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
14 namespace Eigen {
15
16 /** \ingroup LU_Module
17 *
18 * \class PartialPivLU
19 *
20 * \brief LU decomposition of a matrix with partial pivoting, and related features
21 *
22 * \param MatrixType the type of the matrix of which we are computing the LU decomposition
23 *
24 * This class represents a LU decomposition of a \b square \b invertible matrix, with partial pivoting: the matrix A
25 * is decomposed as A = PLU where L is unit-lower-triangular, U is upper-triangular, and P
26 * is a permutation matrix.
27 *
28 * Typically, partial pivoting LU decomposition is only considered numerically stable for square invertible
29 * matrices. Thus LAPACK's dgesv and dgesvx require the matrix to be square and invertible. The present class
30 * does the same. It will assert that the matrix is square, but it won't (actually it can't) check that the
31 * matrix is invertible: it is your task to check that you only use this decomposition on invertible matrices.
32 *
33 * The guaranteed safe alternative, working for all matrices, is the full pivoting LU decomposition, provided
34 * by class FullPivLU.
35 *
36 * This is \b not a rank-revealing LU decomposition. Many features are intentionally absent from this class,
37 * such as rank computation. If you need these features, use class FullPivLU.
38 *
39 * This LU decomposition is suitable to invert invertible matrices. It is what MatrixBase::inverse() uses
40 * in the general case.
41 * On the other hand, it is \b not suitable to determine whether a given matrix is invertible.
42 *
43 * The data of the LU decomposition can be directly accessed through the methods matrixLU(), permutationP().
44 *
45 * \sa MatrixBase::partialPivLu(), MatrixBase::determinant(), MatrixBase::inverse(), MatrixBase::computeInverse(), class FullPivLU
46 */
47 template<typename _MatrixType> class PartialPivLU
48 {
49 public:
50
51 typedef _MatrixType MatrixType;
52 enum {
53 RowsAtCompileTime = MatrixType::RowsAtCompileTime,
54 ColsAtCompileTime = MatrixType::ColsAtCompileTime,
55 Options = MatrixType::Options,
56 MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
57 MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
58 };
59 typedef typename MatrixType::Scalar Scalar;
60 typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
61 typedef typename internal::traits<MatrixType>::StorageKind StorageKind;
62 typedef typename MatrixType::Index Index;
63 typedef PermutationMatrix<RowsAtCompileTime, MaxRowsAtCompileTime> PermutationType;
64 typedef Transpositions<RowsAtCompileTime, MaxRowsAtCompileTime> TranspositionType;
65
66
67 /**
68 * \brief Default Constructor.
69 *
70 * The default constructor is useful in cases in which the user intends to
71 * perform decompositions via PartialPivLU::compute(const MatrixType&).
72 */
73 PartialPivLU();
74
75 /** \brief Default Constructor with memory preallocation
76 *
77 * Like the default constructor but with preallocation of the internal data
78 * according to the specified problem \a size.
79 * \sa PartialPivLU()
80 */
81 PartialPivLU(Index size);
82
83 /** Constructor.
84 *
85 * \param matrix the matrix of which to compute the LU decomposition.
86 *
87 * \warning The matrix should have full rank (e.g. if it's square, it should be invertible).
88 * If you need to deal with non-full rank, use class FullPivLU instead.
89 */
90 PartialPivLU(const MatrixType& matrix);
91
92 PartialPivLU& compute(const MatrixType& matrix);
93
94 /** \returns the LU decomposition matrix: the upper-triangular part is U, the
95 * unit-lower-triangular part is L (at least for square matrices; in the non-square
96 * case, special care is needed, see the documentation of class FullPivLU).
97 *
98 * \sa matrixL(), matrixU()
99 */
matrixLU()100 inline const MatrixType& matrixLU() const
101 {
102 eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
103 return m_lu;
104 }
105
106 /** \returns the permutation matrix P.
107 */
permutationP()108 inline const PermutationType& permutationP() const
109 {
110 eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
111 return m_p;
112 }
113
114 /** This method returns the solution x to the equation Ax=b, where A is the matrix of which
115 * *this is the LU decomposition.
