1 // This file is part of Eigen, a lightweight C++ template library 2 // for linear algebra. 3 // 4 // Copyright (C) 2006-2009 Benoit Jacob <jacob.benoit.1@gmail.com> 5 // 6 // This Source Code Form is subject to the terms of the Mozilla 7 // Public License v. 2.0. If a copy of the MPL was not distributed 8 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/. 9 10 #ifndef EIGEN_LU_H 11 #define EIGEN_LU_H 12 13 namespace Eigen { 14 15 namespace internal { 16 template<typename _MatrixType> struct traits<FullPivLU<_MatrixType> > 17 : traits<_MatrixType> 18 { 19 typedef MatrixXpr XprKind; 20 typedef SolverStorage StorageKind; 21 typedef int StorageIndex; 22 enum { Flags = 0 }; 23 }; 24 25 } // end namespace internal 26 27 /** \ingroup LU_Module 28 * 29 * \class FullPivLU 30 * 31 * \brief LU decomposition of a matrix with complete pivoting, and related features 32 * 33 * \tparam _MatrixType the type of the matrix of which we are computing the LU decomposition 34 * 35 * This class represents a LU decomposition of any matrix, with complete pivoting: the matrix A is 36 * decomposed as \f$ A = P^{-1} L U Q^{-1} \f$ where L is unit-lower-triangular, U is 37 * upper-triangular, and P and Q are permutation matrices. This is a rank-revealing LU 38 * decomposition. The eigenvalues (diagonal coefficients) of U are sorted in such a way that any 39 * zeros are at the end. 40 * 41 * This decomposition provides the generic approach to solving systems of linear equations, computing 42 * the rank, invertibility, inverse, kernel, and determinant. 43 * 44 * This LU decomposition is very stable and well tested with large matrices. However there are use cases where the SVD 45 * decomposition is inherently more stable and/or flexible. For example, when computing the kernel of a matrix, 46 * working with the SVD allows to select the smallest singular values of the matrix, something that 47 * the LU decomposition doesn't see. 48 * 49 * The data of the LU decomposition can be directly accessed through the methods matrixLU(), 50 * permutationP(), permutationQ(). 51 * 52 * As an example, here is how the original matrix can be retrieved: 53 * \include class_FullPivLU.cpp 54 * Output: \verbinclude class_FullPivLU.out 55 * 56 * This class supports the \link InplaceDecomposition inplace decomposition \endlink mechanism. 57 * 58 * \sa MatrixBase::fullPivLu(), MatrixBase::determinant(), MatrixBase::inverse() 59 */ 60 template<typename _MatrixType> class FullPivLU 61 : public SolverBase<FullPivLU<_MatrixType> > 62 { 63 public: 64 typedef _MatrixType MatrixType; 65 typedef SolverBase<FullPivLU> Base; 66 friend class SolverBase<FullPivLU>; 67 68 EIGEN_GENERIC_PUBLIC_INTERFACE(FullPivLU) 69 enum { 70 MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime, 71 MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime 72 }; 73 typedef typename internal::plain_row_type<MatrixType, StorageIndex>::type IntRowVectorType; 74 typedef typename internal::plain_col_type<MatrixType, StorageIndex>::type IntColVectorType; 75 typedef PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime> PermutationQType; 76 typedef PermutationMatrix<RowsAtCompileTime, MaxRowsAtCompileTime> PermutationPType; 77 typedef typename MatrixType::PlainObject PlainObject; 78 79 /** 80 * \brief Default Constructor. 81 * 82 * The default constructor is useful in cases in which the user intends to 83 * perform decompositions via LU::compute(const MatrixType&). 84 */ 85 FullPivLU(); 86 87 /** \brief Default Constructor with memory preallocation 88 * 89 * Like the default constructor but with preallocation of the internal data 90 * according to the specified problem \a size. 91 * \sa FullPivLU() 92 */ 93 FullPivLU(Index rows, Index cols); 94 95 /** Constructor. 96 * 97 * \param matrix the matrix of which to compute the LU decomposition. 98 * It is required to be nonzero. 