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// Copyright (C) 2009 Gael Guennebaud <gael.guennebaud@inria.fr>
6//
7// This Source Code Form is subject to the terms of the Mozilla
8// Public License v. 2.0. If a copy of the MPL was not distributed
9// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
10
11#ifndef EIGEN_PARTIALLU_H
12#define EIGEN_PARTIALLU_H
13
14namespace Eigen {
15
16namespace internal {
17template<typename _MatrixType> struct traits<PartialPivLU<_MatrixType> >
18 : traits<_MatrixType>
19{
20 typedef MatrixXpr XprKind;
21 typedef SolverStorage StorageKind;
22 typedef traits<_MatrixType> BaseTraits;
23 enum {
24 Flags = BaseTraits::Flags & RowMajorBit,
25 CoeffReadCost = Dynamic
26 };
27};
28
29template<typename T,typename Derived>
30struct enable_if_ref;
31// {
32// typedef Derived type;
33// };
34
35template<typename T,typename Derived>
36struct enable_if_ref<Ref<T>,Derived> {
37 typedef Derived type;
38};
39
40} // end namespace internal
41
42/** \ingroup LU_Module
43 *
44 * \class PartialPivLU
45 *
46 * \brief LU decomposition of a matrix with partial pivoting, and related features
47 *
48 * \tparam _MatrixType the type of the matrix of which we are computing the LU decomposition
49 *
50 * This class represents a LU decomposition of a \b square \b invertible matrix, with partial pivoting: the matrix A
51 * is decomposed as A = PLU where L is unit-lower-triangular, U is upper-triangular, and P
52 * is a permutation matrix.
53 *
54 * Typically, partial pivoting LU decomposition is only considered numerically stable for square invertible
55 * matrices. Thus LAPACK's dgesv and dgesvx require the matrix to be square and invertible. The present class
56 * does the same. It will assert that the matrix is square, but it won't (actually it can't) check that the
57 * matrix is invertible: it is your task to check that you only use this decomposition on invertible matrices.
58 *
59 * The guaranteed safe alternative, working for all matrices, is the full pivoting LU decomposition, provided
60 * by class FullPivLU.
61 *
62 * This is \b not a rank-revealing LU decomposition. Many features are intentionally absent from this class,
63 * such as rank computation. If you need these features, use class FullPivLU.
64 *
65 * This LU decomposition is suitable to invert invertible matrices. It is what MatrixBase::inverse() uses
66 * in the general case.
67 * On the other hand, it is \b not suitable to determine whether a given matrix is invertible.
68 *
69 * The data of the LU decomposition can be directly accessed through the methods matrixLU(), permutationP().
70 *
71 * This class supports the \link InplaceDecomposition inplace decomposition \endlink mechanism.
72 *
73 * \sa MatrixBase::partialPivLu(), MatrixBase::determinant(), MatrixBase::inverse(), MatrixBase::computeInverse(), class FullPivLU
74 */
75template<typename _MatrixType> class PartialPivLU
76 : public SolverBase<PartialPivLU<_MatrixType> >
77{
78 public:
79
80 typedef _MatrixType MatrixType;
81 typedef SolverBase<PartialPivLU> Base;
82 EIGEN_GENERIC_PUBLIC_INTERFACE(PartialPivLU)
83 // FIXME StorageIndex defined in EIGEN_GENERIC_PUBLIC_INTERFACE should be int
84 enum {
85 MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
86 MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
87 };
88 typedef PermutationMatrix<RowsAtCompileTime, MaxRowsAtCompileTime> PermutationType;
89 typedef Transpositions<RowsAtCompileTime, MaxRowsAtCompileTime> TranspositionType;
90 typedef typename MatrixType::PlainObject PlainObject;
91
92 /**
93 * \brief Default Constructor.
94 *
95 * The default constructor is useful in cases in which the user intends to
96 * perform decompositions via PartialPivLU::compute(const MatrixType&).
