1 #ifndef SEEN_SYSEQ_H
2 #define SEEN_SYSEQ_H
4 /*
5 * Auxiliary routines to solve systems of linear equations in several variants and sizes.
6 *
7 * Authors:
8 * Maximilian Albert <Anhalter42@gmx.de>
9 *
10 * Copyright (C) 2007 Authors
11 *
12 * Released under GNU GPL, read the file 'COPYING' for more information
13 */
15 #include <algorithm>
16 #include <iostream>
17 #include <iomanip>
18 #include <vector>
19 #include "math.h"
21 namespace SysEq {
23 enum SolutionKind {
24 unique = 0,
25 ambiguous,
26 no_solution,
27 solution_exists // FIXME: remove this; does not yield enough information
28 };
30 inline void explain(SolutionKind sol) {
31 switch (sol) {
32 case SysEq::unique:
33 std::cout << "unique" << std::endl;
34 break;
35 case SysEq::ambiguous:
36 std::cout << "ambiguous" << std::endl;
37 break;
38 case SysEq::no_solution:
39 std::cout << "no solution" << std::endl;
40 break;
41 case SysEq::solution_exists:
42 std::cout << "solution exists" << std::endl;
43 break;
44 }
45 }
47 inline double
48 determinant3x3 (double A[3][3]) {
49 return (A[0][0]*A[1][1]*A[2][2] +
50 A[0][1]*A[1][2]*A[2][0] +
51 A[0][2]*A[1][0]*A[2][1] -
52 A[0][0]*A[1][2]*A[2][1] -
53 A[0][1]*A[1][0]*A[2][2] -
54 A[0][2]*A[1][1]*A[2][0]);
55 }
57 /* Determinant of the 3x3 matrix having a, b, and c as columns */
58 inline double
59 determinant3v (const double a[3], const double b[3], const double c[3]) {
60 return (a[0]*b[1]*c[2] +
61 a[1]*b[2]*c[0] +
62 a[2]*b[0]*c[1] -
63 a[0]*b[2]*c[1] -
64 a[1]*b[0]*c[2] -
65 a[2]*b[1]*c[0]);
66 }
68 /* Copy the elements of A into B */
69 template <int S, int T>
70 inline void copy_mat(double A[S][T], double B[S][T]) {
71 for (int i = 0; i < S; ++i) {
72 for (int j = 0; j < T; ++j) {
73 B[i][j] = A[i][j];
74 }
75 }
76 }
78 template <int S, int T>
79 inline void print_mat (const double A[S][T]) {
80 std::cout.setf(std::ios::left, std::ios::internal);
81 for (int i = 0; i < S; ++i) {
82 for (int j = 0; j < T; ++j) {
83 printf ("%8.2f ", A[i][j]);
84 }
85 std::cout << std::endl;;
86 }
87 }
89 /* Multiplication of two matrices */
90 template <int S, int U, int T>
91 inline void multiply(double A[S][U], double B[U][T], double res[S][T]) {
92 for (int i = 0; i < S; ++i) {
93 for (int j = 0; j < T; ++j) {
94 double sum = 0;
95 for (int k = 0; k < U; ++k) {
96 sum += A[i][k] * B[k][j];
97 }
98 res[i][j] = sum;
99 }
100 }
101 }
103 /*
104 * Multiplication of a matrix with a vector (for convenience, because with the previous
105 * multiplication function we would always have to write v[i][0] for elements of the vector.
106 */
107 template <int S, int T>
108 inline void multiply(double A[S][T], double v[T], double res[S]) {
109 for (int i = 0; i < S; ++i) {
110 double sum = 0;
111 for (int k = 0; k < T; ++k) {
112 sum += A[i][k] * v[k];
113 }
114 res[i] = sum;
115 }
116 }
118 // Remark: Since we are using templates, we cannot separate declarations from definitions (which would
119 // result in linker errors but have to include the definitions here for the following functions.
120 // FIXME: Maybe we should rework all this by using vector<vector<double> > structures for matrices
121 // instead of double[S][T]. This would allow us to avoid templates. Would the performance degrade?
123 /*
124 * Find the element of maximal absolute value in row i that
125 * does not lie in one of the columns given in avoid_cols.
126 */
127 template <int S, int T>
128 static int find_pivot(const double A[S][T], unsigned int i, std::vector<int> const &avoid_cols) {
129 if (i >= S) {
130 return -1;
131 }
132 int pos = -1;
133 double max = 0;
134 for (int j = 0; j < T; ++j) {
135 if (std::find(avoid_cols.begin(), avoid_cols.end(), j) != avoid_cols.end()) {
136 continue; // skip "forbidden" columns
137 }
138 if (fabs(A[i][j]) > max) {
139 pos = j;
140 max = fabs(A[i][j]);
141 }
142 }
143 return pos;
144 }
146 /*
147 * Performs a single 'exchange step' in the Gauss-Jordan algorithm (i.e., swapping variables in the
148 * two vectors).