99 */ 100 template<typename InputType> 101 explicit FullPivLU(const EigenBase<InputType>& matrix); 102 103 /** \brief Constructs a LU factorization from a given matrix 104 * 105 * This overloaded constructor is provided for \link InplaceDecomposition inplace decomposition \endlink when \c MatrixType is a Eigen::Ref. 106 * 107 * \sa FullPivLU(const EigenBase&) 108 */ 109 template<typename InputType> 110 explicit FullPivLU(EigenBase<InputType>& matrix); 111 112 /** Computes the LU decomposition of the given matrix. 113 * 114 * \param matrix the matrix of which to compute the LU decomposition. 115 * It is required to be nonzero. 116 * 117 * \returns a reference to *this 118 */ 119 template<typename InputType> 120 FullPivLU& compute(const EigenBase<InputType>& matrix) { 121 m_lu = matrix.derived(); 122 computeInPlace(); 123 return *this; 124 } 125 126 /** \returns the LU decomposition matrix: the upper-triangular part is U, the 127 * unit-lower-triangular part is L (at least for square matrices; in the non-square 128 * case, special care is needed, see the documentation of class FullPivLU). 129 * 130 * \sa matrixL(), matrixU() 131 */ 132 inline const MatrixType& matrixLU() const 133 { 134 eigen_assert(m_isInitialized && "LU is not initialized."); 135 return m_lu; 136 } 137 138 /** \returns the number of nonzero pivots in the LU decomposition. 139 * Here nonzero is meant in the exact sense, not in a fuzzy sense. 140 * So that notion isn't really intrinsically interesting, but it is 141 * still useful when implementing algorithms. 142 * 143 * \sa rank() 144 */ 145 inline Index nonzeroPivots() const 146 { 147 eigen_assert(m_isInitialized && "LU is not initialized."); 148 return m_nonzero_pivots; 149 } 150 151 /** \returns the absolute value of the biggest pivot, i.e. the biggest 152 * diagonal coefficient of U. 153 */ 154 RealScalar maxPivot() const { return m_maxpivot; } 155 156 /** \returns the permutation matrix P 157 * 158 * \sa permutationQ() 159 */ 160 EIGEN_DEVICE_FUNC inline const PermutationPType& permutationP() const 161 { 162 eigen_assert(m_isInitialized && "LU is not initialized."); 163 return m_p; 164 } 165 166 /** \returns the permutation matrix Q 167 * 168 * \sa permutationP() 169 */ 170 inline const PermutationQType& permutationQ() const 171 { 172 eigen_assert(m_isInitialized && "LU is not initialized."); 173 return m_q; 174 } 175 176 /** \returns the kernel of the matrix, also called its null-space. The columns of the returned matrix 177 * will form a basis of the kernel. 178 * 179 * \note If the kernel has dimension zero, then the returned matrix is a column-vector filled with zeros. 180 * 181 * \note This method has to determine which pivots should be considered nonzero. 182 * For that, it uses the threshold value that you can control by calling 183 * setThreshold(const RealScalar&). 184 * 185 * Example: \include FullPivLU_kernel.cpp 186 * Output: \verbinclude FullPivLU_kernel.out 187 * 188 * \sa image() 189 */ 190 inline const internal::kernel_retval<FullPivLU> kernel() const 191 { 192 eigen_assert(m_isInitialized && "LU is not initialized."); 193 return internal::kernel_retval<FullPivLU>(*this); 194 } 195 196 /** \returns the image of the matrix, also called its column-space. The columns of the returned matrix 197 * will form a basis of the image (column-space). 198 * 199 * \param originalMatrix the original matrix, of which *this is the LU decomposition. 200 * The reason why it is needed to pass it here, is that this allows 201 * a large optimization, as otherwise this method would need to reconstruct it 202 * from the LU decomposition. 203 * 204 * \note If the image has dimension zero, then the returned matrix is a column-vector filled with zeros. 205 * 206 * \note This method has to determine which pivots should be considered nonzero. 207 * For that, it uses the threshold value that you can control by calling 208 * setThreshold(const RealScalar&). 209 * 210 * Example: \include FullPivLU_image.cpp 211 * Output: \verbinclude FullPivLU_image.out 212 * 213 * \sa kernel() 214 */ 215 inline const internal::image_retval<FullPivLU> 216 image(const MatrixType& originalMatrix) const 217 { 218 eigen_assert(m_isInitialized && "LU is not initialized."); 219 return internal::image_retval<FullPivLU>(*this, originalMatrix); 220 } 221 222 #ifdef EIGEN_PARSED_BY_DOXYGEN 223 /** \return a solution x to the equation Ax=b, where A is the matrix of which 224 * *this is the LU decomposition. 