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
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  */
47template<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      */
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      */
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>
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      */
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
170    inline Index rows() const { return m_lu.rows(); }
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
181template<typename MatrixType>
182PartialPivLU<MatrixType>::PartialPivLU()
183  : m_lu(),
184    m_p(),
185    m_rowsTranspositions(),
186    m_det_p(0),
187    m_isInitialized(false)
188{
189}
190
191template<typename MatrixType>
192PartialPivLU<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
201template<typename MatrixType>
202PartialPivLU<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
212namespace internal {
213
214/** \internal This is the blocked version of fullpivlu_unblocked() */
215template<typename Scalar, int StorageOrder, typename PivIndex>
216struct 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    */
239  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  */
373template<typename MatrixType, typename TranspositionType>
374void 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
386template<typename MatrixType>
387PartialPivLU<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
409template<typename MatrixType>
410typename 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. */
419template<typename MatrixType>
420MatrixType 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
435namespace internal {
436
437template<typename _MatrixType, typename Rhs>
438struct 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  */
475template<typename Derived>
476inline const PartialPivLU<typename MatrixBase<Derived>::PlainObject>
477MatrixBase<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  */
491template<typename Derived>
492inline const PartialPivLU<typename MatrixBase<Derived>::PlainObject>
493MatrixBase<Derived>::lu() const
494{
495  return PartialPivLU<PlainObject>(eval());
496}
497#endif
498
499} // end namespace Eigen
500
501#endif // EIGEN_PARTIALLU_H
502