[4ede1e]: itpp / base / algebra / ls_solve.h Maximize Restore History

Download this file

ls_solve.h    412 lines (290 with data), 14.8 kB

  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
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
/*!
* \file
* \brief Definitions of functions for solving linear equation systems
* \author Tony Ottosson
*
* -------------------------------------------------------------------------
*
* Copyright (C) 1995-2010 (see AUTHORS file for a list of contributors)
*
* This file is part of IT++ - a C++ library of mathematical, signal
* processing, speech processing, and communications classes and functions.
*
* IT++ is free software: you can redistribute it and/or modify it under the
* terms of the GNU General Public License as published by the Free Software
* Foundation, either version 3 of the License, or (at your option) any
* later version.
*
* IT++ is distributed in the hope that it will be useful, but WITHOUT ANY
* WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
* FOR A PARTICULAR PURPOSE. See the GNU General Public License for more
* details.
*
* You should have received a copy of the GNU General Public License along
* with IT++. If not, see <http://www.gnu.org/licenses/>.
*
* -------------------------------------------------------------------------
*/
#ifndef LS_SOLVE_H
#define LS_SOLVE_H
#include <itpp/base/mat.h>
namespace itpp
{
/*! \addtogroup linearequations
*/
//!@{
/*! \brief Solve linear equation system by LU factorisation.
Solves the linear system \f$Ax=b\f$, where \f$A\f$ is a \f$n \times n\f$ matrix.
Uses the LAPACK routine DGESV.
*/
bool ls_solve(const mat &A, const vec &b, vec &x);
/*! \brief Solve linear equation system by LU factorisation.
Solves the linear system \f$Ax=b\f$, where \f$A\f$ is a \f$n \times n\f$ matrix.
Uses the LAPACK routine DGESV.
*/
vec ls_solve(const mat &A, const vec &b);
/*! \brief Solve multiple linear equations by LU factorisation.
Solves the linear system \f$AX=B\f$. Here \f$A\f$ is a nonsingular \f$n \times n\f$ matrix.
Uses the LAPACK routine DGESV.
*/
bool ls_solve(const mat &A, const mat &B, mat &X);
/*! \brief Solve multiple linear equations by LU factorisation.
Solves the linear system \f$AX=B\f$. Here \f$A\f$ is a nonsingular \f$n \times n\f$ matrix.
Uses the LAPACK routine DGESV.
*/
mat ls_solve(const mat &A, const mat &B);
/*! \brief Solve linear equation system by LU factorisation.
Solves the linear system \f$Ax=b\f$, where \f$A\f$ is a \f$n \times n\f$ matrix.
Uses the LAPACK routine ZGESV.
*/
bool ls_solve(const cmat &A, const cvec &b, cvec &x);
/*! \brief Solve linear equation system by LU factorisation.
Solves the linear system \f$Ax=b\f$, where \f$A\f$ is a \f$n \times n\f$ matrix.
Uses the LAPACK routine ZGESV.
*/
cvec ls_solve(const cmat &A, const cvec &b);
/*! \brief Solve multiple linear equations by LU factorisation.
Solves the linear system \f$AX=B\f$. Here \f$A\f$ is a nonsingular \f$n \times n\f$ matrix.
Uses the LAPACK routine ZGESV.
*/
bool ls_solve(const cmat &A, const cmat &B, cmat &X);
/*! \brief Solve multiple linear equations by LU factorisation.
Solves the linear system \f$AX=B\f$. Here \f$A\f$ is a nonsingular \f$n \times n\f$ matrix.
Uses the LAPACK routine ZGESV.
*/
cmat ls_solve(const cmat &A, const cmat &B);
/*! \brief Solve linear equation system by Cholesky factorisation.
Solves the linear system \f$Ax=b\f$, where \f$A\f$ is a symmetric positive definite \f$n \times n\f$ matrix.
Uses the LAPACK routine DPOSV.
*/
bool ls_solve_chol(const mat &A, const vec &b, vec &x);
/*! \brief Solve linear equation system by Cholesky factorisation.
Solves the linear system \f$Ax=b\f$, where \f$A\f$ is a symmetric positive definite \f$n \times n\f$ matrix.
Uses the LAPACK routine DPOSV.
*/
vec ls_solve_chol(const mat &A, const vec &b);
/*! \brief Solve linear equation system by Cholesky factorisation.
Solves the linear system \f$AX=B\f$, where \f$A\f$ is a symmetric positive definite \f$n \times n\f$ matrix.
