1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269
|
/* $Id: matrix.c,v 1.8 2003/04/07 16:27:10 ukai Exp $ */
/*
* matrix.h, matrix.c: Liner equation solver using LU decomposition.
*
* by K.Okabe Aug. 1999
*
* LUfactor, LUsolve, Usolve and Lsolve, are based on the functions in
* Meschach Library Version 1.2b.
*/
/**************************************************************************
**
** 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.
**
***************************************************************************/
#include "config.h"
#include "matrix.h"
#include "alloc.h"
/*
* Macros from "fm.h".
*/
#define SWAPD(a,b) { double tmp = a; a = b; b = tmp; }
#define SWAPI(a,b) { int tmp = a; a = b; b = tmp; }
#ifdef HAVE_FLOAT_H
#include <float.h>
#endif /* not HAVE_FLOAT_H */
#if defined(DBL_MAX)
static double Tiny = 10.0 / DBL_MAX;
#elif defined(FLT_MAX)
static double Tiny = 10.0 / FLT_MAX;
#else /* not defined(FLT_MAX) */
static double Tiny = 1.0e-30;
#endif /* not defined(FLT_MAX */
/*
* LUfactor -- gaussian elimination with scaled partial pivoting
* -- Note: returns LU matrix which is A.
*/
int
LUfactor(Matrix A, int *indexarray)
{
int dim = A->dim, i, j, k, i_max, k_max;
Vector scale;
double mx, tmp;
scale = new_vector(dim);
for (i = 0; i < dim; i++)
indexarray[i] = i;
for (i = 0; i < dim; i++) {
mx = 0.;
for (j = 0; j < dim; j++) {
tmp = fabs(M_VAL(A, i, j));
if (mx < tmp)
mx = tmp;
}
scale->ve[i] = mx;
}
k_max = dim - 1;
for (k = 0; k < k_max; k++) {
mx = 0.;
i_max = -1;
for (i = k; i < dim; i++) {
if (fabs(scale->ve[i]) >= Tiny * fabs(M_VAL(A, i, k))) {
tmp = fabs(M_VAL(A, i, k)) / scale->ve[i];
if (mx < tmp) {
mx = tmp;
i_max = i;
}
}
}
if (i_max == -1) {
M_VAL(A, k, k) = 0.;
continue;
}
if (i_max != k) {
SWAPI(indexarray[i_max], indexarray[k]);
for (j = 0; j < dim; j++)
SWAPD(M_VAL(A, i_max, j), M_VAL(A, k, j));
}
for (i = k + 1; i < dim; i++) {
tmp = M_VAL(A, i, k) = M_VAL(A, i, k) / M_VAL(A, k, k);
for (j = k + 1; j < dim; j++)
M_VAL(A, i, j) -= tmp * M_VAL(A, k, j);
}
}
return 0;
}
/*
* LUsolve -- given an LU factorisation in A, solve Ax=b.
*/
int
LUsolve(Matrix A, int *indexarray, Vector b, Vector x)
{
int i, dim = A->dim;
for (i = 0; i < dim; i++)
x->ve[i] = b->ve[indexarray[i]];
if (Lsolve(A, x, x, 1.) == -1 || Usolve(A, x, x, 0.) == -1)
return -1;
return 0;
}
/* m_inverse -- returns inverse of A, provided A is not too rank deficient
* -- uses LU factorisation */
#if 0
Matrix
m_inverse(Matrix A, Matrix out)
{
int *indexarray = NewAtom_N(int, A->dim);
Matrix A1 = new_matrix(A->dim);
m_copy(A, A1);
LUfactor(A1, indexarray);
return LUinverse(A1, indexarray, out);
}
#endif /* 0 */
Matrix
LUinverse(Matrix A, int *indexarray, Matrix out)
{
int i, j, dim = A->dim;
Vector tmp, tmp2;
if (!out)
out = new_matrix(dim);
tmp = new_vector(dim);
tmp2 = new_vector(dim);
for (i = 0; i < dim; i++) {
for (j = 0; j < dim; j++)
tmp->ve[j] = 0.;
tmp->ve[i] = 1.;
if (LUsolve(A, indexarray, tmp, tmp2) == -1)
return NULL;
for (j = 0; j < dim; j++)
M_VAL(out, j, i) = tmp2->ve[j];
}
return out;
}
/*
* Usolve -- back substitution with optional over-riding diagonal
* -- can be in-situ but doesn't need to be.
*/
int
Usolve(Matrix mat, Vector b, Vector out, double diag)
{
int i, j, i_lim, dim = mat->dim;
double sum;
for (i = dim - 1; i >= 0; i--) {
if (b->ve[i] != 0.)
break;
else
out->ve[i] = 0.;
}
i_lim = i;
for (; i >= 0; i--) {
sum = b->ve[i];
for (j = i + 1; j <= i_lim; j++)
sum -= M_VAL(mat, i, j) * out->ve[j];
if (diag == 0.) {
if (fabs(M_VAL(mat, i, i)) <= Tiny * fabs(sum))
return -1;
else
out->ve[i] = sum / M_VAL(mat, i, i);
}
else
out->ve[i] = sum / diag;
}
return 0;
}
/*
* Lsolve -- forward elimination with (optional) default diagonal value.
*/
int
Lsolve(Matrix mat, Vector b, Vector out, double diag)
{
int i, j, i_lim, dim = mat->dim;
double sum;
for (i = 0; i < dim; i++) {
if (b->ve[i] != 0.)
break;
else
out->ve[i] = 0.;
}
i_lim = i;
for (; i < dim; i++) {
sum = b->ve[i];
for (j = i_lim; j < i; j++)
sum -= M_VAL(mat, i, j) * out->ve[j];
if (diag == 0.) {
if (fabs(M_VAL(mat, i, i)) <= Tiny * fabs(sum))
return -1;
else
out->ve[i] = sum / M_VAL(mat, i, i);
}
else
out->ve[i] = sum / diag;
}
return 0;
}
/*
* new_matrix -- generate a nxn matrix.
*/
Matrix
new_matrix(int n)
{
Matrix mat;
mat = New(struct matrix);
mat->dim = n;
mat->me = NewAtom_N(double, n * n);
return mat;
}
/*
* new_matrix -- generate a n-dimension vector.
*/
Vector
new_vector(int n)
{
Vector vec;
vec = New(struct vector);
vec->dim = n;
vec->ve = NewAtom_N(double, n);
return vec;
}
|