225 * 226 * \param b the right-hand-side of the equation to solve. Can be a vector or a matrix, 227 * the only requirement in order for the equation to make sense is that 228 * b.rows()==A.rows(), where A is the matrix of which *this is the LU decomposition. 229 * 230 * \returns a solution. 231 * 232 * \note_about_checking_solutions 233 * 234 * \note_about_arbitrary_choice_of_solution 235 * \note_about_using_kernel_to_study_multiple_solutions 236 * 237 * Example: \include FullPivLU_solve.cpp 238 * Output: \verbinclude FullPivLU_solve.out 239 * 240 * \sa TriangularView::solve(), kernel(), inverse() 241 */ 242 template<typename Rhs> 243 inline const Solve<FullPivLU, Rhs> 244 solve(const MatrixBase<Rhs>& b) const; 245 #endif 246 247 /** \returns an estimate of the reciprocal condition number of the matrix of which \c *this is 248 the LU decomposition. 249 */ 250 inline RealScalar rcond() const 251 { 252 eigen_assert(m_isInitialized && "PartialPivLU is not initialized."); 253 return internal::rcond_estimate_helper(m_l1_norm, *this); 254 } 255 256 /** \returns the determinant of the matrix of which 257 * *this is the LU decomposition. It has only linear complexity 258 * (that is, O(n) where n is the dimension of the square matrix) 259 * as the LU decomposition has already been computed. 260 * 261 * \note This is only for square matrices. 262 * 263 * \note For fixed-size matrices of size up to 4, MatrixBase::determinant() offers 264 * optimized paths. 265 * 266 * \warning a determinant can be very big or small, so for matrices 267 * of large enough dimension, there is a risk of overflow/underflow. 268 * 269 * \sa MatrixBase::determinant() 270 */ 271 typename internal::traits<MatrixType>::Scalar determinant() const; 272 273 /** Allows to prescribe a threshold to be used by certain methods, such as rank(), 274 * who need to determine when pivots are to be considered nonzero. This is not used for the 275 * LU decomposition itself. 276 * 277 * When it needs to get the threshold value, Eigen calls threshold(). By default, this 278 * uses a formula to automatically determine a reasonable threshold. 279 * Once you have called the present method setThreshold(const RealScalar&), 280 * your value is used instead. 281 * 282 * \param threshold The new value to use as the threshold. 283 * 284 * A pivot will be considered nonzero if its absolute value is strictly greater than 285 * \f$ \vert pivot \vert \leqslant threshold \times \vert maxpivot \vert \f$ 286 * where maxpivot is the biggest pivot. 287 * 288 * If you want to come back to the default behavior, call setThreshold(Default_t) 289 */ 290 FullPivLU& setThreshold(const RealScalar& threshold) 291 { 292 m_usePrescribedThreshold = true; 293 m_prescribedThreshold = threshold; 294 return *this; 295 } 296 297 /** Allows to come back to the default behavior, letting Eigen use its default formula for 298 * determining the threshold. 299 * 300 * You should pass the special object Eigen::Default as parameter here. 301 * \code lu.setThreshold(Eigen::Default); \endcode 302 * 303 * See the documentation of setThreshold(const RealScalar&). 304 */ 305 FullPivLU& setThreshold(Default_t) 306 { 307 m_usePrescribedThreshold = false; 308 return *this; 309 } 310 311 /** Returns the threshold that will be used by certain methods such as rank(). 312 * 313 * See the documentation of setThreshold(const RealScalar&). 314 */ 315 RealScalar threshold() const 316 { 317 eigen_assert(m_isInitialized || m_usePrescribedThreshold); 318 return m_usePrescribedThreshold ? m_prescribedThreshold 319 // this formula comes from experimenting (see "LU precision tuning" thread on the list) 320 // and turns out to be identical to Higham's formula used already in LDLt. 321 : NumTraits<Scalar>::epsilon() * RealScalar(m_lu.diagonalSize()); 322 } 323 324 /** \returns the rank of the matrix of which *this is the LU decomposition. 