Uses the LAPACK routine DPOSV.
*/
bool ls_solve_chol(const mat &A, const mat &B, mat &X);
/*! \brief Solve linear equation system by Cholesky factorisation.
Solves the linear system \f$AX=B\f$, where \f$A\f$ is a symmetric positive definite \f$n \times n\f$ matrix.
Uses the LAPACK routine DPOSV.
*/
mat ls_solve_chol(const mat &A, const mat &B);
/*! \brief Solve linear equation system by Cholesky factorisation.
Solves the linear system \f$Ax=b\f$, where \f$A\f$ is a Hermitian positive definite \f$n \times n\f$ matrix.
Uses the LAPACK routine ZPOSV.
*/
bool ls_solve_chol(const cmat &A, const cvec &b, cvec &x);
/*! \brief Solve linear equation system by Cholesky factorisation.
Solves the linear system \f$Ax=b\f$, where \f$A\f$ is a Hermitian positive definite \f$n \times n\f$ matrix.
Uses the LAPACK routine ZPOSV.
*/
cvec ls_solve_chol(const cmat &A, const cvec &b);
/*! \brief Solve linear equation system by Cholesky factorisation.
Solves the linear system \f$AX=B\f$, where \f$A\f$ is a Hermitian positive definite \f$n \times n\f$ matrix.
Uses the LAPACK routine ZPOSV.
*/
bool ls_solve_chol(const cmat &A, const cmat &B, cmat &X);
/*! \brief Solve linear equation system by Cholesky factorisation.
Solves the linear system \f$AX=B\f$, where \f$A\f$ is a Hermitian positive definite \f$n \times n\f$ matrix.
Uses the LAPACK routine ZPOSV.
*/
cmat ls_solve_chol(const cmat &A, const cmat &B);
/*! \brief Solves overdetermined linear equation systems.
Solves the overdetermined linear system \f$Ax=b\f$, where \f$A\f$ is a \f$m \times n\f$ matrix and \f$m \geq n\f$.
Uses QR-factorization and is built upon the LAPACK routine DGELS.
*/
bool ls_solve_od(const mat &A, const vec &b, vec &x);
/*! \brief Solves overdetermined linear equation systems.
Solves the overdetermined linear system \f$Ax=b\f$, where \f$A\f$ is a \f$m \times n\f$ matrix and \f$m \geq n\f$.
Uses QR-factorization and assumes that \f$A\f$ is full rank. Based on the LAPACK routine DGELS.
*/
vec ls_solve_od(const mat &A, const vec &b);
/*! \brief Solves overdetermined linear equation systems.
Solves the overdetermined linear system \f$AX=B\f$, where \f$A\f$ is a \f$m \times n\f$ matrix and \f$m \geq n\f$.
Uses QR-factorization and assumes that \f$A\f$ is full rank. Based on the LAPACK routine DGELS.
*/
bool ls_solve_od(const mat &A, const mat &B, mat &X);
/*! \brief Solves overdetermined linear equation systems.
Solves the overdetermined linear system \f$AX=B\f$, where \f$A\f$ is a \f$m \times n\f$ matrix and \f$m \geq n\f$.
Uses QR-factorization and assumes that \f$A\f$ is full rank. Based on the LAPACK routine DGELS.
*/
mat ls_solve_od(const mat &A, const mat &B);
/*! \brief Solves overdetermined linear equation systems.
Solves the overdetermined linear system \f$Ax=b\f$, where \f$A\f$ is a \f$m \times n\f$ matrix and \f$m \geq n\f$.
Uses QR-factorization and is built upon the LAPACK routine ZGELS.
*/
bool ls_solve_od(const cmat &A, const cvec &b, cvec &x);
/*! \brief Solves overdetermined linear equation systems.
Solves the overdetermined linear system \f$Ax=b\f$, where \f$A\f$ is a \f$m \times n\f$ matrix and \f$m \geq n\f$.
Uses QR-factorization and assumes that \f$A\f$ is full rank. Based on the LAPACK routine ZGELS.
*/
cvec ls_solve_od(const cmat &A, const cvec &b);
/*! \brief Solves overdetermined linear equation systems.
Solves the overdetermined linear system \f$AX=B\f$, where \f$A\f$ is a \f$m \times n\f$ matrix and \f$m \geq n\f$.
Uses QR-factorization and assumes that \f$A\f$ is full rank. Based on the LAPACK routine ZGELS.
