Dogcows Code - chaz/yoink/blob - src/Moof/cml/matrix/inverse.h

   1 /* -*- C++ -*- ------------------------------------------------------------
   2
   3 Copyright (c) 2007 Jesse Anders and Demian Nave http://cmldev.net/
   4
   5 The Configurable Math Library (CML) is distributed under the terms of the
   6 Boost Software License, v1.0 (see cml/LICENSE for details).
   7
   8  *-----------------------------------------------------------------------*/
   9 /** @file
  10  *  @brief Compute the inverse of a matrix by LU factorization.
  11  */
  12
  13 #ifndef matrix_inverse_h
  14 #define matrix_inverse_h
  15
  16 #include <cml/matrix/lu.h>
  17
  18 namespace cml {
  19 namespace detail {
  20
  21 /* Need to use a functional, since template functions cannot be
  22  * specialized. _tag is used to specialize based upon dimension:
  23  */
  24 template<typename MatT, int _tag> struct inverse_f;
  25
  26 /* @todo: Reciprocal optimization for division by determinant.
  27  */
  28
  29 /* 2x2 inverse.  Despite being marked for fixed_size matrices, this can
  30  * be used for dynamic-sized ones also:
  31  */
  32 template<typename MatT>
  33 struct inverse_f<MatT,2>
  34 {
  35     typename MatT::temporary_type operator()(const MatT& M) const
  36     {
  37         typedef typename MatT::temporary_type temporary_type;
  38         typedef typename temporary_type::value_type value_type;
  39
  40         /* Matrix containing the inverse: */
  41         temporary_type Z;
  42         cml::et::detail::Resize(Z,2,2);
  43
  44         /* Compute determinant and inverse: */
  45         value_type D = value_type(1) / (M(0,0)*M(1,1) - M(0,1)*M(1,0));
  46         Z(0,0) =   M(1,1)*D; Z(0,1) = - M(0,1)*D;
  47         Z(1,0) = - M(1,0)*D; Z(1,1) =   M(0,0)*D;
  48
  49         return Z;
  50     }
  51 };
  52
  53 /* 3x3 inverse.  Despite being marked for fixed_size matrices, this can
  54  * be used for dynamic-sized ones also:
  55  */
  56 template<typename MatT>
  57 struct inverse_f<MatT,3>
  58 {
  59     /*     [00 01 02]
  60      * M = [10 11 12]
  61      *     [20 21 22]
  62      */
  63     typename MatT::temporary_type operator()(const MatT& M) const
  64     {
  65         /* Shorthand. */
  66         typedef typename MatT::value_type value_type;
  67
  68         /* Compute cofactors for each entry: */
  69         value_type m_00 = M(1,1)*M(2,2) - M(1,2)*M(2,1);
  70         value_type m_01 = M(1,2)*M(2,0) - M(1,0)*M(2,2);
  71         value_type m_02 = M(1,0)*M(2,1) - M(1,1)*M(2,0);
  72
  73         value_type m_10 = M(0,2)*M(2,1) - M(0,1)*M(2,2);
  74         value_type m_11 = M(0,0)*M(2,2) - M(0,2)*M(2,0);
  75         value_type m_12 = M(0,1)*M(2,0) - M(0,0)*M(2,1);
  76
  77         value_type m_20 = M(0,1)*M(1,2) - M(0,2)*M(1,1);
  78         value_type m_21 = M(0,2)*M(1,0) - M(0,0)*M(1,2);