325 * 326 * \note This method has to determine which pivots should be considered nonzero. 327 * For that, it uses the threshold value that you can control by calling 328 * setThreshold(const RealScalar&). 329 */ 330 inline Index rank() const 331 { 332 using std::abs; 333 eigen_assert(m_isInitialized && "LU is not initialized."); 334 RealScalar premultiplied_threshold = abs(m_maxpivot) * threshold(); 335 Index result = 0; 336 for(Index i = 0; i < m_nonzero_pivots; ++i) 337 result += (abs(m_lu.coeff(i,i)) > premultiplied_threshold); 338 return result; 339 } 340 341 /** \returns the dimension of the kernel of the matrix of which *this is the LU decomposition. 342 * 343 * \note This method has to determine which pivots should be considered nonzero. 344 * For that, it uses the threshold value that you can control by calling 345 * setThreshold(const RealScalar&). 346 */ 347 inline Index dimensionOfKernel() const 348 { 349 eigen_assert(m_isInitialized && "LU is not initialized."); 350 return cols() - rank(); 351 } 352 353 /** \returns true if the matrix of which *this is the LU decomposition represents an injective 354 * linear map, i.e. has trivial kernel; false otherwise. 355 * 356 * \note This method has to determine which pivots should be considered nonzero. 357 * For that, it uses the threshold value that you can control by calling 358 * setThreshold(const RealScalar&). 359 */ 360 inline bool isInjective() const 361 { 362 eigen_assert(m_isInitialized && "LU is not initialized."); 363 return rank() == cols(); 364 } 365 366 /** \returns true if the matrix of which *this is the LU decomposition represents a surjective 367 * linear map; false otherwise. 368 * 369 * \note This method has to determine which pivots should be considered nonzero. 370 * For that, it uses the threshold value that you can control by calling 371 * setThreshold(const RealScalar&). 372 */ 373 inline bool isSurjective() const 374 { 375 eigen_assert(m_isInitialized && "LU is not initialized."); 376 return rank() == rows(); 377 } 378 379 /** \returns true if the matrix of which *this is the LU decomposition is invertible. 380 * 381 * \note This method has to determine which pivots should be considered nonzero. 382 * For that, it uses the threshold value that you can control by calling 383 * setThreshold(const RealScalar&). 384 */ 385 inline bool isInvertible() const 386 { 387 eigen_assert(m_isInitialized && "LU is not initialized."); 388 return isInjective() && (m_lu.rows() == m_lu.cols()); 389 } 390 391 /** \returns the inverse of the matrix of which *this is the LU decomposition. 392 * 393 * \note If this matrix is not invertible, the returned matrix has undefined coefficients. 394 * Use isInvertible() to first determine whether this matrix is invertible. 395 * 396 * \sa MatrixBase::inverse() 397 */ 398 inline const Inverse<FullPivLU> inverse() const 399 { 400 eigen_assert(m_isInitialized && "LU is not initialized."); 401 eigen_assert(m_lu.rows() == m_lu.cols() && "You can't take the inverse of a non-square matrix!"); 402 return Inverse<FullPivLU>(*this); 403 } 404 405 MatrixType reconstructedMatrix() const; 406 407 EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR 408 inline Index rows() const EIGEN_NOEXCEPT { return m_lu.rows(); } 409 EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR 410 inline Index cols() const EIGEN_NOEXCEPT { return m_lu.cols(); } 411 412 #ifndef EIGEN_PARSED_BY_DOXYGEN 413 template<typename RhsType, typename DstType> 414 void _solve_impl(const RhsType &rhs, DstType &dst) const; 415 416 template<bool Conjugate, typename RhsType, typename DstType> 417 void _solve_impl_transposed(const RhsType &rhs, DstType &dst) const; 418 #endif 419 420 protected: 421 422 static void check_template_parameters() 423 { 424 EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar); 425 } 426 427 void computeInPlace(); 428 429 MatrixType m_lu; 430 PermutationPType m_p; 431 PermutationQType m_q; 432 IntColVectorType