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
43/** \ingroup LU_Module
44 *
45 * \class PartialPivLU
46 *
47 * \brief LU decomposition of a matrix with partial pivoting, and related features
48 *
49 * \tparam _MatrixType the type of the matrix of which we are computing the LU decomposition
50 *
51 * This class represents a LU decomposition of a \b square \b invertible matrix, with partial pivoting: the matrix A
52 * is decomposed as A = PLU where L is unit-lower-triangular, U is upper-triangular, and P
53 * is a permutation matrix.
54 *
55 * Typically, partial pivoting LU decomposition is only considered numerically stable for square invertible
56 * matrices. Thus LAPACK's dgesv and dgesvx require the matrix to be square and invertible. The present class
57 * does the same. It will assert that the matrix is square, but it won't (actually it can't) check that the
58 * matrix is invertible: it is your task to check that you only use this decomposition on invertible matrices.
59 *
60 * The guaranteed safe alternative, working for all matrices, is the full pivoting LU decomposition, provided
61 * by class FullPivLU.
62 *
63 * This is \b not a rank-revealing LU decomposition. Many features are intentionally absent from this class,
64 * such as rank computation. If you need these features, use class FullPivLU.
65 *
66 * This LU decomposition is suitable to invert invertible matrices. It is what MatrixBase::inverse() uses
67 * in the general case.
68 * On the other hand, it is \b not suitable to determine whether a given matrix is invertible.
69 *
70 * The data of the LU decomposition can be directly accessed through the methods matrixLU(), permutationP().
71 *
72 * This class supports the \link InplaceDecomposition inplace decomposition \endlink mechanism.
73 *
74 * \sa MatrixBase::partialPivLu(), MatrixBase::determinant(), MatrixBase::inverse(), MatrixBase::computeInverse(), class FullPivLU
75 */
76template<typename _MatrixType> class PartialPivLU
77 : public SolverBase<PartialPivLU<_MatrixType> >
78{
79 public:
80
81 typedef _MatrixType MatrixType;
82 typedef SolverBase<PartialPivLU> Base;
83 friend class SolverBase<PartialPivLU>;
84
85 EIGEN_GENERIC_PUBLIC_INTERFACE(PartialPivLU)
86 enum {
87 MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
88 MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
89 };
90 typedef PermutationMatrix<RowsAtCompileTime, MaxRowsAtCompileTime> PermutationType;
91 typedef Transpositions<RowsAtCompileTime, MaxRowsAtCompileTime> TranspositionType;
92 typedef typename MatrixType::PlainObject PlainObject;
93
94 /**
95 * \brief Default Constructor.
96 *
97 * The default constructor is useful in cases in which the user intends to
98 * perform decompositions via PartialPivLU::compute(const MatrixType&).
99 */
100 PartialPivLU();
101
102 /** \brief Default Constructor with memory preallocation
103 *
104 * Like the default constructor but with preallocation of the internal data
105 * according to the specified problem \a size.
106 * \sa PartialPivLU()
107 */
108 explicit PartialPivLU(Index size);
109
110 /** Constructor.
111 *
112 * \param matrix the matrix of which to compute the LU decomposition.
113 *
114 * \warning The matrix should have full rank (e.g. if it's square, it should be invertible).
115 * If you need to deal with non-full rank, use class FullPivLU instead.
116 */
117 template<typename InputType>
118 explicit PartialPivLU(const EigenBase<InputType>& matrix);
119
120 /** Constructor for \link InplaceDecomposition inplace decomposition \endlink
121 *
122 * \param matrix the matrix of which to compute the LU decomposition.
123 *
124 * \warning The matrix should have full rank (e.g. if it's square, it should be invertible).
125 * If you need to deal with non-full rank, use class FullPivLU instead.
126 */
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
137 /** \returns the LU decomposition matrix: the upper-triangular part is U, the
138 * unit-lower-triangular part is L (at least for square matrices; in the non-square
139 * case, special care is needed, see the documentation of class FullPivLU).
140 *
141 * \sa matrixL(), matrixU()
142 */
143 inline const MatrixType& matrixLU() const
144 {
145 eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
146 return m_lu;
147 }
148
149 /** \returns the permutation matrix P.
150 */
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
158 /** This method returns the solution x to the equation Ax=b, where A is the matrix of which
159 * *this is the LU decomposition.
160 *
161 * \param b the right-hand-side of the equation to solve. Can be a vector or a matrix,
162 * the only requirement in order for the equation to make sense is that
163 * b.rows()==A.rows(), where A is the matrix of which *this is the LU decomposition.
164 *
165 * \returns the solution.
166 *
167 * Example: \include PartialPivLU_solve.cpp
168 * Output: \verbinclude PartialPivLU_solve.out
169 *
170 * Since this PartialPivLU class assumes anyway that the matrix A is invertible, the solution
171 * theoretically exists and is unique regardless of b.
172 *
173 * \sa TriangularView::solve(), inverse(), computeInverse()
174 */
175 template<typename Rhs>
176 inline const Solve<PartialPivLU, Rhs>
177 solve(const MatrixBase<Rhs>& b) const;
178 #endif
179
180 /** \returns an estimate of the reciprocal condition number of the matrix of which \c *this is
181 the LU decomposition.
