKPfactor -- Bunch-Kaufman-Parlett factorisation of A in-situ
-- A is factored into the form P'AP = MDM' where
P is a permutation matrix, M lower triangular and D is block
diagonal with blocks of size 1 or 2
-- P is stored in pivot; blocks[i]==i iff D[i][i] is a block */
MAT *BKPfactor(A,pivot,blocks)
MAT *A;
PERM *pivot, *blocks;
{
int i, j, k, n, onebyone, r;
Real **A_me, aii, aip1, aip1i, lambda, sigma, tmp;
Real det, s, t;
if ( ! A || ! pivot || ! blocks )
error(E_NULL,"BKPfactor");
if ( A->m != A->n )
error(E_SQUARE,"BKPfactor");
if ( A->m != pivot->size || pivot->size != blocks->size )
error(E_SIZES,"BKPfactor");
n = A->n;
A_me = A->me;
px_ident(pivot); px_ident(blocks);
for ( i = 0; i < n; i = onebyone ? i+1 : i+2 )
{
/* printf("# Stage: %d\n",i); */
aii = fabs(m_entry(A,i,i));
lambda = 0.0; r = (i+1 < n) ? i+1 : i;
for ( k = i+1; k < n; k++ )
{
tmp = fabs(m_entry(A,i,k));
if ( tmp >= lambda )
{
lambda = tmp;
r = k;
}
}
/* printf("# lambda = %g, r = %d\n", lambda, r); */
/* printf("# |A[%d][%d]| = %g\n",r,r,fabs(m_entry(A,r,r))); */
/* determine if 1x1 or 2x2 block, and do pivoting if needed */
if ( aii >= alpha*lambda )
{
onebyone = TRUE;
goto dopivot;
}
/* compute sigma */
sigma = 0.0;
for ( k = i; k < n; k++ )
{
if ( k == r )
continue;
tmp = ( k > r ) ? fabs(m_entry(A,r,k)) :
fabs(m_entry(A,k,r));
if ( tmp > sigma )
sigma = tmp;
}
if ( aii*sigma >= alpha*sqr(lambda) )
onebyone = TRUE;
else if ( fabs(m_entry(A,r,r)) >= alpha*sigma )
{
/* printf("# Swapping rows/cols %d and %d\n",i,r); */
interchange(A,i,r);
px_transp(pivot,i,r);
onebyone = TRUE;
}
else
{
/* printf("# Swapping rows/cols %d and %d\n",i+1,r); */
interchange(A,i+1,r);
px_transp(pivot,i+1,r);
px_transp(blocks,i,i+1);
onebyone = FALSE;
}
/* printf("onebyone = %s\n",btos(onebyone)); */
/* printf("# Matrix so far (@checkpoint A) =\n"); */
/* m_output(A); */
/* printf("# pivot =\n"); px_output(pivot); */
/* printf("# blocks =\n"); px_output(blocks); */
dopivot:
if ( onebyone )
{ /* do one by one block */
if ( m_entry(A,i,i) != 0.0 )
{
aii = m_entry(A,i,i);
for ( j = i+1; j < n; j++ )
{
tmp = m_entry(A,i,j)/aii;
for ( k = j; k < n; k++ )
m_sub_val(A,j,k,tmp*m_entry(A,i,k));
m_set_val(A,i,j,tmp);
}
}
}
else /* onebyone == FALSE */
{ /* do two by two block */
det = m_entry(A,i,i)*m_entry(A,i+1,i+1)-sqr(m_entry(A,i,i+1));
/* Must have det < 0 */
/* printf("# det = %g\n",det); */
aip1i = m_entry(A,i,i+1)/det;
aii = m_entry(A,i,i)/det;
aip1 = m_entry(A,i+1,i+1)/det;
for ( j = i+2; j < n; j++ )
{
s = - aip1i*m_entry(A,i+1,j) + aip1*m_entry(A,i,j);
t = - aip1i*m_entry(A,i,j) + aii*m_entry(A,i+1,j);