m_rowsTranspositions; 433 IntRowVectorType m_colsTranspositions; 434 Index m_nonzero_pivots; 435 RealScalar m_l1_norm; 436 RealScalar m_maxpivot, m_prescribedThreshold; 437 signed char m_det_pq; 438 bool m_isInitialized, m_usePrescribedThreshold; 439 }; 440 441 template<typename MatrixType> 442 FullPivLU<MatrixType>::FullPivLU() 443 : m_isInitialized(false), m_usePrescribedThreshold(false) 444 { 445 } 446 447 template<typename MatrixType> 448 FullPivLU<MatrixType>::FullPivLU(Index rows, Index cols) 449 : m_lu(rows, cols), 450 m_p(rows), 451 m_q(cols), 452 m_rowsTranspositions(rows), 453 m_colsTranspositions(cols), 454 m_isInitialized(false), 455 m_usePrescribedThreshold(false) 456 { 457 } 458 459 template<typename MatrixType> 460 template<typename InputType> 461 FullPivLU<MatrixType>::FullPivLU(const EigenBase<InputType>& matrix) 462 : m_lu(matrix.rows(), matrix.cols()), 463 m_p(matrix.rows()), 464 m_q(matrix.cols()), 465 m_rowsTranspositions(matrix.rows()), 466 m_colsTranspositions(matrix.cols()), 467 m_isInitialized(false), 468 m_usePrescribedThreshold(false) 469 { 470 compute(matrix.derived()); 471 } 472 473 template<typename MatrixType> 474 template<typename InputType> 475 FullPivLU<MatrixType>::FullPivLU(EigenBase<InputType>& matrix) 476 : m_lu(matrix.derived()), 477 m_p(matrix.rows()), 478 m_q(matrix.cols()), 479 m_rowsTranspositions(matrix.rows()), 480 m_colsTranspositions(matrix.cols()), 481 m_isInitialized(false), 482 m_usePrescribedThreshold(false) 483 { 484 computeInPlace(); 485 } 486 487 template<typename MatrixType> 488 void FullPivLU<MatrixType>::computeInPlace() 489 { 490 check_template_parameters(); 491 492 // the permutations are stored as int indices, so just to be sure: 493 eigen_assert(m_lu.rows()<=NumTraits<int>::highest() && m_lu.cols()<=NumTraits<int>::highest()); 494 495 m_l1_norm = m_lu.cwiseAbs().colwise().sum().maxCoeff(); 496 497 const Index size = m_lu.diagonalSize(); 498 const Index rows = m_lu.rows(); 499 const Index cols = m_lu.cols(); 500 501 // will store the transpositions, before we accumulate them at the end. 502 // can't accumulate on-the-fly because that will be done in reverse order for the rows. 503 m_rowsTranspositions.resize(m_lu.rows()); 504 m_colsTranspositions.resize(m_lu.cols()); 505 Index number_of_transpositions = 0; // number of NONTRIVIAL transpositions, i.e. m_rowsTranspositions[i]!=i 506 507 m_nonzero_pivots = size; // the generic case is that in which all pivots are nonzero (invertible case) 508 m_maxpivot = RealScalar(0); 509 510 for(Index k = 0; k < size; ++k) 511 { 512 // First, we need to find the pivot. 513 514 // biggest coefficient in the remaining bottom-right corner (starting at row k, col k) 515 Index row_of_biggest_in_corner, col_of_biggest_in_corner; 516 typedef internal::scalar_score_coeff_op<Scalar> Scoring; 517 typedef typename Scoring::result_type Score; 518 Score biggest_in_corner; 519 biggest_in_corner = m_lu.bottomRightCorner(rows-k, cols-k) 520 .unaryExpr(Scoring()) 521 .maxCoeff(&row_of_biggest_in_corner, &col_of_biggest_in_corner); 522 row_of_biggest_in_corner += k; // correct the values! since they were computed in the corner, 523 col_of_biggest_in_corner += k; // need to add k to them. 524 525 if(biggest_in_corner==Score(0)) 526 { 527 // before exiting, make sure to initialize the still uninitialized transpositions 528 // in a sane state without destroying what we already have. 529 m_nonzero_pivots = k; 530 for(Index i = k; i < size; ++i) 531 { 532 m_rowsTranspositions.coeffRef(i) = internal::convert_index<StorageIndex>(i); 533 m_colsTranspositions.coeffRef(i) = internal::convert_index<StorageIndex>(i); 534 } 535 break; 536 } 537 538 RealScalar abs_pivot = internal::abs_knowing_score<Scalar>()(m_lu(row_of_biggest_in_corner, col_of_biggest_in_corner), biggest_in_corner); 539 if(abs_pivot > m_maxpivot) m_maxpivot = abs_pivot; 540 541 // Now that we've found the pivot, we need to apply the row/col swaps to 542 // bring it to the location (k,k). 