  79         value_type m_22 = M(0,0)*M(1,1) - M(0,1)*M(1,0);
  80
  81         /* Compute determinant from the minors: */
  82         value_type D =
  83             value_type(1) / (M(0,0)*m_00 + M(0,1)*m_01 + M(0,2)*m_02);
  84
  85         /* Matrix containing the inverse: */
  86         typename MatT::temporary_type Z;
  87         cml::et::detail::Resize(Z,3,3);
  88
  89         /* Assign the inverse as (1/D) * (cofactor matrix)^T: */
  90         Z(0,0) = m_00*D;  Z(0,1) = m_10*D;  Z(0,2) = m_20*D;
  91         Z(1,0) = m_01*D;  Z(1,1) = m_11*D;  Z(1,2) = m_21*D;
  92         Z(2,0) = m_02*D;  Z(2,1) = m_12*D;  Z(2,2) = m_22*D;
  93
  94         return Z;
  95     }
  96 };
  97
  98 /* 4x4 inverse.  Despite being marked for fixed_size matrices, this can
  99  * be used for dynamic-sized ones also:
 100  */
 101 template<typename MatT>
 102 struct inverse_f<MatT,4>
 103 {
 104     /*     [00 01 02 03]
 105      * M = [10 11 12 13]
 106      *     [20 21 22 23]
 107      *     [30 31 32 33]
 108      *
 109      *       |11 12 13|         |10 12 13|
 110      * C00 = |21 22 23|   C01 = |20 22 23|
 111      *       |31 32 33|         |30 32 33|
 112      *
 113      *       |10 11 13|         |10 11 12|
 114      * C02 = |20 21 23|   C03 = |20 21 22|
 115      *       |30 31 33|         |30 31 32|
 116      */
 117     typename MatT::temporary_type operator()(const MatT& M) const
 118     {
 119         /* Shorthand. */
 120         typedef typename MatT::value_type value_type;
 121
 122         /* Common cofactors, rows 0,1: */
 123         value_type m_22_33_23_32 = M(2,2)*M(3,3) - M(2,3)*M(3,2);
 124         value_type m_23_30_20_33 = M(2,3)*M(3,0) - M(2,0)*M(3,3);
 125         value_type m_20_31_21_30 = M(2,0)*M(3,1) - M(2,1)*M(3,0);
 126         value_type m_21_32_22_31 = M(2,1)*M(3,2) - M(2,2)*M(3,1);
 127         value_type m_23_31_21_33 = M(2,3)*M(3,1) - M(2,1)*M(3,3);
 128         value_type m_20_32_22_30 = M(2,0)*M(3,2) - M(2,2)*M(3,0);
 129
 130         /* Compute minors: */
 131         value_type d00
 132             = M(1,1)*m_22_33_23_32+M(1,2)*m_23_31_21_33+M(1,3)*m_21_32_22_31;
 133
 134         value_type d01
 135             = M(1,0)*m_22_33_23_32+M(1,2)*m_23_30_20_33+M(1,3)*m_20_32_22_30;
 136
 137         value_type d02
 138             = M(1,0)*-m_23_31_21_33+M(1,1)*m_23_30_20_33+M(1,3)*m_20_31_21_30;
 139
 140         value_type d03
 141             = M(1,0)*m_21_32_22_31+M(1,1)*-m_20_32_22_30+M(1,2)*m_20_31_21_30;
 142
 143         /* Compute minors: */
 144         value_type d10
 145             = M(0,1)*m_22_33_23_32+M(0,2)*m_23_31_21_33+M(0,3)*m_21_32_22_31;
 146
 147         value_type d11
 148             = M(0,0)*m_22_33_23_32+M(0,2)*m_23_30_20_33+M(0,3)*m_20_32_22_30;
 149
 150         value_type d12
 151             = M(0,0)*-m_23_31_21_33+M(0,1)*m_23_30_20_33+M(0,3)*m_20_31_21_30;
 152
 153         value_type d13