116 *
117 * \param b the right-hand-side of the equation to solve. Can be a vector or a matrix,
118 * the only requirement in order for the equation to make sense is that
119 * b.rows()==A.rows(), where A is the matrix of which *this is the LU decomposition.
120 *
121 * \returns the solution.
122 *
123 * Example: \include PartialPivLU_solve.cpp
124 * Output: \verbinclude PartialPivLU_solve.out
125 *
126 * Since this PartialPivLU class assumes anyway that the matrix A is invertible, the solution
127 * theoretically exists and is unique regardless of b.
128 *
129 * \sa TriangularView::solve(), inverse(), computeInverse()
130 */
131 template<typename Rhs>
132 inline const internal::solve_retval<PartialPivLU, Rhs>
solve(const MatrixBase<Rhs> & b)133 solve(const MatrixBase<Rhs>& b) const
134 {
135 eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
136 return internal::solve_retval<PartialPivLU, Rhs>(*this, b.derived());
137 }
138
139 /** \returns the inverse of the matrix of which *this is the LU decomposition.
140 *
141 * \warning The matrix being decomposed here is assumed to be invertible. If you need to check for
142 * invertibility, use class FullPivLU instead.
143 *
144 * \sa MatrixBase::inverse(), LU::inverse()
145 */
inverse()146 inline const internal::solve_retval<PartialPivLU,typename MatrixType::IdentityReturnType> inverse() const
147 {
148 eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
149 return internal::solve_retval<PartialPivLU,typename MatrixType::IdentityReturnType>
150 (*this, MatrixType::Identity(m_lu.rows(), m_lu.cols()));
151 }
152
153 /** \returns the determinant of the matrix of which
154 * *this is the LU decomposition. It has only linear complexity
155 * (that is, O(n) where n is the dimension of the square matrix)
156 * as the LU decomposition has already been computed.
157 *
158 * \note For fixed-size matrices of size up to 4, MatrixBase::determinant() offers
159 * optimized paths.
160 *
161 * \warning a determinant can be very big or small, so for matrices
162 * of large enough dimension, there is a risk of overflow/underflow.
163 *
164 * \sa MatrixBase::determinant()
165 */
166 typename internal::traits<MatrixType>::Scalar determinant() const;
167
168 MatrixType reconstructedMatrix() const;
169
rows()170 inline Index rows() const { return m_lu.rows(); }
cols()171 inline Index cols() const { return m_lu.cols(); }
172
173 protected:
174 MatrixType m_lu;
175 PermutationType m_p;
176 TranspositionType m_rowsTranspositions;
177 Index m_det_p;
178 bool m_isInitialized;
179 };
180
181 template<typename MatrixType>
PartialPivLU()182 PartialPivLU<MatrixType>::PartialPivLU()
183 : m_lu(),
184 m_p(),
185 m_rowsTranspositions(),
186 m_det_p(0),
187 m_isInitialized(false)
188 {
189 }
190
191 template<typename MatrixType>
PartialPivLU(Index size)192 PartialPivLU<MatrixType>::PartialPivLU(Index size)
193 : m_lu(size, size),
194 m_p(size),
195 m_rowsTranspositions(size),
196 m_det_p(0),
197 m_isInitialized(false)
198 {
199 }
200
201 template<typename MatrixType>
PartialPivLU(const MatrixType & matrix)202 PartialPivLU<MatrixType>::PartialPivLU(const MatrixType& matrix)
203 : m_lu(matrix.rows(), matrix.rows()),
204 m_p(matrix.rows()),
205 m_rowsTranspositions(matrix.rows()),
206 m_det_p(0),
207 m_isInitialized(false)
208 {
209 compute(matrix);
210 }
211
212 namespace internal {
213
214 /** \internal This is the blocked version of fullpivlu_unblocked() */
215 template<typename Scalar, int StorageOrder, typename PivIndex>
216 struct partial_lu_impl
217 {
218 // FIXME add a stride to Map, so that the following mapping becomes easier,
219 // another option would be to create an expression being able to automatically
220 // warp any Map, Matrix, and Block expressions as a unique type, but since that's exactly
221 // a Map + stride, why not adding a stride to Map, and convenient ctors from a Matrix,
222 // and Block.