543 544 m_rowsTranspositions.coeffRef(k) = internal::convert_index<StorageIndex>(row_of_biggest_in_corner); 545 m_colsTranspositions.coeffRef(k) = internal::convert_index<StorageIndex>(col_of_biggest_in_corner); 546 if(k != row_of_biggest_in_corner) { 547 m_lu.row(k).swap(m_lu.row(row_of_biggest_in_corner)); 548 ++number_of_transpositions; 549 } 550 if(k != col_of_biggest_in_corner) { 551 m_lu.col(k).swap(m_lu.col(col_of_biggest_in_corner)); 552 ++number_of_transpositions; 553 } 554 555 // Now that the pivot is at the right location, we update the remaining 556 // bottom-right corner by Gaussian elimination. 557 558 if(k<rows-1) 559 m_lu.col(k).tail(rows-k-1) /= m_lu.coeff(k,k); 560 if(k<size-1) 561 m_lu.block(k+1,k+1,rows-k-1,cols-k-1).noalias() -= m_lu.col(k).tail(rows-k-1) * m_lu.row(k).tail(cols-k-1); 562 } 563 564 // the main loop is over, we still have to accumulate the transpositions to find the 565 // permutations P and Q 566 567 m_p.setIdentity(rows); 568 for(Index k = size-1; k >= 0; --k) 569 m_p.applyTranspositionOnTheRight(k, m_rowsTranspositions.coeff(k)); 570 571 m_q.setIdentity(cols); 572 for(Index k = 0; k < size; ++k) 573 m_q.applyTranspositionOnTheRight(k, m_colsTranspositions.coeff(k)); 574 575 m_det_pq = (number_of_transpositions%2) ? -1 : 1; 576 577 m_isInitialized = true; 578 } 579 580 template<typename MatrixType> 581 typename internal::traits<MatrixType>::Scalar FullPivLU<MatrixType>::determinant() const 582 { 583 eigen_assert(m_isInitialized && "LU is not initialized."); 584 eigen_assert(m_lu.rows() == m_lu.cols() && "You can't take the determinant of a non-square matrix!"); 585 return Scalar(m_det_pq) * Scalar(m_lu.diagonal().prod()); 586 } 587 588 /** \returns the matrix represented by the decomposition, 589 * i.e., it returns the product: \f$ P^{-1} L U Q^{-1} \f$. 590 * This function is provided for debug purposes. */ 591 template<typename MatrixType> 592 MatrixType FullPivLU<MatrixType>::reconstructedMatrix() const 593 { 594 eigen_assert(m_isInitialized && "LU is not initialized."); 595 const Index smalldim = (std::min)(m_lu.rows(), m_lu.cols()); 596 // LU 597 MatrixType res(m_lu.rows(),m_lu.cols()); 598 // FIXME the .toDenseMatrix() should not be needed... 599 res = m_lu.leftCols(smalldim) 600 .template triangularView<UnitLower>().toDenseMatrix() 601 * m_lu.topRows(smalldim) 602 .template triangularView<Upper>().toDenseMatrix(); 603 604 // P^{-1}(LU) 605 res = m_p.inverse() * res; 606 607 // (P^{-1}LU)Q^{-1} 608 res = res * m_q.inverse(); 609 610 return res; 611 } 612 613 /********* Implementation of kernel() **************************************************/ 614 615 namespace internal { 616 template<typename _MatrixType> 617 struct kernel_retval<FullPivLU<_MatrixType> > 618 : kernel_retval_base<FullPivLU<_MatrixType> > 619 { 620 EIGEN_MAKE_KERNEL_HELPERS(FullPivLU<_MatrixType>) 621 622 enum { MaxSmallDimAtCompileTime = EIGEN_SIZE_MIN_PREFER_FIXED( 623 MatrixType::MaxColsAtCompileTime, 624 MatrixType::MaxRowsAtCompileTime) 625 }; 626 627 template<typename Dest> void evalTo(Dest& dst) const 628 { 629 using std::abs; 630 const Index cols = dec().matrixLU().cols(), dimker = cols - rank(); 631 if(dimker == 0) 632 { 633 // The Kernel is just {0}, so it doesn't have a basis properly speaking, but let's 634 // avoid crashing/asserting as that depends on floating point calculations. Let's 635 // just return a single column vector filled with zeros. 636 dst.setZero(); 637 return; 638 } 639 640 /* Let us use the following lemma: 641 * 642 * Lemma: If the matrix A has the LU decomposition PAQ = LU, 643 * then Ker A = Q(Ker U). 644 * 645 * Proof: trivial: just keep in mind that P, Q, L are invertible. 