 154             = M(0,0)*m_21_32_22_31+M(0,1)*-m_20_32_22_30+M(0,2)*m_20_31_21_30;
 155
 156         /* Common cofactors, rows 2,3: */
 157         value_type m_02_13_03_12 = M(0,2)*M(1,3) - M(0,3)*M(1,2);
 158         value_type m_03_10_00_13 = M(0,3)*M(1,0) - M(0,0)*M(1,3);
 159         value_type m_00_11_01_10 = M(0,0)*M(1,1) - M(0,1)*M(1,0);
 160         value_type m_01_12_02_11 = M(0,1)*M(1,2) - M(0,2)*M(1,1);
 161         value_type m_03_11_01_13 = M(0,3)*M(1,1) - M(0,1)*M(1,3);
 162         value_type m_00_12_02_10 = M(0,0)*M(1,2) - M(0,2)*M(1,0);
 163
 164         /* Compute minors (uses row 3 as the multipliers instead of row 0,
 165          * which uses the same signs as row 0):
 166          */
 167         value_type d20
 168             = M(3,1)*m_02_13_03_12+M(3,2)*m_03_11_01_13+M(3,3)*m_01_12_02_11;
 169
 170         value_type d21
 171             = M(3,0)*m_02_13_03_12+M(3,2)*m_03_10_00_13+M(3,3)*m_00_12_02_10;
 172
 173         value_type d22
 174             = M(3,0)*-m_03_11_01_13+M(3,1)*m_03_10_00_13+M(3,3)*m_00_11_01_10;
 175
 176         value_type d23
 177             = M(3,0)*m_01_12_02_11+M(3,1)*-m_00_12_02_10+M(3,2)*m_00_11_01_10;
 178
 179         /* Compute minors: */
 180         value_type d30
 181             = M(2,1)*m_02_13_03_12+M(2,2)*m_03_11_01_13+M(2,3)*m_01_12_02_11;
 182
 183         value_type d31
 184             = M(2,0)*m_02_13_03_12+M(2,2)*m_03_10_00_13+M(2,3)*m_00_12_02_10;
 185
 186         value_type d32
 187             = M(2,0)*-m_03_11_01_13+M(2,1)*m_03_10_00_13+M(2,3)*m_00_11_01_10;
 188
 189         value_type d33
 190             = M(2,0)*m_01_12_02_11+M(2,1)*-m_00_12_02_10+M(2,2)*m_00_11_01_10;
 191
 192         /* Finally, compute determinant from the minors, and assign the
 193          * inverse as (1/D) * (cofactor matrix)^T:
 194          */
 195         typename MatT::temporary_type Z;
 196         cml::et::detail::Resize(Z,4,4);
 197
 198         value_type D = value_type(1) /
 199             (M(0,0)*d00 - M(0,1)*d01 + M(0,2)*d02 - M(0,3)*d03);
 200         Z(0,0) = +d00*D; Z(0,1) = -d10*D; Z(0,2) = +d20*D; Z(0,3) = -d30*D;
 201         Z(1,0) = -d01*D; Z(1,1) = +d11*D; Z(1,2) = -d21*D; Z(1,3) = +d31*D;
 202         Z(2,0) = +d02*D; Z(2,1) = -d12*D; Z(2,2) = +d22*D; Z(2,3) = -d32*D;
 203         Z(3,0) = -d03*D; Z(3,1) = +d13*D; Z(3,2) = -d23*D; Z(3,3) = +d33*D;
 204
 205         return Z;
 206     }
 207 };
 208
 209 /* If more extensive general linear algebra functionality is offered in
 210  * future versions it may be useful to make the elementary row and column
 211  * operations separate functions. For now they're simply performed in place,
 212  * but the commented-out lines of code show where the calls to these functions
 213  * should go if and when they become available.
 214  */
 215
 216 /* @todo: In-place version, and address memory allocation for pivot vector.