97 */
98 PartialPivLU();
99
100 /** \brief Default Constructor with memory preallocation
101 *
102 * Like the default constructor but with preallocation of the internal data
103 * according to the specified problem \a size.
104 * \sa PartialPivLU()
105 */
106 explicit PartialPivLU(Index size);
107
108 /** Constructor.
109 *
110 * \param matrix the matrix of which to compute the LU decomposition.
111 *
112 * \warning The matrix should have full rank (e.g. if it's square, it should be invertible).
113 * If you need to deal with non-full rank, use class FullPivLU instead.
114 */
115 template<typename InputType>
116 explicit PartialPivLU(const EigenBase<InputType>& matrix);
117
118 /** Constructor for \link InplaceDecomposition inplace decomposition \endlink
119 *
120 * \param matrix the matrix of which to compute the LU decomposition.
121 *
122 * \warning The matrix should have full rank (e.g. if it's square, it should be invertible).
123 * If you need to deal with non-full rank, use class FullPivLU instead.
124 */
125 template<typename InputType>
126 explicit PartialPivLU(EigenBase<InputType>& matrix);
127
128 template<typename InputType>
129 PartialPivLU& compute(const EigenBase<InputType>& matrix) {
130 m_lu = matrix.derived();
131 compute();
132 return *this;
133 }
134
135 /** \returns the LU decomposition matrix: the upper-triangular part is U, the
136 * unit-lower-triangular part is L (at least for square matrices; in the non-square
137 * case, special care is needed, see the documentation of class FullPivLU).
138 *
139 * \sa matrixL(), matrixU()
140 */
141 inline const MatrixType& matrixLU() const
142 {
143 eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
144 return m_lu;
145 }
146
147 /** \returns the permutation matrix P.
148 */
149 inline const PermutationType& permutationP() const
150 {
151 eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
152 return m_p;
153 }
154
155 /** This method returns the solution x to the equation Ax=b, where A is the matrix of which
156 * *this is the LU decomposition.
157 *
158 * \param b the right-hand-side of the equation to solve. Can be a vector or a matrix,
159 * the only requirement in order for the equation to make sense is that
160 * b.rows()==A.rows(), where A is the matrix of which *this is the LU decomposition.
161 *
162 * \returns the solution.
163 *
164 * Example: \include PartialPivLU_solve.cpp
165 * Output: \verbinclude PartialPivLU_solve.out
166 *
167 * Since this PartialPivLU class assumes anyway that the matrix A is invertible, the solution
168 * theoretically exists and is unique regardless of b.
169 *
170 * \sa TriangularView::solve(), inverse(), computeInverse()
171 */
172 // FIXME this is a copy-paste of the base-class member to add the isInitialized assertion.
173 template<typename Rhs>
174 inline const Solve<PartialPivLU, Rhs>
175 solve(const MatrixBase<Rhs>& b) const
176 {
177 eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
178 return Solve<PartialPivLU, Rhs>(*this, b.derived());
179 }
180
181 /** \returns an estimate of the reciprocal condition number of the matrix of which \c *this is
182 the LU decomposition.
183 */
184 inline RealScalar rcond() const
185 {
186 eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
187 return internal::rcond_estimate_helper(m_l1_norm, *this);
188 }
189
190 /** \returns the inverse of the matrix of which *this is the LU decomposition.
191 *
192 * \warning The matrix being decomposed here is assumed to be invertible. If you need to check for
193 * invertibility, use class FullPivLU instead.
194 *
195 * \sa MatrixBase::inverse(), LU::inverse()
196 */
197 inline const Inverse<PartialPivLU> inverse() const
198 {
199 eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
200 return Inverse<PartialPivLU>(*this);
201 }
202
203 /** \returns the determinant of the matrix of which
204 * *this is the LU decomposition. It has only linear complexity
205 * (that is, O(n) where n is the dimension of the square matrix)
206 * as the LU decomposition has already been computed.
207 *
208 * \note For fixed-size matrices of size up to 4, MatrixBase::determinant() offers
209 * optimized paths.
