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Eigen  3.4.0
 
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PartialPivLU.h
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 int StorageIndex;
23 typedef traits<_MatrixType> BaseTraits;
24 enum {
25 Flags = BaseTraits::Flags & RowMajorBit,
26 CoeffReadCost = Dynamic
27 };
28};
29
30template<typename T,typename Derived>
31struct enable_if_ref;
32// {
33// typedef Derived type;
34// };
35
36template<typename T,typename Derived>
37struct enable_if_ref<Ref<T>,Derived> {
38 typedef Derived type;
39};
40
41} // end namespace internal
42
76template<typename _MatrixType> class PartialPivLU
77 : public SolverBase<PartialPivLU<_MatrixType> >
78{
79 public:
80
81 typedef _MatrixType MatrixType;
83 friend class SolverBase<PartialPivLU>;
84
85 EIGEN_GENERIC_PUBLIC_INTERFACE(PartialPivLU)
86 enum {
87 MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
88 MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
89 };
92 typedef typename MatrixType::PlainObject PlainObject;
93
100 PartialPivLU();
101
108 explicit PartialPivLU(Index size);
109
117 template<typename InputType>
118 explicit PartialPivLU(const EigenBase<InputType>& matrix);
119
127 template<typename InputType>
128 explicit PartialPivLU(EigenBase<InputType>& matrix);
129
130 template<typename InputType>
131 PartialPivLU& compute(const EigenBase<InputType>& matrix) {
132 m_lu = matrix.derived();
133 compute();
134 return *this;
135 }
136
143 inline const MatrixType& matrixLU() const
144 {
145 eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
146 return m_lu;
147 }
148
151 inline const PermutationType& permutationP() const
152 {
153 eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
154 return m_p;
155 }
156
157 #ifdef EIGEN_PARSED_BY_DOXYGEN
175 template<typename Rhs>
176 inline const Solve<PartialPivLU, Rhs>
177 solve(const MatrixBase<Rhs>& b) const;
178 #endif
179
183 inline RealScalar rcond() const
184 {
185 eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
186 return internal::rcond_estimate_helper(m_l1_norm, *this);
187 }
188
196 inline const Inverse<PartialPivLU> inverse() const
197 {
198 eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
199 return Inverse<PartialPivLU>(*this);
200 }
201
215 Scalar determinant() const;
216
217 MatrixType reconstructedMatrix() const;
218
219 EIGEN_CONSTEXPR inline Index rows() const EIGEN_NOEXCEPT { return m_lu.rows(); }
220 EIGEN_CONSTEXPR inline Index cols() const EIGEN_NOEXCEPT { return m_lu.cols(); }
221
222 #ifndef EIGEN_PARSED_BY_DOXYGEN
223 template<typename RhsType, typename DstType>
224 EIGEN_DEVICE_FUNC
225 void _solve_impl(const RhsType &rhs, DstType &dst) const {
226 /* The decomposition PA = LU can be rewritten as A = P^{-1} L U.
227 * So we proceed as follows:
228 * Step 1: compute c = Pb.
229 * Step 2: replace c by the solution x to Lx = c.
230 * Step 3: replace c by the solution x to Ux = c.
231 */
232
233 // Step 1
234 dst = permutationP() * rhs;
235
236 // Step 2
237 m_lu.template triangularView<UnitLower>().solveInPlace(dst);
238
239 // Step 3
240 m_lu.template triangularView<Upper>().solveInPlace(dst);
241 }
242
243 template<bool Conjugate, typename RhsType, typename DstType>
244 EIGEN_DEVICE_FUNC
245 void _solve_impl_transposed(const RhsType &rhs, DstType &dst) const {
246 /* The decomposition PA = LU can be rewritten as A^T = U^T L^T P.
247 * So we proceed as follows:
248 * Step 1: compute c as the solution to L^T c = b
249 * Step 2: replace c by the solution x to U^T x = c.
250 * Step 3: update c = P^-1 c.