*/
bool ls_solve_od(const cmat &A, const cmat &B, cmat &X);
/*! \brief Solves overdetermined linear equation systems.
Solves the overdetermined linear system \f$AX=B\f$, where \f$A\f$ is a \f$m \times n\f$ matrix and \f$m \geq n\f$.
Uses QR-factorization and assumes that \f$A\f$ is full rank. Based on the LAPACK routine ZGELS.
*/
cmat ls_solve_od(const cmat &A, const cmat &B);
/*! \brief Solves underdetermined linear equation systems.
Solves the underdetermined linear system \f$Ax=b\f$, where \f$A\f$ is a \f$m \times n\f$ matrix and \f$m \leq n\f$.
Uses LQ-factorization and is built upon the LAPACK routine DGELS.
*/
bool ls_solve_ud(const mat &A, const vec &b, vec &x);
/*! \brief Solves overdetermined linear equation systems.
Solves the underdetermined linear system \f$Ax=b\f$, where \f$A\f$ is a \f$m \times n\f$ matrix and \f$m \leq n\f$.
Uses LQ-factorization and assumes that \f$A\f$ is full rank. Based on the LAPACK routine DGELS.
*/
vec ls_solve_ud(const mat &A, const vec &b);
/*! \brief Solves underdetermined linear equation systems.
Solves the underdetermined linear system \f$AX=B\f$, where \f$A\f$ is a \f$m \times n\f$ matrix and \f$m \leq n\f$.
Uses LQ-factorization and assumes that \f$A\f$ is full rank. Based on the LAPACK routine DGELS.
*/
bool ls_solve_ud(const mat &A, const mat &B, mat &X);
/*! \brief Solves underdetermined linear equation systems.
Solves the underdetermined linear system \f$AX=B\f$, where \f$A\f$ is a \f$m \times n\f$ matrix and \f$m \leq n\f$.
Uses LQ-factorization and assumes that \f$A\f$ is full rank. Based on the LAPACK routine DGELS.
*/
mat ls_solve_ud(const mat &A, const mat &B);
/*! \brief Solves underdetermined linear equation systems.
Solves the underdetermined linear system \f$Ax=b\f$, where \f$A\f$ is a \f$m \times n\f$ matrix and \f$m \leq n\f$.
Uses LQ-factorization and is built upon the LAPACK routine ZGELS.
*/
bool ls_solve_ud(const cmat &A, const cvec &b, cvec &x);
/*! \brief Solves overdetermined linear equation systems.
Solves the underdetermined linear system \f$Ax=b\f$, where \f$A\f$ is a \f$m \times n\f$ matrix and \f$m \leq n\f$.
Uses LQ-factorization and assumes that \f$A\f$ is full rank. Based on the LAPACK routine ZGELS.
*/
cvec ls_solve_ud(const cmat &A, const cvec &b);
/*! \brief Solves underdetermined linear equation systems.
Solves the underdetermined linear system \f$AX=B\f$, where \f$A\f$ is a \f$m \times n\f$ matrix and \f$m \leq n\f$.
Uses LQ-factorization and assumes that \f$A\f$ is full rank. Based on the LAPACK routine ZGELS.
*/
bool ls_solve_ud(const cmat &A, const cmat &B, cmat &X);
/*! \brief Solves underdetermined linear equation systems.
Solves the underdetermined linear system \f$AX=B\f$, where \f$A\f$ is a \f$m \times n\f$ matrix and \f$m \leq n\f$.
Uses LQ-factorization and assumes that \f$A\f$ is full rank. Based on the LAPACK routine ZGELS.
*/
cmat ls_solve_ud(const cmat &A, const cmat &B);
/*! \brief A general linear equation system solver.
Tries to emulate the backslash operator in Matlab by calling
ls_solve(A,b,x), ls_solve_od(A,b,x) or ls_solve_ud(A,b,x)
*/
bool backslash(const mat &A, const vec &b, vec &x);
/*! \brief A general linear equation system solver.
Tries to emulate the backslash operator in Matlab by calling
ls_solve(A,b), ls_solve_od(A,b) or ls_solve_ud(A,b)
*/
vec backslash(const mat &A, const vec &b);
/*! \brief A general linear equation system solver.
Tries to emulate the backslash operator in Matlab by calling
ls_solve(A,B,X), ls_solve_od(A,B,X), or ls_solve_ud(A,B,X).
*/
bool backslash(const mat &A, const mat &B, mat &X);
/*! \brief A general linear equation system solver.
