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
|
/* automatically generated by sed scripts from the c source named below: */
/* cddproj.c: Polyhedral Projections in cddlib
written by Komei Fukuda, fukuda@cs.mcgill.ca
Version 0.94, Aug. 4, 2005
*/
/* cddlib : C-library of the double description method for
computing all vertices and extreme rays of the polyhedron
P= {x : b - A x >= 0}.
Please read COPYING (GNU General Public Licence) and
the manual cddlibman.tex for detail.
*/
#include "setoper.h" /* set operation library header (Ver. June 1, 2000 or later) */
#include "cdd_f.h"
#include <stdio.h>
#include <stdlib.h>
#include <time.h>
#include <math.h>
#include <string.h>
ddf_MatrixPtr ddf_BlockElimination(ddf_MatrixPtr M, ddf_colset delset, ddf_ErrorType *error)
/* Eliminate the variables (columns) delset by
the Block Elimination with ddf_DoubleDescription algorithm.
Given (where y is to be eliminated):
c1 + A1 x + B1 y >= 0
c2 + A2 x + B2 y = 0
1. First construct the dual system: z1^T B1 + z2^T B2 = 0, z1 >= 0.
2. Compute the generators of the dual.
3. Then take the linear combination of the original system with each generator.
4. Remove redundant inequalies.
*/
{
ddf_MatrixPtr Mdual=NULL, Mproj=NULL, Gdual=NULL;
ddf_rowrange i,h,m,mproj,mdual,linsize;
ddf_colrange j,k,d,dproj,ddual,delsize;
ddf_colindex delindex;
myfloat temp,prod;
ddf_PolyhedraPtr dualpoly;
ddf_ErrorType err=ddf_NoError;
ddf_boolean localdebug=ddf_FALSE;
*error=ddf_NoError;
m= M->rowsize;
d= M->colsize;
delindex=(long*)calloc(d+1,sizeof(long));
ddf_init(temp);
ddf_init(prod);
k=0; delsize=0;
for (j=1; j<=d; j++){
if (set_member(j, delset)){
k++; delsize++;
delindex[k]=j; /* stores the kth deletion column index */
}
}
if (localdebug) ddf_WriteMatrix(stdout, M);
linsize=set_card(M->linset);
ddual=m+1;
mdual=delsize + m - linsize; /* #equalitions + dimension of z1 */
/* setup the dual matrix */
Mdual=ddf_CreateMatrix(mdual, ddual);
Mdual->representation=ddf_Inequality;
for (i = 1; i <= delsize; i++){
set_addelem(Mdual->linset,i); /* equality */
for (j = 1; j <= m; j++) {
ddf_set(Mdual->matrix[i-1][j], M->matrix[j-1][delindex[i]-1]);
}
}
k=0;
for (i = 1; i <= m; i++){
if (!set_member(i, M->linset)){
/* set nonnegativity for the dual variable associated with
each non-linearity inequality. */
k++;
ddf_set(Mdual->matrix[delsize+k-1][i], ddf_one);
}
}
/* 2. Compute the generators of the dual system. */
dualpoly=ddf_DDMatrix2Poly(Mdual, &err);
Gdual=ddf_CopyGenerators(dualpoly);
/* 3. Take the linear combination of the original system with each generator. */
dproj=d-delsize;
mproj=Gdual->rowsize;
Mproj=ddf_CreateMatrix(mproj, dproj);
Mproj->representation=ddf_Inequality;
set_copy(Mproj->linset, Gdual->linset);
for (i=1; i<=mproj; i++){
k=0;
for (j=1; j<=d; j++){
if (!set_member(j, delset)){
k++; /* new index of the variable x_j */
ddf_set(prod, ddf_purezero);
for (h = 1; h <= m; h++){
ddf_mul(temp,M->matrix[h-1][j-1],Gdual->matrix[i-1][h]);
ddf_add(prod,prod,temp);
}
ddf_set(Mproj->matrix[i-1][k-1],prod);
}
}
}
if (localdebug) printf("Size of the projection system: %ld x %ld\n", mproj, dproj);
ddf_FreePolyhedra(dualpoly);
free(delindex);
ddf_clear(temp);
ddf_clear(prod);
ddf_FreeMatrix(Mdual);
ddf_FreeMatrix(Gdual);
return Mproj;
}
ddf_MatrixPtr ddf_FourierElimination(ddf_MatrixPtr M,ddf_ErrorType *error)
/* Eliminate the last variable (column) from the given H-matrix using
the standard Fourier Elimination.
