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
|
/*=============================================================================
matrix
===============================================================================
Matrix math.
=============================================================================*/
#include <assert.h>
#include <math.h>
#include <stdlib.h>
#include <stdio.h>
#include "pm_c_util.h"
#include "mallocvar.h"
#include "nstring.h"
#include "matrix.h"
static double const epsilon = 1e-10;
static void
swap(double * const aP,
double * const bP) {
double const oldA = *aP;
*aP = *bP;
*bP = oldA;
}
static void
initializeWorkMatrices(unsigned int const n,
double ** const aInit,
const double * const cInit,
double *** const aP,
double ** const cP,
const char ** const errorP) {
/*----------------------------------------------------------------------------
Allocate memory for an n x n matrix, initialize it to the value of
aInit[], and return it as *aP.
Allocate memory for an n x 1 matrix, initialize it to the value of
cInit[], and return it as *cP.
-----------------------------------------------------------------------------*/
double ** a;
double * c;
MALLOCARRAY2(a, n, n);
if (a == NULL)
pm_asprintf(errorP, "Could not get memory for a %u x %u matrix", n, n);
else {
unsigned int i;
for (i = 0; i < n; ++i) {
unsigned int j;
for (j = 0; j < n; ++j)
a[i][j] = aInit[i][j];
}
MALLOCARRAY(c, n);
if (c == NULL)
pm_asprintf(errorP, "Could not get memory for a %u x 1 matrix", n);
else {
unsigned int i;
for (i = 0; i < n; ++i)
c[i] = cInit[i];
*errorP = NULL;
}
if (*errorP)
free(a);
}
*aP = a;
*cP = c;
}
static void
findLargestIthCoeff(unsigned int const n,
double ** const a,
unsigned int const i,
unsigned int * const istarP,
const char ** const errorP) {
/*----------------------------------------------------------------------------
Among the 'i'th and following rows in 'a' (which has 'n' total rows),
find the one with the largest 'i'th column.
And it had better be greater than zero; if not, we fail (return *errorP
non-null).
Return its index as *istarP.
-----------------------------------------------------------------------------*/
double maxSoFar;
unsigned int maxIdx;
unsigned int ii;
for (ii = i, maxSoFar = 0.0; ii < n; ++ii) {
double const thisA = fabs(a[ii][i]);
if (thisA >= maxSoFar) {
maxIdx = ii;
maxSoFar = thisA;
}
}
if (maxSoFar < epsilon) {
const char * const baseMsg = "Matrix equation has no unique solution";
if (pm_have_float_format())
pm_asprintf(errorP, "%s. (debug: coeff %u %e < %e)",
baseMsg, i, maxSoFar, epsilon);
else
pm_asprintf(errorP, "%s", baseMsg);
} else {
*istarP = maxIdx;
*errorP = NULL;
}
}
static void
eliminateOneUnknown(unsigned int const i,
unsigned int const n,
double ** const a,
double * const c,
const char ** const errorP) {
unsigned int maxRow;
findLargestIthCoeff(n, a, i, &maxRow, errorP);
if (!*errorP) {
/* swap rows 'i' and 'maxRow' in 'a' and 'c', so that the ith
row has the largest ith coefficient.
*/
unsigned int j;
for (j = 0; j < n; j++)
swap(&a[maxRow][j], &a[i][j]);
swap(&c[maxRow], &c[i]);
/* Combine rows so that the ith coefficient in every row below
the ith is zero.
*/
{
unsigned int ii;
for (ii = i+1; ii < n; ++ii) {
double const multiplier = a[ii][i] / a[i][i];
/* This is what we multiply the whole ith row by to make
its ith coefficient equal to that in the iith row.
*/
unsigned int j;
/* Combine ith row into iith row so that the ith coefficient
in the iith is zero.
*/
c[ii] = c[ii] - multiplier * c[i];
for (j = 0; j < n; ++j)
a[ii][j] = a[ii][j] - multiplier * a[i][j];
assert(a[ii][i] < epsilon);
}
}
*errorP = NULL;
}
}
void
pm_solvelineareq(double ** const aArg,
double * const x,
double * const cArg,
unsigned int const n,
const char ** const errorP) {
/*----------------------------------------------------------------------------
Solve the matrix equation 'a' * 'x' = 'c' for 'x'.
'n' is the dimension of the matrices. 'a' is 'n' x 'n',
while 'x' and 'c' are 'n' x 1.
-----------------------------------------------------------------------------*/
/* We use Gaussian reduction. */
double ** a;
double * c;
initializeWorkMatrices(n, aArg, cArg, &a, &c, errorP);
if (!*errorP) {
unsigned int i;
for (i = 0, *errorP = NULL; i < n && !*errorP; ++i)
eliminateOneUnknown(i, n, a, c, errorP);
if (!*errorP) {
/* a[] now has all zeros in the lower left triangle. */
/* Work from the bottom up to solve for the unknowns x[], from
the a and c rows in question and all the x[] below it
*/
unsigned int k;
for (k = 0; k < n; ++k) {
unsigned int const m = n - k - 1;
unsigned int j;
double xwork;
for (j = m+1, xwork = c[m]; j < n; ++j)
xwork -= a[m][j] * x[j];
x[m] = xwork / a[m][m];
}
}
}
pm_freearray2((void**)a);
free(c);
}
|