210 *
211 * \warning a determinant can be very big or small, so for matrices
212 * of large enough dimension, there is a risk of overflow/underflow.
213 *
214 * \sa MatrixBase::determinant()
215 */
216 Scalar determinant() const;
217
218 MatrixType reconstructedMatrix() const;
219
220 inline Index rows() const { return m_lu.rows(); }
221 inline Index cols() const { return m_lu.cols(); }
222
223 #ifndef EIGEN_PARSED_BY_DOXYGEN
224 template<typename RhsType, typename DstType>
225 EIGEN_DEVICE_FUNC
226 void _solve_impl(const RhsType &rhs, DstType &dst) const {
227 /* The decomposition PA = LU can be rewritten as A = P^{-1} L U.
228 * So we proceed as follows:
229 * Step 1: compute c = Pb.
230 * Step 2: replace c by the solution x to Lx = c.
231 * Step 3: replace c by the solution x to Ux = c.
232 */
233
234 eigen_assert(rhs.rows() == m_lu.rows());
235
236 // Step 1
237 dst = permutationP() * rhs;
238
239 // Step 2
240 m_lu.template triangularView<UnitLower>().solveInPlace(dst);
241
242 // Step 3
243 m_lu.template triangularView<Upper>().solveInPlace(dst);
244 }
245
246 template<bool Conjugate, typename RhsType, typename DstType>
247 EIGEN_DEVICE_FUNC
248 void _solve_impl_transposed(const RhsType &rhs, DstType &dst) const {
249 /* The decomposition PA = LU can be rewritten as A = P^{-1} L U.
250 * So we proceed as follows:
251 * Step 1: compute c = Pb.
252 * Step 2: replace c by the solution x to Lx = c.
253 * Step 3: replace c by the solution x to Ux = c.
254 */
255
256 eigen_assert(rhs.rows() == m_lu.cols());
257
258 if (Conjugate) {
259 // Step 1
260 dst = m_lu.template triangularView<Upper>().adjoint().solve(rhs);
261 // Step 2
262 m_lu.template triangularView<UnitLower>().adjoint().solveInPlace(dst);
263 } else {
264 // Step 1
265 dst = m_lu.template triangularView<Upper>().transpose().solve(rhs);
266 // Step 2
267 m_lu.template triangularView<UnitLower>().transpose().solveInPlace(dst);
268 }
269 // Step 3
270 dst = permutationP().transpose() * dst;
271 }
272 #endif
273
274 protected:
275
276 static void check_template_parameters()
277 {
278 EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
279 }
280
281 void compute();
282
283 MatrixType m_lu;
284 PermutationType m_p;
285 TranspositionType m_rowsTranspositions;
286 RealScalar m_l1_norm;
287 signed char m_det_p;
288 bool m_isInitialized;
289};
290
291template<typename MatrixType>
292PartialPivLU<MatrixType>::PartialPivLU()
293 : m_lu(),
294 m_p(),
295 m_rowsTranspositions(),
296 m_l1_norm(0),
297 m_det_p(0),
298 m_isInitialized(false)
299{
300}
301
302template<typename MatrixType>
303PartialPivLU<MatrixType>::PartialPivLU(Index size)
304 : m_lu(size, size),
305 m_p(size),
306 m_rowsTranspositions(size),
307 m_l1_norm(0),
308 m_det_p(0),
309 m_isInitialized(false)
310{
311}
312
313template<typename MatrixType>
314template<typename InputType>
315PartialPivLU<MatrixType>::PartialPivLU(const EigenBase<InputType>& matrix)
316 : m_lu(matrix.rows(),matrix.cols()),
317 m_p(matrix.rows()),
318 m_rowsTranspositions(matrix.rows()),
319 m_l1_norm(0),
320 m_det_p(0),
321 m_isInitialized(false)
322{
323 compute(matrix.derived());
324}
325
326template<typename MatrixType>
327template<typename InputType>
328PartialPivLU<MatrixType>::PartialPivLU(EigenBase<InputType>& matrix)
329 : m_lu(matrix.derived()),
330 m_p(matrix.rows()),
331 m_rowsTranspositions(matrix.rows()),
332 m_l1_norm(0),
333 m_det_p(0),
334 m_isInitialized(false)
335{
336 compute();
337}
338
339namespace internal {
340
341/** \internal This is the blocked version of fullpivlu_unblocked() */
342template<typename Scalar, int StorageOrder, typename PivIndex>
343struct partial_lu_impl
344{
345 // FIXME add a stride to Map, so that the following mapping becomes easier,
346 // another option would be to create an expression being able to automatically
347 // warp any Map, Matrix, and Block expressions as a unique type, but since that's exactly
348 // a Map + stride, why not adding a stride to Map, and convenient ctors from a Matrix,
349 // and Block.
