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