149 */
150 template <int S, int T>
151 static void gauss_jordan_step (double A[S][T], int row, int col) {
152 double piv = A[row][col]; // pivot element
153 /* adapt the entries of the matrix, first outside the pivot row/column */
154 for (int k = 0; k < S; ++k) {
155 if (k == row) continue;
156 for (int l = 0; l < T; ++l) {
157 if (l == col) continue;
158 A[k][l] -= A[k][col] * A[row][l] / piv;
159 }
160 }
161 /* now adapt the pivot column ... */
162 for (int k = 0; k < S; ++k) {
163 if (k == row) continue;
164 A[k][col] /= piv;
165 }
166 /* and the pivot row */
167 for (int l = 0; l < T; ++l) {
168 if (l == col) continue;
169 A[row][l] /= -piv;
170 }
171 /* finally, set the element at the pivot position itself */
172 A[row][col] = 1/piv;
173 }
175 /*
176 * Perform Gauss-Jordan elimination on the matrix A, optionally avoiding a given column during pivot search
177 */
178 template <int S, int T>
179 static std::vector<int> gauss_jordan (double A[S][T], int avoid_col = -1) {
180 std::vector<int> cols_used;
181 if (avoid_col != -1) {
182 cols_used.push_back (avoid_col);
183 }
184 int col;
185 for (int i = 0; i < S; ++i) {
186 /* for each row find a pivot element of maximal absolute value, skipping the columns that were used before */
187 col = find_pivot<S,T>(A, i, cols_used);
188 cols_used.push_back(col);
189 if (col == -1) {
190 // no non-zero elements in the row
191 return cols_used;
192 }
194 /* if pivot search was successful we can perform a Gauss-Jordan step */
195 gauss_jordan_step<S,T> (A, i, col);
196 }
197 if (avoid_col != -1) {
198 // since the columns that were used will be needed later on, we need to clean up the column vector
199 cols_used.erase(cols_used.begin());
200 }
201 return cols_used;
202 }
204 /* compute the modified value that x[index] needs to assume so that in the end we have x[index]/x[T-1] = val */
205 template <int S, int T>
206 static double projectify (std::vector<int> const &cols, const double B[S][T], const double x[T],
207 const int index, const double val) {
208 double val_proj = 0.0;
209 if (index != -1) {
210 int c = -1;
211 for (int i = 0; i < S; ++i) {
212 if (cols[i] == T-1) {
213 c = i;
214 break;
215 }
216 }
217 if (c == -1) {
218 std::cout << "Something is wrong. Rethink!!" << std::endl;
219 return SysEq::no_solution;
220 }
222 double sp = 0;
223 for (int j = 0; j < T; ++j) {
224 if (j == index) continue;
225 sp += B[c][j] * x[j];
226 }
227 double mu = 1 - val * B[c][index];
228 if (fabs(mu) < 1E-6) {
229 std::cout << "No solution since adapted value is too close to zero" << std::endl;
230 return SysEq::no_solution;
231 }
232 val_proj = sp*val/mu;
233 } else {
234 val_proj = val; // FIXME: Is this correct?
235 }
236 return val_proj;
237 }
239 /**
240 * Solve the linear system of equations \a A * \a x = \a v where we additionally stipulate
241 * \a x[\a index] = \a val if \a index is not -1. The system is solved using Gauss-Jordan
242 * elimination so that we can gracefully handle the case that zero or infinitely many
243 * solutions exist.
244 *
245 * Since our application will be to finding preimages of projective mappings, we provide
246 * an additional argument \a proj. If this is true, we find a solution of
247 * \a x[\a index]/\a x[\T - 1] = \a val insted (i.e., we want the corresponding coordinate
248 * of the _affine image_ of the point with homogeneous coordinate vector \a x to be equal
249 * to \a val.
250 *
251 * Remark: We don't need this but it would be relatively simple to let the calling function
252 * prescripe the value of _multiple_ components of the solution vector instead of only a single one.
253 */
254 template <int S, int T> SolutionKind gaussjord_solve (double A[S][T], double x[T], double v[S],
255 int index = -1, double val = 0.0, bool proj = false) {
256 double B[S][T];
257 //copy_mat<S,T>(A,B);
258 SysEq::copy_mat<S,T>(A,B);
259 std::vector<int> cols = gauss_jordan<S,T>(B, index);
260 if (std::find(cols.begin(), cols.end(), -1) != cols.end()) {
261 // pivot search failed for some row so the system is not solvable
262 return SysEq::no_solution;
263 }
265 /* the vector x is filled with the coefficients of the desired solution vector at appropriate places;
266 * the other components are set to zero, and we additionally set x[index] = val if applicable
267 */
268 std::vector<int>::iterator k;
269 for (int j = 0; j < S; ++j) {
270 x[cols[j]] = v[j];
271 }
272 for (int j = 0; j < T; ++j) {
273 k = std::find(cols.begin(), cols.end(), j);
274 if (k == cols.end()) {
275 x[j] = 0;
276 }
277 }
279 // we need to adapt the value if we we are in the "projective case" (see above)
280 double val_new = (proj ? projectify<S,T>(cols, B, x, index, val) : val);
282 if (index != -1 && index >= 0 && index < T) {
283 // we want the specified coefficient of the solution vector to have a given value
284 x[index] = val_new;
285 }
287 /* the final solution vector is now obtained as the product B*x, where B is the matrix
288 * obtained by Gauss-Jordan manipulation of A; we use w as an auxiliary vector and
289 * afterwards copy the result back to x
290 */
291 double w[S];
292 SysEq::multiply<S,T>(B,x,w);
293 for (int j = 0; j < S; ++j) {
294 x[cols[j]] = w[j];
295 }
297 if (S + (index == -1 ? 0 : 1) == T) {
298 return SysEq::unique;
299 } else {
300 return SysEq::ambiguous;
301 }
302 }
304 } // namespace SysEq
306 #endif /* __SYSEQ_H__ */
308 /*
309 Local Variables:
310 mode:c++
311 c-file-style:"stroustrup"
312 c-file-offsets:((innamespace . 0)(inline-open . 0)(case-label . +))
313 indent-tabs-mode:nil
314 fill-column:99
315 End:
316 */
317 // vim: filetype=cpp:expandtab:shiftwidth=4:tabstop=8:softtabstop=4:encoding=utf-8:textwidth=99 :