182 */
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
189 /** \returns the inverse of the matrix of which *this is the LU decomposition.
190 *
191 * \warning The matrix being decomposed here is assumed to be invertible. If you need to check for
192 * invertibility, use class FullPivLU instead.
193 *
194 * \sa MatrixBase::inverse(), LU::inverse()
195 */
196 inline const Inverse<PartialPivLU> inverse() const
197 {
198 eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
199 return Inverse<PartialPivLU>(*this);
200 }
201
202 /** \returns the determinant of the matrix of which
203 * *this is the LU decomposition. It has only linear complexity
204 * (that is, O(n) where n is the dimension of the square matrix)
205 * as the LU decomposition has already been computed.
206 *
207 * \note For fixed-size matrices of size up to 4, MatrixBase::determinant() offers
208 * optimized paths.
209 *
210 * \warning a determinant can be very big or small, so for matrices
211 * of large enough dimension, there is a risk of overflow/underflow.
212 *
213 * \sa MatrixBase::determinant()
214 */
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>
284PartialPivLU<MatrixType>::PartialPivLU()
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>
295PartialPivLU<MatrixType>::PartialPivLU(Index size)
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>
307PartialPivLU<MatrixType>::PartialPivLU(const EigenBase<InputType>& matrix)
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>
320PartialPivLU<MatrixType>::PartialPivLU(EigenBase<InputType>& matrix)
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)
327{
328 compute();
329}
330
331namespace internal {
332
333/** \internal This is the blocked version of fullpivlu_unblocked() */
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;
343 typedef Matrix<Scalar, ActualSizeAtCompileTime, ActualSizeAtCompileTime, StorageOrder> MatrixType;
344 typedef Ref<MatrixType> MatrixTypeRef;
345 typedef Ref<Matrix<Scalar, Dynamic, Dynamic, StorageOrder> > BlockType;
346 typedef typename MatrixType::RealScalar RealScalar;
347
348 /** \internal performs the LU decomposition in-place of the matrix \a lu
349 * using an unblocked algorithm.
350 *
351 * In addition, this function returns the row transpositions in the
352 * vector \a row_transpositions which must have a size equal to the number
353 * of columns of the matrix \a lu, and an integer \a nb_transpositions
354 * which returns the actual number of transpositions.
355 *
356 * \returns The index of the first pivot which is exactly zero if any, or a negative number otherwise.
357 */
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)(a: rows,b: 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>(idx: rows-k-1);
373 int rcols = internal::convert_index<int>(idx: 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
415 /** \internal performs the LU decomposition in-place of the matrix represented
416 * by the variables \a rows, \a cols, \a lu_data, and \a lu_stride using a
417 * recursive, blocked algorithm.
418 *
419 * In addition, this function returns the row transpositions in the
420 * vector \a row_transpositions which must have a size equal to the number
421 * of columns of the matrix \a lu, and an integer \a nb_transpositions
422 * which returns the actual number of transpositions.
423 *
424 * \returns The index of the first pivot which is exactly zero if any, or a negative number otherwise.
425 *
426 * \note This very low level interface using pointers, etc. is to:
427 * 1 - reduce the number of instantiations to the strict minimum
428 * 2 - avoid infinite recursion of the instantiations with Block<Block<Block<...> > >
429 */
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)(a: rows,b: 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)(a: (std::max)(a: blockSize,b: Index(8)), b: 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)(a: size-k,b: 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(rows: trows+bs, cols: bs, lu_data: &lu.coeffRef(k,k), luStride,
474 row_transpositions: row_transpositions+k, nb_transpositions&: nb_transpositions_in_panel, maxBlockSize: 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
502/** \internal performs the LU decomposition with partial pivoting in-place.
503 */
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
558/** \returns the matrix represented by the decomposition,
559 * i.e., it returns the product: P^{-1} L U.
560 * This function is provided for debug purpose. */
561template<typename MatrixType>
562MatrixType PartialPivLU<MatrixType>::reconstructedMatrix() const
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
594/** \lu_module
595 *
596 * \return the partial-pivoting LU decomposition of \c *this.
597 *
598 * \sa class PartialPivLU
599 */
600template<typename Derived>
601inline const PartialPivLU<typename MatrixBase<Derived>::PlainObject>
602MatrixBase<Derived>::partialPivLu() const
603{
604 return PartialPivLU<PlainObject>(eval());
605}
606
607/** \lu_module
608 *
609 * Synonym of partialPivLu().
610 *
611 * \return the partial-pivoting LU decomposition of \c *this.
612 *
613 * \sa class PartialPivLU
614 */
615template<typename Derived>
616inline const PartialPivLU<typename MatrixBase<Derived>::PlainObject>
617MatrixBase<Derived>::lu() const
618{
619 return PartialPivLU<PlainObject>(eval());
620}
621
622} // end namespace Eigen
623
624#endif // EIGEN_PARTIALLU_H
625

source code of qtmultimedia/src/3rdparty/eigen/Eigen/src/LU/PartialPivLU.h