Tries to emulate the backslash operator in Matlab by calling
ls_solve(A,B), ls_solve_od(A,B), or ls_solve_ud(A,B).
*/
mat backslash(const mat &A, const mat &B);
/*! \brief A general linear equation system solver.
Tries to emulate the backslash operator in Matlab by calling
ls_solve(A,b,x), ls_solve_od(A,b,x) or ls_solve_ud(A,b,x)
*/
bool backslash(const cmat &A, const cvec &b, cvec &x);
/*! \brief A general linear equation system solver.
Tries to emulate the backslash operator in Matlab by calling
ls_solve(A,b), ls_solve_od(A,b) or ls_solve_ud(A,b)
*/
cvec backslash(const cmat &A, const cvec &b);
/*! \brief A general linear equation system solver.
Tries to emulate the backslash operator in Matlab by calling
ls_solve(A,B,X), ls_solve_od(A,B,X), or ls_solve_ud(A,B,X).
*/
bool backslash(const cmat &A, const cmat &B, cmat &X);
/*! \brief A general linear equation system solver.
Tries to emulate the backslash operator in Matlab by calling
ls_solve(A,B), ls_solve_od(A,B), or ls_solve_ud(A,B).
*/
cmat backslash(const cmat &A, const cmat &B);
/*! \brief Forward substitution of square matrix.
Solves Lx=b, where L is a lower triangular n by n matrix.
Assumes that L is nonsingular. Requires n^2 flops.
Uses Alg. 3.1.1 in Golub & van Loan "Matrix computations", 3rd ed., p. 89.
*/
vec forward_substitution(const mat &L, const vec &b);
/*! \brief Forward substitution of square matrix.
Solves Lx=b, where L is a lower triangular n by n matrix.
Assumes that L is nonsingular. Requires n^2 flops.
Uses Alg. 3.1.1 in Golub & van Loan "Matrix computations", 3rd ed., p. 89.
*/
void forward_substitution(const mat &L, const vec &b, vec &x);
/*! \brief Forward substitution of band matrices.
Solves Lx=b, where L is a lower triangular n by n band-matrix with lower
bandwidth p.
Assumes that L is nonsingular. Requires about 2np flops (if n >> p).
Uses Alg. 4.3.2 in Golub & van Loan "Matrix computations", 3rd ed., p. 153.
*/
vec forward_substitution(const mat &L, int p, const vec &b);
/*! \brief Forward substitution of band matrices.
Solves Lx=b, where L is a lower triangular n by n band-matrix with
lower bandwidth p.
Assumes that L is nonsingular. Requires about 2np flops (if n >> p).
Uses Alg. 4.3.2 in Golub & van Loan "Matrix computations", 3rd ed., p. 153.
*/
void forward_substitution(const mat &L, int p, const vec &b, vec &x);
/*! \brief Backward substitution of square matrix.
Solves Ux=b, where U is a upper triangular n by n matrix.
Assumes that U is nonsingular. Requires n^2 flops.
Uses Alg. 3.1.2 in Golub & van Loan "Matrix computations", 3rd ed., p. 89.
*/
vec backward_substitution(const mat &U, const vec &b);
/*! \brief Backward substitution of square matrix.
Solves Ux=b, where U is a upper triangular n by n matrix.
Assumes that U is nonsingular. Requires n^2 flops.
Uses Alg. 3.1.2 in Golub & van Loan "Matrix computations", 3rd ed., p. 89.
*/
void backward_substitution(const mat &U, const vec &b, vec &x);
/*! \brief Backward substitution of band matrix.
Solves Ux=b, where U is a upper triangular n by n matrix band-matrix with
upper bandwidth q.
Assumes that U is nonsingular. Requires about 2nq flops (if n >> q).
Uses Alg. 4.3.3 in Golub & van Loan "Matrix computations", 3rd ed., p. 153.
*/
vec backward_substitution(const mat &U, int q, const vec &b);
/*! \brief Backward substitution of band matrix.
Solves Ux=b, where U is a upper triangular n by n matrix band-matrix with
upper bandwidth q.
Assumes that U is nonsingular. Requires about 2nq flops (if n >> q).
Uses Alg. 4.3.3 in Golub & van Loan "Matrix computations", 3rd ed., p. 153.
*/
void backward_substitution(const mat &U, int q, const vec &b, vec &x);
//!@}
} //namespace itpp
#endif // #ifndef LS_SOLVE_H