 217  */
 218
 219 /* General NxN inverse by Gauss-Jordan elimination with full pivoting:  */
 220 template<typename MatT, int _tag>
 221 struct inverse_f
 222 {
 223     typename MatT::temporary_type operator()(const MatT& M) const
 224     {
 225         /* Shorthand. */
 226         typedef typename MatT::value_type value_type;
 227
 228         /* Size of matrix */
 229         size_t N = M.rows();
 230
 231         /* Matrix containing the inverse: */
 232         typename MatT::temporary_type Z;
 233         cml::et::detail::Resize(Z,N,N);
 234         Z = M;
 235
 236         /* For tracking pivots */
 237         std::vector<size_t> row_index(N);
 238         std::vector<size_t> col_index(N);
 239         std::vector<size_t> pivoted(N,0);
 240
 241         /* For each column */
 242         for (size_t i = 0; i < N; ++i) {
 243
 244             /* Find the pivot */
 245             size_t row = 0, col = 0;
 246             value_type max = value_type(0);
 247             for (size_t j = 0; j < N; ++j) {
 248                 if (!pivoted[j]) {
 249                     for (size_t k = 0; k < N; ++k) {
 250                         if (!pivoted[k]) {
 251                             value_type mag = std::fabs(Z(j,k));
 252                             if (mag > max) {
 253                                 max = mag;
 254                                 row = j;
 255                                 col = k;
 256                             }
 257                         }
 258                     }
 259                 }
 260             }
 261
 262             /* TODO: Check max against epsilon here to catch singularity */
 263
 264             row_index[i] = row;
 265             col_index[i] = col;
 266
 267             /* Swap rows if necessary */
 268             if (row != col) {
 269                 /*Z.row_op_swap(row,col);*/
 270                 for (size_t j = 0; j < Z.cols(); ++j) {
 271                     std::swap(Z(row,j),Z(col,j));
 272                 }
 273             }
 274
 275             /* Process pivot row */
 276             pivoted[col] = true;
 277             value_type pivot = Z(col,col);
 278             Z(col,col) = value_type(1);
 279             /*Z.row_op_mult(col,value_type(1)/pivot);*/
 280             value_type k = value_type(1)/pivot;
 281             for (size_t j = 0; j < Z.cols(); ++j) {
 282                 Z(col,j) *= k;
 283             }
 284
 285             /* Process other rows */
 286             for (size_t j = 0; j < N; ++j) {
 287                 if (j != col) {
 288                     value_type mult = -Z(j,col);
 289                     Z(j,col) = value_type(0);
 290                     /*Z.row_op_add_mult(col,j,mult);*/
 291                     for (size_t k = 0; k < Z.cols(); ++k) {
 292                         Z(j,k) += mult * Z(col,k);
 293                     }
 294                 }
 295             }
 296         }
 297
 298         /* Swap columns if necessary */
 299         for (int i = N-1; i >= 0; --i) {
 300             if (row_index[i] != col_index[i]) {
 301                 /*Z.col_op_swap(row_index[i],col_index[i]);*/
 302                 for (size_t j = 0; j < Z.rows(); ++j) {
 303                     std::swap(Z(j,row_index[i]),Z(j,col_index[i]));
 304                 }
 305             }
 306         }
 307
 308         /* Return result */
 309         return Z;
 310     }
 311 };
 312
 313 /* Inversion by LU factorization is turned off for now due to lack of
 314  * pivoting in the implementation, but we may switch back to it at some future
 315  * time.