251 */
252
253 eigen_assert(rhs.rows() == m_lu.cols());
254
255 // Step 1
256 dst = m_lu.template triangularView<Upper>().transpose()
257 .template conjugateIf<Conjugate>().solve(rhs);
258 // Step 2
259 m_lu.template triangularView<UnitLower>().transpose()
260 .template conjugateIf<Conjugate>().solveInPlace(dst);
261 // Step 3
262 dst = permutationP().transpose() * dst;
263 }
264 #endif
265
266 protected:
267
268 static void check_template_parameters()
269 {
270 EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
271 }
272
273 void compute();
274
275 MatrixType m_lu;
276 PermutationType m_p;
277 TranspositionType m_rowsTranspositions;
278 RealScalar m_l1_norm;
279 signed char m_det_p;
280 bool m_isInitialized;
281};
282
283template<typename MatrixType>
285 : m_lu(),
286 m_p(),
287 m_rowsTranspositions(),
288 m_l1_norm(0),
289 m_det_p(0),
290 m_isInitialized(false)
291{
292}
293
294template<typename MatrixType>
296 : m_lu(size, size),
297 m_p(size),
298 m_rowsTranspositions(size),
299 m_l1_norm(0),
300 m_det_p(0),
301 m_isInitialized(false)
302{
303}
304
305template<typename MatrixType>
306template<typename InputType>
308 : m_lu(matrix.rows(),matrix.cols()),
309 m_p(matrix.rows()),
310 m_rowsTranspositions(matrix.rows()),
311 m_l1_norm(0),
312 m_det_p(0),
313 m_isInitialized(false)
314{
315 compute(matrix.derived());
316}
317
318template<typename MatrixType>
319template<typename InputType>
321 : m_lu(matrix.derived()),
322 m_p(matrix.rows()),
323 m_rowsTranspositions(matrix.rows()),
324 m_l1_norm(0),
325 m_det_p(0),
326 m_isInitialized(false)
328 compute();
330
331namespace internal {
332
334template<typename Scalar, int StorageOrder, typename PivIndex, int SizeAtCompileTime=Dynamic>
335struct partial_lu_impl
336{
337 static const int UnBlockedBound = 16;
338 static const bool UnBlockedAtCompileTime = SizeAtCompileTime!=Dynamic && SizeAtCompileTime<=UnBlockedBound;
339 static const int ActualSizeAtCompileTime = UnBlockedAtCompileTime ? SizeAtCompileTime : Dynamic;
340 // Remaining rows and columns at compile-time:
341 static const int RRows = SizeAtCompileTime==2 ? 1 : Dynamic;
342 static const int RCols = SizeAtCompileTime==2 ? 1 : Dynamic;
344 typedef Ref<MatrixType> MatrixTypeRef;
346 typedef typename MatrixType::RealScalar RealScalar;
347
358 static Index unblocked_lu(MatrixTypeRef& lu, PivIndex* row_transpositions, PivIndex& nb_transpositions)
359 {
360 typedef scalar_score_coeff_op<Scalar> Scoring;
361 typedef typename Scoring::result_type Score;
362 const Index rows = lu.rows();
363 const Index cols = lu.cols();
364 const Index size = (std::min)(rows,cols);
365 // For small compile-time matrices it is worth processing the last row separately:
366 // speedup: +100% for 2x2, +10% for others.
367 const Index endk = UnBlockedAtCompileTime ? size-1 : size;
368 nb_transpositions = 0;
369 Index first_zero_pivot = -1;
370 for(Index k = 0; k < endk; ++k)
371 {
372 int rrows = internal::convert_index<int>(rows-k-1);
373 int rcols = internal::convert_index<int>(cols-k-1);
374
375 Index row_of_biggest_in_col;
376 Score biggest_in_corner
377 = lu.col(k).tail(rows-k).unaryExpr(Scoring()).maxCoeff(&row_of_biggest_in_col);
378 row_of_biggest_in_col += k;
379
380 row_transpositions[k] = PivIndex(row_of_biggest_in_col);
381
382 if(biggest_in_corner != Score(0))
383 {
384 if(k != row_of_biggest_in_col)
385 {
386 lu.row(k).swap(lu.row(row_of_biggest_in_col));
387 ++nb_transpositions;
388 }
389
390 lu.col(k).tail(fix<RRows>(rrows)) /= lu.coeff(k,k);
391 }
392 else if(first_zero_pivot==-1)
393 {
394 // the pivot is exactly zero, we record the index of the first pivot which is exactly 0,
395 // and continue the factorization such we still have A = PLU
396 first_zero_pivot = k;
397 }
398
399 if(k<rows-1)
400 lu.bottomRightCorner(fix<RRows>(rrows),fix<RCols>(rcols)).noalias() -= lu.col(k).tail(fix<RRows>(rrows)) * lu.row(k).tail(fix<RCols>(rcols));
401 }
402
403 // special handling of the last entry
404 if(UnBlockedAtCompileTime)
405 {
406 Index k = endk;
407 row_transpositions[k] = PivIndex(k);
408 if (Scoring()(lu(k, k)) == Score(0) && first_zero_pivot == -1)