for ( k = j; k < n; k++ )
m_sub_val(A,j,k,m_entry(A,i,k)*s + m_entry(A,i+1,k)*t);
m_set_val(A,i,j,s);
m_set_val(A,i+1,j,t);
}
}
/* printf("# Matrix so far (@checkpoint B) =\n"); */
/* m_output(A); */
/* printf("# pivot =\n"); px_output(pivot); */
/* printf("# blocks =\n"); px_output(blocks); */
}
/* set lower triangular half */
for ( i = 0; i < A->m; i++ )
for ( j = 0; j < i; j++ )
m_set_val(A,i,j,m_entry(A,j,i));
return A;
}
/* BKPsolve -- solves A.x = b where A has been factored a la BKPfactor()
-- returns x, which is created if NULL */
VEC *BKPsolve(A,pivot,block,b,x)
MAT *A;
PERM *pivot, *block;
VEC *b, *x;
{
static VEC *tmp=VNULL; /* dummy storage needed */
int i, j, n, onebyone;
Real **A_me, a11, a12, a22, b1, b2, det, sum, *tmp_ve, tmp_diag;
if ( ! A || ! pivot || ! block || ! b )
error(E_NULL,"BKPsolve");
if ( A->m != A->n )
error(E_SQUARE,"BKPsolve");
n = A->n;
if ( b->dim != n || pivot->size != n || block->size != n )
error(E_SIZES,"BKPsolve");
x = v_resize(x,n);
tmp = v_resize(tmp,n);
MEM_STAT_REG(tmp,TYPE_VEC);
A_me = A->me; tmp_ve = tmp->ve;
px_vec(pivot,b,tmp);
/* solve for lower triangular part */
for ( i = 0; i < n; i++ )
{
sum = v_entry(tmp,i);
if ( block->pe[i] < i )
for ( j = 0; j < i-1; j++ )
sum -= m_entry(A,i,j)*v_entry(tmp,j);
else
for ( j = 0; j < i; j++ )
sum -= m_entry(A,i,j)*v_entry(tmp,j);
v_set_val(tmp,i,sum);
}
/* printf("# BKPsolve: solving L part: tmp =\n"); v_output(tmp); */
/* solve for diagonal part */
for ( i = 0; i < n; i = onebyone ? i+1 : i+2 )
{
onebyone = ( block->pe[i] == i );
if ( onebyone )
{
tmp_diag = m_entry(A,i,i);
if ( tmp_diag == 0.0 )
error(E_SING,"BKPsolve");
/* tmp_ve[i] /= tmp_diag; */
v_set_val(tmp,i,v_entry(tmp,i) / tmp_diag);
}
else
{
a11 = m_entry(A,i,i);
a22 = m_entry(A,i+1,i+1);
a12 = m_entry(A,i+1,i);
b1 = v_entry(tmp,i); b2 = v_entry(tmp,i+1);
det = a11*a22-a12*a12; /* < 0 : see BKPfactor() */
if ( det == 0.0 )
error(E_SING,"BKPsolve");
det = 1/det;
v_set_val(tmp,i,det*(a22*b1-a12*b2));
v_set_val(tmp,i+1,det*(a11*b2-a12*b1));
}
}
/* printf("# BKPsolve: solving D part: tmp =\n"); v_output(tmp); */
/* solve for transpose of lower traingular part */
for ( i = n-1; i >= 0; i-- )
{ /* use symmetry of factored form to get stride 1 */
sum = v_entry(tmp,i);
if ( block->pe[i] > i )
for ( j = i+2; j < n; j++ )
sum -= m_entry(A,i,j)*v_entry(tmp,j);
else
for ( j = i+1; j < n; j++ )
sum -= m_entry(A,i,j)*v_entry(tmp,j);
v_set_val(tmp,i,sum);
}
/* printf("# BKPsolve: solving L^T part: tmp =\n");v_output(tmp); */
/* and do final permutation */
x = pxinv_vec(pivot,tmp,x);
return x;
}