 316  */
 317
 318 #if 0
 319
 320 /* General NxN inverse by LU factorization:  */
 321 template<typename MatT, int _tag>
 322 struct inverse_f
 323 {
 324     typename MatT::temporary_type operator()(const MatT& M) const
 325     {
 326         /* Shorthand. */
 327         typedef typename MatT::value_type value_type;
 328
 329         /* Compute LU factorization: */
 330         size_t N = M.rows();
 331         typename MatT::temporary_type LU;
 332         cml::et::detail::Resize(LU,N,N);
 333         LU = lu(M);
 334
 335         /* Matrix containing the inverse: */
 336         typename MatT::temporary_type Z;
 337         cml::et::detail::Resize(Z,N,N);
 338
 339         typename MatT::col_vector_type v, x;
 340         cml::et::detail::Resize(v,N);
 341         cml::et::detail::Resize(x,N);
 342         for(size_t i = 0; i < N; ++i)
 343             v[i] = value_type(0);
 344         /* XXX Need a fill() function here. */
 345
 346         /* Use lu_solve to solve M*x = v for x, where v = [0 ... 1 ... 0]^T: */
 347         for(size_t i = 0; i < N; ++i) {
 348             v[i] = 1.;
 349             x = lu_solve(LU,v);
 350
 351             /* x is column i of the inverse of LU: */
 352             for(size_t k = 0; k < N; ++ k) {
 353                 Z(k,i) = x[k];
 354             }
 355             v[i] = 0.;
 356         }
 357
 358         return Z;
 359     }
 360
 361 };
 362
 363 #endif
 364
 365 /* Note: force_NxN is for checking general NxN inversion against the special-
 366  * case 2x2, 3x3 and 4x4 code. I'm leaving it in for now since we may need to
 367  * test the NxN code further if the implementation changes. At some future
 368  * time when the implementation is stable, everything related to force_NxN can
 369  * be taken out.
 370  */
 371
 372 /* Note: Commenting the force_NxN stuff out, but leaving the code here in
 373  * case we need to do more testing in the future.
 374  */
 375
 376 /* Generator for the inverse functional for fixed-size matrices: */
 377 template<typename MatT> typename MatT::temporary_type
 378 inverse(const MatT& M, fixed_size_tag/*, bool force_NxN*/)
 379 {
 380     /* Require a square matrix: */
 381     cml::et::CheckedSquare(M, fixed_size_tag());
 382
 383     /*
 384     if (force_NxN) {
 385         return inverse_f<MatT,0>()(M);
 386     } else {
 387     */
 388     return inverse_f<MatT,MatT::array_rows>()(M);
 389     /*
 390     }
 391     */
 392 }
 393
 394 /* Generator for the inverse functional for dynamic-size matrices: */
 395 template<typename MatT> typename MatT::temporary_type
 396 inverse(const MatT& M, dynamic_size_tag/*, bool force_NxN*/)
 397 {
 398     /* Require a square matrix: */
 399     cml::et::CheckedSquare(M, dynamic_size_tag());
 400
 401     /*
 402     if (force_NxN) {
 403         return inverse_f<MatT,0>()(M);
 404     } else {
 405     */
 406     /* Dispatch based upon the matrix dimension: */
 407     switch(M.rows()) {
 408         case 2:  return inverse_f<MatT,2>()(M);     //   2x2
 409         case 3:  return inverse_f<MatT,3>()(M);     //   3x3
 410         case 4:  return inverse_f<MatT,4>()(M);     //   4x4
 411         default: return inverse_f<MatT,0>()(M);     // > 4x4 (or 1x1)
 412     }
 413     /*
 414     }
 415     */
 416 }
 417
 418 } // namespace detail
 419
 420 /** Inverse of a matrix. */
 421 template<typename E, class AT, typename BO, typename L> inline
 422 typename matrix<E,AT,BO,L>::temporary_type
 423 inverse(const matrix<E,AT,BO,L>& M/*, bool force_NxN = false*/)
 424 {
 425     typedef typename matrix<E,AT,BO,L>::size_tag size_tag;
 426     return detail::inverse(M,size_tag()/*,force_NxN*/);
 427 }
 428
 429 /** Inverse of a matrix expression. */
 430 template<typename XprT> inline
 431 typename et::MatrixXpr<XprT>::temporary_type
 432 inverse(const et::MatrixXpr<XprT>& e/*, bool force_NxN = false*/)
 433 {
 434     typedef typename et::MatrixXpr<XprT>::size_tag size_tag;
 435     return detail::inverse(e,size_tag()/*,force_NxN*/);
 436 }
 437
 438 } // namespace cml
 439
 440 #endif
 441
 442 // -------------------------------------------------------------------------
 443 // vim:ft=cpp