409 first_zero_pivot = k;
410 }
411
412 return first_zero_pivot;
413 }
414
430 static Index blocked_lu(Index rows, Index cols, Scalar* lu_data, Index luStride, PivIndex* row_transpositions, PivIndex& nb_transpositions, Index maxBlockSize=256)
431 {
432 MatrixTypeRef lu = MatrixType::Map(lu_data,rows, cols, OuterStride<>(luStride));
433
434 const Index size = (std::min)(rows,cols);
435
436 // if the matrix is too small, no blocking:
437 if(UnBlockedAtCompileTime || size<=UnBlockedBound)
438 {
439 return unblocked_lu(lu, row_transpositions, nb_transpositions);
440 }
441
442 // automatically adjust the number of subdivisions to the size
443 // of the matrix so that there is enough sub blocks:
444 Index blockSize;
445 {
446 blockSize = size/8;
447 blockSize = (blockSize/16)*16;
448 blockSize = (std::min)((std::max)(blockSize,Index(8)), maxBlockSize);
449 }
450
451 nb_transpositions = 0;
452 Index first_zero_pivot = -1;
453 for(Index k = 0; k < size; k+=blockSize)
454 {
455 Index bs = (std::min)(size-k,blockSize); // actual size of the block
456 Index trows = rows - k - bs; // trailing rows
457 Index tsize = size - k - bs; // trailing size
458
459 // partition the matrix:
460 // A00 | A01 | A02
461 // lu = A_0 | A_1 | A_2 = A10 | A11 | A12
462 // A20 | A21 | A22
463 BlockType A_0 = lu.block(0,0,rows,k);
464 BlockType A_2 = lu.block(0,k+bs,rows,tsize);
465 BlockType A11 = lu.block(k,k,bs,bs);
466 BlockType A12 = lu.block(k,k+bs,bs,tsize);
467 BlockType A21 = lu.block(k+bs,k,trows,bs);
468 BlockType A22 = lu.block(k+bs,k+bs,trows,tsize);
469
470 PivIndex nb_transpositions_in_panel;
471 // recursively call the blocked LU algorithm on [A11^T A21^T]^T
472 // with a very small blocking size:
473 Index ret = blocked_lu(trows+bs, bs, &lu.coeffRef(k,k), luStride,
474 row_transpositions+k, nb_transpositions_in_panel, 16);
475 if(ret>=0 && first_zero_pivot==-1)
476 first_zero_pivot = k+ret;
477
478 nb_transpositions += nb_transpositions_in_panel;
479 // update permutations and apply them to A_0
480 for(Index i=k; i<k+bs; ++i)
481 {
482 Index piv = (row_transpositions[i] += internal::convert_index<PivIndex>(k));
483 A_0.row(i).swap(A_0.row(piv));
484 }
485
486 if(trows)
487 {
488 // apply permutations to A_2
489 for(Index i=k;i<k+bs; ++i)
490 A_2.row(i).swap(A_2.row(row_transpositions[i]));
491
492 // A12 = A11^-1 A12
493 A11.template triangularView<UnitLower>().solveInPlace(A12);
494
495 A22.noalias() -= A21 * A12;
496 }
497 }
498 return first_zero_pivot;
499 }
500};
501
504template<typename MatrixType, typename TranspositionType>
505void partial_lu_inplace(MatrixType& lu, TranspositionType& row_transpositions, typename TranspositionType::StorageIndex& nb_transpositions)
506{
507 // Special-case of zero matrix.
508 if (lu.rows() == 0 || lu.cols() == 0) {
509 nb_transpositions = 0;
510 return;
511 }
512 eigen_assert(lu.cols() == row_transpositions.size());
513 eigen_assert(row_transpositions.size() < 2 || (&row_transpositions.coeffRef(1)-&row_transpositions.coeffRef(0)) == 1);
514
515 partial_lu_impl
516 < typename MatrixType::Scalar, MatrixType::Flags&RowMajorBit?RowMajor:ColMajor,
517 typename TranspositionType::StorageIndex,
518 EIGEN_SIZE_MIN_PREFER_FIXED(MatrixType::RowsAtCompileTime,MatrixType::ColsAtCompileTime)>
519 ::blocked_lu(lu.rows(), lu.cols(), &lu.coeffRef(0,0), lu.outerStride(), &row_transpositions.coeffRef(0), nb_transpositions);
520}
521
522} // end namespace internal
523
524template<typename MatrixType>
525void PartialPivLU<MatrixType>::compute()
526{
527 check_template_parameters();
528
529 // the row permutation is stored as int indices, so just to be sure:
530 eigen_assert(m_lu.rows()<NumTraits<int>::highest());
531
532 if(m_lu.cols()>0)
533 m_l1_norm = m_lu.cwiseAbs().colwise().sum().maxCoeff();
534 else
535 m_l1_norm = RealScalar(0);
536
537 eigen_assert(m_lu.rows() == m_lu.cols() && "PartialPivLU is only for square (and moreover invertible) matrices");
538 const Index size = m_lu.rows();
539
540 m_rowsTranspositions.resize(size);
541
542 typename TranspositionType::StorageIndex nb_transpositions;