*/
{
ddf_MatrixPtr Mnew=NULL;
ddf_rowrange i,inew,ip,in,iz,m,mpos=0,mneg=0,mzero=0,mnew;
ddf_colrange j,d,dnew;
ddf_rowindex posrowindex, negrowindex,zerorowindex;
myfloat temp1,temp2;
ddf_boolean localdebug=ddf_FALSE;
*error=ddf_NoError;
m= M->rowsize;
d= M->colsize;
if (d<=1){
*error=ddf_ColIndexOutOfRange;
if (localdebug) {
printf("The number of column is too small: %ld for Fourier's Elimination.\n",d);
}
goto _L99;
}
if (M->representation==ddf_Generator){
*error=ddf_NotAvailForV;
if (localdebug) {
printf("Fourier's Elimination cannot be applied to a V-polyhedron.\n");
}
goto _L99;
}
if (set_card(M->linset)>0){
*error=ddf_CannotHandleLinearity;
if (localdebug) {
printf("The Fourier Elimination function does not handle equality in this version.\n");
}
goto _L99;
}
/* Create temporary spaces to be removed at the end of this function */
posrowindex=(long*)calloc(m+1,sizeof(long));
negrowindex=(long*)calloc(m+1,sizeof(long));
zerorowindex=(long*)calloc(m+1,sizeof(long));
ddf_init(temp1);
ddf_init(temp2);
for (i = 1; i <= m; i++) {
if (ddf_Positive(M->matrix[i-1][d-1])){
mpos++;
posrowindex[mpos]=i;
} else if (ddf_Negative(M->matrix[i-1][d-1])) {
mneg++;
negrowindex[mneg]=i;
} else {
mzero++;
zerorowindex[mzero]=i;
}
} /*of i*/
if (localdebug) {
ddf_WriteMatrix(stdout, M);
printf("No of (+ - 0) rows = (%ld, %ld, %ld)\n", mpos,mneg, mzero);
}
/* The present code generates so many redundant inequalities and thus
is quite useless, except for very small examples
*/
mnew=mzero+mpos*mneg; /* the total number of rows after elimination */
dnew=d-1;
Mnew=ddf_CreateMatrix(mnew, dnew);
ddf_CopyArow(Mnew->rowvec, M->rowvec, dnew);
/* set_copy(Mnew->linset,M->linset); */
Mnew->numbtype=M->numbtype;
Mnew->representation=M->representation;
Mnew->objective=M->objective;
/* Copy the inequalities independent of x_d to the top of the new matrix. */
for (iz = 1; iz <= mzero; iz++){
for (j = 1; j <= dnew; j++) {
ddf_set(Mnew->matrix[iz-1][j-1], M->matrix[zerorowindex[iz]-1][j-1]);
}
}
/* Create the new inequalities by combining x_d positive and negative ones. */
inew=mzero; /* the index of the last x_d zero inequality */
for (ip = 1; ip <= mpos; ip++){
for (in = 1; in <= mneg; in++){
inew++;
ddf_neg(temp1, M->matrix[negrowindex[in]-1][d-1]);
for (j = 1; j <= dnew; j++) {
ddf_LinearComb(temp2,M->matrix[posrowindex[ip]-1][j-1],temp1,\
M->matrix[negrowindex[in]-1][j-1],\
M->matrix[posrowindex[ip]-1][d-1]);
ddf_set(Mnew->matrix[inew-1][j-1],temp2);
}
ddf_Normalize(dnew,Mnew->matrix[inew-1]);
}
}
free(posrowindex);
free(negrowindex);
free(zerorowindex);
ddf_clear(temp1);
ddf_clear(temp2);
_L99:
return Mnew;
}
/* end of cddproj.c */
|