223 typedef Map<Matrix<Scalar, Dynamic, Dynamic, StorageOrder> > MapLU;
224 typedef Block<MapLU, Dynamic, Dynamic> MatrixType;
225 typedef Block<MatrixType,Dynamic,Dynamic> BlockType;
226 typedef typename MatrixType::RealScalar RealScalar;
227 typedef typename MatrixType::Index Index;
228
229 /** \internal performs the LU decomposition in-place of the matrix \a lu
230 * using an unblocked algorithm.
231 *
232 * In addition, this function returns the row transpositions in the
233 * vector \a row_transpositions which must have a size equal to the number
234 * of columns of the matrix \a lu, and an integer \a nb_transpositions
235 * which returns the actual number of transpositions.
236 *
237 * \returns The index of the first pivot which is exactly zero if any, or a negative number otherwise.
238 */
unblocked_lupartial_lu_impl239 static Index unblocked_lu(MatrixType& lu, PivIndex* row_transpositions, PivIndex& nb_transpositions)
240 {
241 const Index rows = lu.rows();
242 const Index cols = lu.cols();
243 const Index size = (std::min)(rows,cols);
244 nb_transpositions = 0;
245 Index first_zero_pivot = -1;
246 for(Index k = 0; k < size; ++k)
247 {
248 Index rrows = rows-k-1;
249 Index rcols = cols-k-1;
250
251 Index row_of_biggest_in_col;
252 RealScalar biggest_in_corner
253 = lu.col(k).tail(rows-k).cwiseAbs().maxCoeff(&row_of_biggest_in_col);
254 row_of_biggest_in_col += k;
255
256 row_transpositions[k] = PivIndex(row_of_biggest_in_col);
257
258 if(biggest_in_corner != RealScalar(0))
259 {
260 if(k != row_of_biggest_in_col)
261 {
262 lu.row(k).swap(lu.row(row_of_biggest_in_col));
263 ++nb_transpositions;
264 }
265
266 // FIXME shall we introduce a safe quotient expression in cas 1/lu.coeff(k,k)
267 // overflow but not the actual quotient?
268 lu.col(k).tail(rrows) /= lu.coeff(k,k);
269 }
270 else if(first_zero_pivot==-1)
271 {
272 // the pivot is exactly zero, we record the index of the first pivot which is exactly 0,
273 // and continue the factorization such we still have A = PLU
274 first_zero_pivot = k;
275 }
276
277 if(k<rows-1)
278 lu.bottomRightCorner(rrows,rcols).noalias() -= lu.col(k).tail(rrows) * lu.row(k).tail(rcols);
279 }
280 return first_zero_pivot;
281 }
282
283 /** \internal performs the LU decomposition in-place of the matrix represented
284 * by the variables \a rows, \a cols, \a lu_data, and \a lu_stride using a
285 * recursive, blocked algorithm.
286 *
287 * In addition, this function returns the row transpositions in the
288 * vector \a row_transpositions which must have a size equal to the number
289 * of columns of the matrix \a lu, and an integer \a nb_transpositions
290 * which returns the actual number of transpositions.
291 *
292 * \returns The index of the first pivot which is exactly zero if any, or a negative number otherwise.