646 */ 647 648 /* Thus, all we need to do is to compute Ker U, and then apply Q. 649 * 650 * U is upper triangular, with eigenvalues sorted so that any zeros appear at the end. 651 * Thus, the diagonal of U ends with exactly 652 * dimKer zero's. Let us use that to construct dimKer linearly 653 * independent vectors in Ker U. 654 */ 655 656 Matrix<Index, Dynamic, 1, 0, MaxSmallDimAtCompileTime, 1> pivots(rank()); 657 RealScalar premultiplied_threshold = dec().maxPivot() * dec().threshold(); 658 Index p = 0; 659 for(Index i = 0; i < dec().nonzeroPivots(); ++i) 660 if(abs(dec().matrixLU().coeff(i,i)) > premultiplied_threshold) 661 pivots.coeffRef(p++) = i; 662 eigen_internal_assert(p == rank()); 663 664 // we construct a temporaty trapezoid matrix m, by taking the U matrix and 665 // permuting the rows and cols to bring the nonnegligible pivots to the top of 666 // the main diagonal. We need that to be able to apply our triangular solvers. 667 // FIXME when we get triangularView-for-rectangular-matrices, this can be simplified 668 Matrix<typename MatrixType::Scalar, Dynamic, Dynamic, MatrixType::Options, 669 MaxSmallDimAtCompileTime, MatrixType::MaxColsAtCompileTime> 670 m(dec().matrixLU().block(0, 0, rank(), cols)); 671 for(Index i = 0; i < rank(); ++i) 672 { 673 if(i) m.row(i).head(i).setZero(); 674 m.row(i).tail(cols-i) = dec().matrixLU().row(pivots.coeff(i)).tail(cols-i); 675 } 676 m.block(0, 0, rank(), rank()); 677 m.block(0, 0, rank(), rank()).template triangularView<StrictlyLower>().setZero(); 678 for(Index i = 0; i < rank(); ++i) 679 m.col(i).swap(m.col(pivots.coeff(i))); 680 681 // ok, we have our trapezoid matrix, we can apply the triangular solver. 682 // notice that the math behind this suggests that we should apply this to the 683 // negative of the RHS, but for performance we just put the negative sign elsewhere, see below. 684 m.topLeftCorner(rank(), rank()) 685 .template triangularView<Upper>().solveInPlace( 686 m.topRightCorner(rank(), dimker) 687 ); 688 689 // now we must undo the column permutation that we had applied! 690 for(Index i = rank()-1; i >= 0; --i) 691 m.col(i).swap(m.col(pivots.coeff(i))); 692 693 // see the negative sign in the next line, that's what we were talking about above. 694 for(Index i = 0; i < rank(); ++i) dst.row(dec().permutationQ().indices().coeff(i)) = -m.row(i).tail(dimker); 695 for(Index i = rank(); i < cols; ++i) dst.row(dec().permutationQ().indices().coeff(i)).setZero(); 696 for(Index k = 0; k < dimker; ++k) dst.coeffRef(dec().permutationQ().indices().coeff(rank()+k), k) = Scalar(1); 697 } 698 }; 699 700 /***** Implementation of image() *****************************************************/ 701 702 template<typename _MatrixType> 703 struct image_retval<FullPivLU<_MatrixType> > 704 : image_retval_base<FullPivLU<_MatrixType> > 705 { 706 EIGEN_MAKE_IMAGE_HELPERS(FullPivLU<_MatrixType>) 707 708 enum { MaxSmallDimAtCompileTime = EIGEN_SIZE_MIN_PREFER_FIXED( 709 MatrixType::MaxColsAtCompileTime, 710 MatrixType::MaxRowsAtCompileTime) 711 }; 712 713 template<typename Dest> void evalTo(Dest& dst) const 714 { 715 using std::abs; 716 if(rank() == 0) 717 { 718 // The Image is just {0}, so it doesn't have a basis properly speaking, but let's 719 // avoid crashing/asserting as that depends on floating point calculations. Let's 720 // just return a single column vector filled with zeros. 721 dst.setZero(); 722 return; 723 } 724 725 Matrix<Index, Dynamic, 1, 0, MaxSmallDimAtCompileTime, 1> pivots(rank()); 726 RealScalar premultiplied_threshold = dec().maxPivot() * dec().threshold(); 727 Index p = 0; 728 for(Index i = 0; i < dec().nonzeroPivots(); ++i) 729 if(abs(dec().matrixLU().coeff(i,i)) > premultiplied_threshold) 730 pivots.coeffRef(p++) = i; 731 eigen_internal_assert(p == rank()); 732 733 for(Index i = 0; i < rank(); ++i) 734 dst.col(i) = originalMatrix().col(dec().permutationQ().indices().coeff(pivots.coeff(i))); 735 } 736 }; 737 738 /***** Implementation of solve() *****************************************************/ 739 740 } // end namespace internal 741 742 #ifndef EIGEN_PARSED_BY_DOXYGEN 743 template<typename _MatrixType> 744 template<typename RhsType, typename DstType> 745 void FullPivLU<_MatrixType>::_solve_impl(const RhsType &rhs, DstType &dst) const 746 { 747 /* The decomposition PAQ = LU can be rewritten as A = P^{-1} L U Q^{-1}. 748 * So we proceed as follows: 749 * Step 1: compute c = P * rhs. 750 * Step 2: replace c by the solution x to Lx = c. Exists because L is invertible. 