543 internal::partial_lu_inplace(m_lu, m_rowsTranspositions, nb_transpositions);
544 m_det_p = (nb_transpositions%2) ? -1 : 1;
545
546 m_p = m_rowsTranspositions;
547
548 m_isInitialized = true;
549}
550
551template<typename MatrixType>
552typename PartialPivLU<MatrixType>::Scalar PartialPivLU<MatrixType>::determinant() const
553{
554 eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
555 return Scalar(m_det_p) * m_lu.diagonal().prod();
556}
557
561template<typename MatrixType>
563{
564 eigen_assert(m_isInitialized && "LU is not initialized.");
565 // LU
566 MatrixType res = m_lu.template triangularView<UnitLower>().toDenseMatrix()
567 * m_lu.template triangularView<Upper>();
568
569 // P^{-1}(LU)
570 res = m_p.inverse() * res;
571
572 return res;
573}
574
575/***** Implementation details *****************************************************/
576
577namespace internal {
578
579/***** Implementation of inverse() *****************************************************/
580template<typename DstXprType, typename MatrixType>
581struct Assignment<DstXprType, Inverse<PartialPivLU<MatrixType> >, internal::assign_op<typename DstXprType::Scalar,typename PartialPivLU<MatrixType>::Scalar>, Dense2Dense>
582{
583 typedef PartialPivLU<MatrixType> LuType;
584 typedef Inverse<LuType> SrcXprType;
585 static void run(DstXprType &dst, const SrcXprType &src, const internal::assign_op<typename DstXprType::Scalar,typename LuType::Scalar> &)
586 {
587 dst = src.nestedExpression().solve(MatrixType::Identity(src.rows(), src.cols()));
588 }
589};
590} // end namespace internal
591
592/******** MatrixBase methods *******/
593
600template<typename Derived>
601inline const PartialPivLU<typename MatrixBase<Derived>::PlainObject>
603{
604 return PartialPivLU<PlainObject>(eval());
605}
606
615template<typename Derived>
618{
619 return PartialPivLU<PlainObject>(eval());
620}
621
622} // end namespace Eigen
623
624#endif // EIGEN_PARTIALLU_H
Expression of the inverse of another expression.
Definition: Inverse.h:44
Base class for all dense matrices, vectors, and expressions.
Definition: MatrixBase.h:50
The matrix class, also used for vectors and row-vectors.
Definition: Matrix.h:180
Convenience specialization of Stride to specify only an outer stride See class Map for some examples.
Definition: Stride.h:107
LU decomposition of a matrix with partial pivoting, and related features.
Definition: PartialPivLU.h:78
const Inverse< PartialPivLU > inverse() const
Definition: PartialPivLU.h:196
const Solve< PartialPivLU, Rhs > solve(const MatrixBase< Rhs > &b) const
const PermutationType & permutationP() const
Definition: PartialPivLU.h:151
RealScalar rcond() const
Definition: PartialPivLU.h:183
Scalar determinant() const
Definition: PartialPivLU.h:552
PartialPivLU()
Default Constructor.
Definition: PartialPivLU.h:284
const MatrixType & matrixLU() const
Definition: PartialPivLU.h:143
MatrixType reconstructedMatrix() const
Definition: PartialPivLU.h:562
InverseReturnType transpose() const
Definition: PermutationMatrix.h:191
Permutation matrix.
Definition: PermutationMatrix.h:298
static ConstMapType Map(const Scalar *data)
Definition: PlainObjectBase.h:644
A matrix or vector expression mapping an existing expression.
Definition: Ref.h:283
Pseudo expression representing a solving operation.
Definition: Solve.h:63
A base class for matrix decomposition and solvers.
Definition: SolverBase.h:69
Represents a sequence of transpositions (row/column interchange)
Definition: Transpositions.h:156
@ ColMajor
Definition: Constants.h:319
@ RowMajor
Definition: Constants.h:321
const unsigned int RowMajorBit
Definition: Constants.h:66
Namespace containing all symbols from the Eigen library.
Definition: Core:141
EIGEN_DEFAULT_DENSE_INDEX_TYPE Index
The Index type as used for the API.
Definition: Meta.h:74
const int Dynamic
Definition: Constants.h:22
Definition: EigenBase.h:30
Eigen::Index Index
The interface type of indices.
Definition: EigenBase.h:39
Derived & derived()
Definition: EigenBase.h:46
EIGEN_CONSTEXPR Index size() const EIGEN_NOEXCEPT
Definition: EigenBase.h:67