350 typedef Map<Matrix<Scalar, Dynamic, Dynamic, StorageOrder> > MapLU;
351 typedef Block<MapLU, Dynamic, Dynamic> MatrixType;
352 typedef Block<MatrixType,Dynamic,Dynamic> BlockType;
353 typedef typename MatrixType::RealScalar RealScalar;
354
355 /** \internal performs the LU decomposition in-place of the matrix \a lu
356 * using an unblocked algorithm.
357 *
358 * In addition, this function returns the row transpositions in the
359 * vector \a row_transpositions which must have a size equal to the number
360 * of columns of the matrix \a lu, and an integer \a nb_transpositions
361 * which returns the actual number of transpositions.
362 *
363 * \returns The index of the first pivot which is exactly zero if any, or a negative number otherwise.
364 */
365 static Index unblocked_lu(MatrixType& lu, PivIndex* row_transpositions, PivIndex& nb_transpositions)
366 {
367 typedef scalar_score_coeff_op<Scalar> Scoring;
368 typedef typename Scoring::result_type Score;
369 const Index rows = lu.rows();
370 const Index cols = lu.cols();
371 const Index size = (std::min)(rows,cols);
372 nb_transpositions = 0;
373 Index first_zero_pivot = -1;
374 for(Index k = 0; k < size; ++k)
375 {
376 Index rrows = rows-k-1;
377 Index rcols = cols-k-1;
378
379 Index row_of_biggest_in_col;
380 Score biggest_in_corner
381 = lu.col(k).tail(rows-k).unaryExpr(Scoring()).maxCoeff(&row_of_biggest_in_col);
382 row_of_biggest_in_col += k;
383
384 row_transpositions[k] = PivIndex(row_of_biggest_in_col);
385
386 if(biggest_in_corner != Score(0))
387 {
388 if(k != row_of_biggest_in_col)
389 {
390 lu.row(k).swap(lu.row(row_of_biggest_in_col));
391 ++nb_transpositions;
392 }
393
394 // FIXME shall we introduce a safe quotient expression in cas 1/lu.coeff(k,k)
395 // overflow but not the actual quotient?
396 lu.col(k).tail(rrows) /= lu.coeff(k,k);
397 }
398 else if(first_zero_pivot==-1)
399 {
400 // the pivot is exactly zero, we record the index of the first pivot which is exactly 0,
401 // and continue the factorization such we still have A = PLU
402 first_zero_pivot = k;
403 }
404
405 if(k<rows-1)
406 lu.bottomRightCorner(rrows,rcols).noalias() -= lu.col(k).tail(rrows) * lu.row(k).tail(rcols);
407 }
408 return first_zero_pivot;
409 }
410
411 /** \internal performs the LU decomposition in-place of the matrix represented
412 * by the variables \a rows, \a cols, \a lu_data, and \a lu_stride using a
413 * recursive, blocked algorithm.
414 *
415 * In addition, this function returns the row transpositions in the
416 * vector \a row_transpositions which must have a size equal to the number
417 * of columns of the matrix \a lu, and an integer \a nb_transpositions
418 * which returns the actual number of transpositions.