751 * Step 3: replace c by the solution x to Ux = c. May or may not exist. 752 * Step 4: result = Q * c; 753 */ 754 755 const Index rows = this->rows(), 756 cols = this->cols(), 757 nonzero_pivots = this->rank(); 758 const Index smalldim = (std::min)(rows, cols); 759 760 if(nonzero_pivots == 0) 761 { 762 dst.setZero(); 763 return; 764 } 765 766 typename RhsType::PlainObject c(rhs.rows(), rhs.cols()); 767 768 // Step 1 769 c = permutationP() * rhs; 770 771 // Step 2 772 m_lu.topLeftCorner(smalldim,smalldim) 773 .template triangularView<UnitLower>() 774 .solveInPlace(c.topRows(smalldim)); 775 if(rows>cols) 776 c.bottomRows(rows-cols) -= m_lu.bottomRows(rows-cols) * c.topRows(cols); 777 778 // Step 3 779 m_lu.topLeftCorner(nonzero_pivots, nonzero_pivots) 780 .template triangularView<Upper>() 781 .solveInPlace(c.topRows(nonzero_pivots)); 782 783 // Step 4 784 for(Index i = 0; i < nonzero_pivots; ++i) 785 dst.row(permutationQ().indices().coeff(i)) = c.row(i); 786 for(Index i = nonzero_pivots; i < m_lu.cols(); ++i) 787 dst.row(permutationQ().indices().coeff(i)).setZero(); 788 } 789 790 template<typename _MatrixType> 791 template<bool Conjugate, typename RhsType, typename DstType> 792 void FullPivLU<_MatrixType>::_solve_impl_transposed(const RhsType &rhs, DstType &dst) const 793 { 794 /* The decomposition PAQ = LU can be rewritten as A = P^{-1} L U Q^{-1}, 795 * and since permutations are real and unitary, we can write this 796 * as A^T = Q U^T L^T P, 797 * So we proceed as follows: 798 * Step 1: compute c = Q^T rhs. 799 * Step 2: replace c by the solution x to U^T x = c. May or may not exist. 800 * Step 3: replace c by the solution x to L^T x = c. 801 * Step 4: result = P^T c. 802 * If Conjugate is true, replace "^T" by "^*" above. 803 */ 804 805 const Index rows = this->rows(), cols = this->cols(), 806 nonzero_pivots = this->rank(); 807 const Index smalldim = (std::min)(rows, cols); 808 809 if(nonzero_pivots == 0) 810 { 811 dst.setZero(); 812 return; 813 } 814 815 typename RhsType::PlainObject c(rhs.rows(), rhs.cols()); 816 817 // Step 1 818 c = permutationQ().inverse() * rhs; 819 820 // Step 2 821 m_lu.topLeftCorner(nonzero_pivots, nonzero_pivots) 822 .template triangularView<Upper>() 823 .transpose() 824 .template conjugateIf<Conjugate>() 825 .solveInPlace(c.topRows(nonzero_pivots)); 826 827 // Step 3 828 m_lu.topLeftCorner(smalldim, smalldim) 829 .template triangularView<UnitLower>() 830 .transpose() 831 .template conjugateIf<Conjugate>() 832 .solveInPlace(c.topRows(smalldim)); 833 834 // Step 4 835 PermutationPType invp = permutationP().inverse().eval(); 836 for(Index i = 0; i < smalldim; ++i) 837 dst.row(invp.indices().coeff(i)) = c.row(i); 838 for(Index i = smalldim; i < rows; ++i) 839 dst.row(invp.indices().coeff(i)).setZero(); 840 } 841 842 #endif 843 844 namespace internal { 845 846 847 /***** Implementation of inverse() *****************************************************/ 848 template<typename DstXprType, typename MatrixType> 849 struct Assignment<DstXprType, Inverse<FullPivLU<MatrixType> >, internal::assign_op<typename DstXprType::Scalar,typename FullPivLU<MatrixType>::Scalar>, Dense2Dense> 850 { 851 typedef FullPivLU<MatrixType> LuType; 852 typedef Inverse<LuType> SrcXprType; 853 static void run(DstXprType &dst, const SrcXprType &src, const internal::assign_op<typename DstXprType::Scalar,typename MatrixType::Scalar> &) 854 { 855 dst = src.nestedExpression().solve(MatrixType::Identity(src.rows(), src.cols())); 856 } 857 }; 858 } // end namespace internal 859 860 /******* MatrixBase methods *****************************************************************/ 861 862 /** \lu_module 863 * 864 * \return the full-pivoting LU decomposition of \c *this. 865 * 866 * \sa class FullPivLU 867 */ 868 template<typename Derived> 869 inline const FullPivLU<typename MatrixBase<Derived>::PlainObject> 870 MatrixBase<Derived>::fullPivLu() const 871 { 872 return FullPivLU<PlainObject>(eval()); 873 } 874 875 } // end namespace Eigen 876 877 #endif // EIGEN_LU_H 878