293 *
294 * \note This very low level interface using pointers, etc. is to:
295 * 1 - reduce the number of instanciations to the strict minimum
296 * 2 - avoid infinite recursion of the instanciations with Block<Block<Block<...> > >
297 */
298 static Index blocked_lu(Index rows, Index cols, Scalar* lu_data, Index luStride, PivIndex* row_transpositions, PivIndex& nb_transpositions, Index maxBlockSize=256)
299 {
300 MapLU lu1(lu_data,StorageOrder==RowMajor?rows:luStride,StorageOrder==RowMajor?luStride:cols);
301 MatrixType lu(lu1,0,0,rows,cols);
302
303 const Index size = (std::min)(rows,cols);
304
305 // if the matrix is too small, no blocking:
306 if(size<=16)
307 {
308 return unblocked_lu(lu, row_transpositions, nb_transpositions);
309 }
310
311 // automatically adjust the number of subdivisions to the size
312 // of the matrix so that there is enough sub blocks:
313 Index blockSize;
314 {
315 blockSize = size/8;
316 blockSize = (blockSize/16)*16;
317 blockSize = (std::min)((std::max)(blockSize,Index(8)), maxBlockSize);
318 }
319
320 nb_transpositions = 0;
321 Index first_zero_pivot = -1;
322 for(Index k = 0; k < size; k+=blockSize)
323 {
324 Index bs = (std::min)(size-k,blockSize); // actual size of the block
325 Index trows = rows - k - bs; // trailing rows
326 Index tsize = size - k - bs; // trailing size
327
328 // partition the matrix:
329 // A00 | A01 | A02
330 // lu = A_0 | A_1 | A_2 = A10 | A11 | A12
331 // A20 | A21 | A22
332 BlockType A_0(lu,0,0,rows,k);
333 BlockType A_2(lu,0,k+bs,rows,tsize);
334 BlockType A11(lu,k,k,bs,bs);
335 BlockType A12(lu,k,k+bs,bs,tsize);
336 BlockType A21(lu,k+bs,k,trows,bs);
337 BlockType A22(lu,k+bs,k+bs,trows,tsize);
338
339 PivIndex nb_transpositions_in_panel;
340 // recursively call the blocked LU algorithm on [A11^T A21^T]^T
341 // with a very small blocking size:
342 Index ret = blocked_lu(trows+bs, bs, &lu.coeffRef(k,k), luStride,
343 row_transpositions+k, nb_transpositions_in_panel, 16);
344 if(ret>=0 && first_zero_pivot==-1)
345 first_zero_pivot = k+ret;
346
347 nb_transpositions += nb_transpositions_in_panel;
348 // update permutations and apply them to A_0
349 for(Index i=k; i<k+bs; ++i)
350 {
351 Index piv = (row_transpositions[i] += k);
352 A_0.row(i).swap(A_0.row(piv));
353 }
354
355 if(trows)
356 {
357 // apply permutations to A_2
358 for(Index i=k;i<k+bs; ++i)
359 A_2.row(i).swap(A_2.row(row_transpositions[i]));
360
361 // A12 = A11^-1 A12
362 A11.template triangularView<UnitLower>().solveInPlace(A12);
363
364 A22.noalias() -= A21 * A12;
365 }
366 }
367 return first_zero_pivot;
368 }
369 };
370
371 /** \internal performs the LU decomposition with partial pivoting in-place.
372 */
373 template<typename MatrixType, typename TranspositionType>
partial_lu_inplace(MatrixType & lu,TranspositionType & row_transpositions,typename TranspositionType::Index & nb_transpositions)374 void partial_lu_inplace(MatrixType& lu, TranspositionType& row_transpositions, typename TranspositionType::Index& nb_transpositions)
375 {
376 eigen_assert(lu.cols() == row_transpositions.size());
377 eigen_assert((&row_transpositions.coeffRef(1)-&row_transpositions.coeffRef(0)) == 1);
378
379 partial_lu_impl
380 <typename MatrixType::Scalar, MatrixType::Flags&RowMajorBit?RowMajor:ColMajor, typename TranspositionType::Index>
381 ::blocked_lu(lu.rows(), lu.cols(), &lu.coeffRef(0,0), lu.outerStride(), &row_transpositions.coeffRef(0), nb_transpositions);
382 }
383
384 } // end namespace internal
385
386 template<typename MatrixType>
compute(const MatrixType & matrix)387 PartialPivLU<MatrixType>& PartialPivLU<MatrixType>::compute(const MatrixType& matrix)
388 {
389 // the row permutation is stored as int indices, so just to be sure:
390 eigen_assert(matrix.rows()<NumTraits<int>::highest());
391
392 m_lu = matrix;
393
394 eigen_assert(matrix.rows() == matrix.cols() && "PartialPivLU is only for square (and moreover invertible) matrices");
395 const Index size = matrix.rows();
396
397 m_rowsTranspositions.resize(size);
398
399 typename TranspositionType::Index nb_transpositions;
400 internal::partial_lu_inplace(m_lu, m_rowsTranspositions, nb_transpositions);
401 m_det_p = (nb_transpositions%2) ? -1 : 1;
402
403 m_p = m_rowsTranspositions;
404
405 m_isInitialized = true;
406 return *this;
407 }
408
409 template<typename MatrixType>
determinant()410 typename internal::traits<MatrixType>::Scalar PartialPivLU<MatrixType>::determinant() const
411 {
412 eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
413 return Scalar(m_det_p) * m_lu.diagonal().prod();
414 }
415
416 /** \returns the matrix represented by the decomposition,
417 * i.e., it returns the product: P^{-1} L U.