419 *
420 * \returns The index of the first pivot which is exactly zero if any, or a negative number otherwise.
421 *
422 * \note This very low level interface using pointers, etc. is to:
423 * 1 - reduce the number of instanciations to the strict minimum
424 * 2 - avoid infinite recursion of the instanciations with Block<Block<Block<...> > >
425 */
426 static Index blocked_lu(Index rows, Index cols, Scalar* lu_data, Index luStride, PivIndex* row_transpositions, PivIndex& nb_transpositions, Index maxBlockSize=256)
427 {
428 MapLU lu1(lu_data,StorageOrder==RowMajor?rows:luStride,StorageOrder==RowMajor?luStride:cols);
429 MatrixType lu(lu1,0,0,rows,cols);
430
431 const Index size = (std::min)(rows,cols);
432
433 // if the matrix is too small, no blocking:
434 if(size<=16)
435 {
436 return unblocked_lu(lu, row_transpositions, nb_transpositions);
437 }
438
439 // automatically adjust the number of subdivisions to the size
440 // of the matrix so that there is enough sub blocks:
441 Index blockSize;
442 {
443 blockSize = size/8;
444 blockSize = (blockSize/16)*16;
445 blockSize = (std::min)((std::max)(blockSize,Index(8)), maxBlockSize);
446 }
447
448 nb_transpositions = 0;
449 Index first_zero_pivot = -1;
450 for(Index k = 0; k < size; k+=blockSize)
451 {
452 Index bs = (std::min)(size-k,blockSize); // actual size of the block
453 Index trows = rows - k - bs; // trailing rows
454 Index tsize = size - k - bs; // trailing size
455
456 // partition the matrix:
457 // A00 | A01 | A02
458 // lu = A_0 | A_1 | A_2 = A10 | A11 | A12
459 // A20 | A21 | A22
460 BlockType A_0(lu,0,0,rows,k);
461 BlockType A_2(lu,0,k+bs,rows,tsize);
462 BlockType A11(lu,k,k,bs,bs);
463 BlockType A12(lu,k,k+bs,bs,tsize);
464 BlockType A21(lu,k+bs,k,trows,bs);
465 BlockType A22(lu,k+bs,k+bs,trows,tsize);
466
467 PivIndex nb_transpositions_in_panel;
468 // recursively call the blocked LU algorithm on [A11^T A21^T]^T
469 // with a very small blocking size:
470 Index ret = blocked_lu(trows+bs, bs, &lu.coeffRef(k,k), luStride,
471 row_transpositions+k, nb_transpositions_in_panel, 16);
472 if(ret>=0 && first_zero_pivot==-1)
473 first_zero_pivot = k+ret;
474
475 nb_transpositions += nb_transpositions_in_panel;
476 // update permutations and apply them to A_0
477 for(Index i=k; i<k+bs; ++i)
478 {
479 Index piv = (row_transpositions[i] += internal::convert_index<PivIndex>(k));
480 A_0.row(i).swap(A_0.row(piv));
481 }
482
483 if(trows)
484 {
485 // apply permutations to A_2
486 for(Index i=k;i<k+bs; ++i)
487 A_2.row(i).swap(A_2.row(row_transpositions[i]));
488
489 // A12 = A11^-1 A12
490 A11.template triangularView<UnitLower>().solveInPlace(A12);
491
492 A22.noalias() -= A21 * A12;
493 }
494 }
495 return first_zero_pivot;
496 }
497};
498
499/** \internal performs the LU decomposition with partial pivoting in-place.
