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      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 (at) gmail.com>
      5 // Copyright (C) 2009 Gael Guennebaud <gael.guennebaud (at) 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       */
    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 
    181 template<typename MatrixType>
    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>
    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>
    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     */
    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   */
    373 template<typename MatrixType, typename TranspositionType>
    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>
    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>
    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>
    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