418 * This function is provided for debug purpose. */
419 template<typename MatrixType>
reconstructedMatrix()420 MatrixType PartialPivLU<MatrixType>::reconstructedMatrix() const
421 {
422 eigen_assert(m_isInitialized && "LU is not initialized.");
423 // LU
424 MatrixType res = m_lu.template triangularView<UnitLower>().toDenseMatrix()
425 * m_lu.template triangularView<Upper>();
426
427 // P^{-1}(LU)
428 res = m_p.inverse() * res;
429
430 return res;
431 }
432
433 /***** Implementation of solve() *****************************************************/
434
435 namespace internal {
436
437 template<typename _MatrixType, typename Rhs>
438 struct solve_retval<PartialPivLU<_MatrixType>, Rhs>
439 : solve_retval_base<PartialPivLU<_MatrixType>, Rhs>
440 {
441 EIGEN_MAKE_SOLVE_HELPERS(PartialPivLU<_MatrixType>,Rhs)
442
443 template<typename Dest> void evalTo(Dest& dst) const
444 {
445 /* The decomposition PA = LU can be rewritten as A = P^{-1} L U.
446 * So we proceed as follows:
447 * Step 1: compute c = Pb.
448 * Step 2: replace c by the solution x to Lx = c.
449 * Step 3: replace c by the solution x to Ux = c.
450 */
451
452 eigen_assert(rhs().rows() == dec().matrixLU().rows());
453
454 // Step 1
455 dst = dec().permutationP() * rhs();
456
457 // Step 2
458 dec().matrixLU().template triangularView<UnitLower>().solveInPlace(dst);
459
460 // Step 3
461 dec().matrixLU().template triangularView<Upper>().solveInPlace(dst);
462 }
463 };
464
465 } // end namespace internal
466
467 /******** MatrixBase methods *******/
468
469 /** \lu_module
470 *
471 * \return the partial-pivoting LU decomposition of \c *this.
472 *
473 * \sa class PartialPivLU
474 */
475 template<typename Derived>
476 inline const PartialPivLU<typename MatrixBase<Derived>::PlainObject>
477 MatrixBase<Derived>::partialPivLu() const
478 {
479 return PartialPivLU<PlainObject>(eval());
480 }
481
482 #if EIGEN2_SUPPORT_STAGE > STAGE20_RESOLVE_API_CONFLICTS
483 /** \lu_module
484 *
485 * Synonym of partialPivLu().
486 *
487 * \return the partial-pivoting LU decomposition of \c *this.
488 *
489 * \sa class PartialPivLU
490 */
491 template<typename Derived>
492 inline const PartialPivLU<typename MatrixBase<Derived>::PlainObject>
493 MatrixBase<Derived>::lu() const
494 {
495 return PartialPivLU<PlainObject>(eval());
496 }
497 #endif
498
499 } // end namespace Eigen
500
501 #endif // EIGEN_PARTIALLU_H
502