500 */
501template<typename MatrixType, typename TranspositionType>
502void partial_lu_inplace(MatrixType& lu, TranspositionType& row_transpositions, typename TranspositionType::StorageIndex& nb_transpositions)
503{
504 eigen_assert(lu.cols() == row_transpositions.size());
505 eigen_assert((&row_transpositions.coeffRef(1)-&row_transpositions.coeffRef(0)) == 1);
506
507 partial_lu_impl
508 <typename MatrixType::Scalar, MatrixType::Flags&RowMajorBit?RowMajor:ColMajor, typename TranspositionType::StorageIndex>
509 ::blocked_lu(lu.rows(), lu.cols(), &lu.coeffRef(0,0), lu.outerStride(), &row_transpositions.coeffRef(0), nb_transpositions);
510}
511
512} // end namespace internal
513
514template<typename MatrixType>
515void PartialPivLU<MatrixType>::compute()
516{
517 check_template_parameters();
518
519 // the row permutation is stored as int indices, so just to be sure:
520 eigen_assert(m_lu.rows()<NumTraits<int>::highest());
521
522 m_l1_norm = m_lu.cwiseAbs().colwise().sum().maxCoeff();
523
524 eigen_assert(m_lu.rows() == m_lu.cols() && "PartialPivLU is only for square (and moreover invertible) matrices");
525 const Index size = m_lu.rows();
526
527 m_rowsTranspositions.resize(size);
528
529 typename TranspositionType::StorageIndex nb_transpositions;
530 internal::partial_lu_inplace(m_lu, m_rowsTranspositions, nb_transpositions);
531 m_det_p = (nb_transpositions%2) ? -1 : 1;
532
533 m_p = m_rowsTranspositions;
534
535 m_isInitialized = true;
536}
537
538template<typename MatrixType>
539typename PartialPivLU<MatrixType>::Scalar PartialPivLU<MatrixType>::determinant() const
540{
541 eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
542 return Scalar(m_det_p) * m_lu.diagonal().prod();
543}
544
545/** \returns the matrix represented by the decomposition,
546 * i.e., it returns the product: P^{-1} L U.
547 * This function is provided for debug purpose. */
548template<typename MatrixType>
549MatrixType PartialPivLU<MatrixType>::reconstructedMatrix() const
550{
551 eigen_assert(m_isInitialized && "LU is not initialized.");
552 // LU
553 MatrixType res = m_lu.template triangularView<UnitLower>().toDenseMatrix()
554 * m_lu.template triangularView<Upper>();
555
556 // P^{-1}(LU)
557 res = m_p.inverse() * res;
558
559 return res;
560}
561
562/***** Implementation details *****************************************************/
563
564namespace internal {
565
566/***** Implementation of inverse() *****************************************************/
567template<typename DstXprType, typename MatrixType>
568struct Assignment<DstXprType, Inverse<PartialPivLU<MatrixType> >, internal::assign_op<typename DstXprType::Scalar,typename PartialPivLU<MatrixType>::Scalar>, Dense2Dense>
569{
570 typedef PartialPivLU<MatrixType> LuType;
571 typedef Inverse<LuType> SrcXprType;
572 static void run(DstXprType &dst, const SrcXprType &src, const internal::assign_op<typename DstXprType::Scalar,typename LuType::Scalar> &)
573 {
574 dst = src.nestedExpression().solve(MatrixType::Identity(src.rows(), src.cols()));
575 }
576};
577} // end namespace internal
578
579/******** MatrixBase methods *******/
580
581/** \lu_module
582 *
583 * \return the partial-pivoting LU decomposition of \c *this.
584 *
585 * \sa class PartialPivLU
586 */
587template<typename Derived>
588inline const PartialPivLU<typename MatrixBase<Derived>::PlainObject>
589MatrixBase<Derived>::partialPivLu() const
590{
591 return PartialPivLU<PlainObject>(eval());
592}
593
594/** \lu_module
595 *
596 * Synonym of partialPivLu().
597 *
598 * \return the partial-pivoting LU decomposition of \c *this.
599 *
600 * \sa class PartialPivLU
601 */
602template<typename Derived>
603inline const PartialPivLU<typename MatrixBase<Derived>::PlainObject>
604MatrixBase<Derived>::lu() const
605{
606 return PartialPivLU<PlainObject>(eval());
607}
608
609} // end namespace Eigen
610
611#endif // EIGEN_PARTIALLU_H
612