Eigen/src/LU/PartialLU.h - mirror - Git at Google

 // This file is part of Eigen, a lightweight C++ template library
 // for linear algebra.
 //
 // Copyright (C) 2006-2009 Benoit Jacob <jacob.benoit.1@gmail.com>
 //
 // Eigen is free software; you can redistribute it and/or
 // modify it under the terms of the GNU Lesser General Public
 // License as published by the Free Software Foundation; either
 // version 3 of the License, or (at your option) any later version.
 //
 // Alternatively, you can redistribute it and/or
 // modify it under the terms of the GNU General Public License as
 // published by the Free Software Foundation; either version 2 of
 // the License, or (at your option) any later version.
 //
 // Eigen is distributed in the hope that it will be useful, but WITHOUT ANY
 // WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
 // FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the
 // GNU General Public License for more details.
 //
 // You should have received a copy of the GNU Lesser General Public
 // License and a copy of the GNU General Public License along with
 // Eigen. If not, see <http://www.gnu.org/licenses/>.

 #ifndef EIGEN_PARTIALLU_H
 #define EIGEN_PARTIALLU_H

 /** \ingroup LU_Module
   *
   * \class PartialLU
   *
   * \brief LU decomposition of a matrix with partial pivoting, and related features
   *
   * \param MatrixType the type of the matrix of which we are computing the LU decomposition
   *
   * This class represents a LU decomposition of a \b square \b invertible matrix, with partial pivoting: the matrix A
   * is decomposed as A = PLU where L is unit-lower-triangular, U is upper-triangular, and P
   * is a permutation matrix.
   *
   * Typically, partial pivoting LU decomposition is only considered numerically stable for square invertible matrices.
   * So in this class, we plainly require that and take advantage of that to do some simplifications and optimizations.
   * This class will assert that the matrix is square, but it won't (actually it can't) check that the matrix is invertible:
   * it is your task to check that you only use this decomposition on invertible matrices.
   *
   * The guaranteed safe alternative, working for all matrices, is the full pivoting LU decomposition, provided by class LU.
   *
   * This is \b not a rank-revealing LU decomposition. Many features are intentionally absent from this class,
   * such as rank computation. If you need these features, use class LU.
   *
   * This LU decomposition is suitable to invert invertible matrices. It is what MatrixBase::inverse() uses. On the other hand,
   * it is \b not suitable to determine whether a given matrix is invertible.
   *
   * The data of the LU decomposition can be directly accessed through the methods matrixLU(), permutationP().
   *
   * \sa MatrixBase::partialLu(), MatrixBase::determinant(), MatrixBase::inverse(), MatrixBase::computeInverse(), class LU
   */
 template<typename MatrixType> class PartialLU
 {
   public:

     typedef typename MatrixType::Scalar Scalar;
     typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
     typedef Matrix<int, 1, MatrixType::ColsAtCompileTime> IntRowVectorType;
     typedef Matrix<int, MatrixType::RowsAtCompileTime, 1> IntColVectorType;
     typedef Matrix<Scalar, 1, MatrixType::ColsAtCompileTime> RowVectorType;
     typedef Matrix<Scalar, MatrixType::RowsAtCompileTime, 1> ColVectorType;

     enum { MaxSmallDimAtCompileTime = EIGEN_ENUM_MIN(
              MatrixType::MaxColsAtCompileTime,
              MatrixType::MaxRowsAtCompileTime)
     };

     /**
     * \brief Default Constructor.
     *
     * The default constructor is useful in cases in which the user intends to
     * perform decompositions via PartialLU::compute(const MatrixType&).
     */
     PartialLU();

     /** Constructor.
       *
       * \param matrix the matrix of which to compute the LU decomposition.
       *
       * \warning The matrix should have full rank (e.g. if it's square, it should be invertible).
       * If you need to deal with non-full rank, use class LU instead.
       */
     PartialLU(const MatrixType& matrix);

     PartialLU& compute(const MatrixType& matrix);

     /** \returns the LU decomposition matrix: the upper-triangular part is U, the
       * unit-lower-triangular part is L (at least for square matrices; in the non-square
       * case, special care is needed, see the documentation of class LU).
       *
       * \sa matrixL(), matrixU()
       */
     inline const MatrixType& matrixLU() const
     {
       ei_assert(m_isInitialized && "PartialLU is not initialized.");
       return m_lu;
     }

     /** \returns a vector of integers, whose size is the number of rows of the matrix being decomposed,
       * representing the P permutation i.e. the permutation of the rows. For its precise meaning,
       * see the examples given in the documentation of class LU.
       */
     inline const IntColVectorType& permutationP() const
     {
       ei_assert(m_isInitialized && "PartialLU is not initialized.");
       return m_p;
     }

     /** This method finds the solution x to the equation Ax=b, where A is the matrix of which
       * *this is the LU decomposition. Since if this partial pivoting decomposition the matrix is assumed
       * to have full rank, such a solution is assumed to exist and to be unique.
       *
       * \warning Again, if your matrix may not have full rank, use class LU instead. See LU::solve().
       *
       * \param b the right-hand-side of the equation to solve. Can be a vector or a matrix,
       *          the only requirement in order for the equation to make sense is that
       *          b.rows()==A.rows(), where A is the matrix of which *this is the LU decomposition.
       * \param result a pointer to the vector or matrix in which to store the solution, if any exists.
       *          Resized if necessary, so that result->rows()==A.cols() and result->cols()==b.cols().
       *          If no solution exists, *result is left with undefined coefficients.
       *
       * Example: \include PartialLU_solve.cpp
       * Output: \verbinclude PartialLU_solve.out
       *
       * \sa TriangularView::solve(), inverse(), computeInverse()
       */
     template<typename OtherDerived, typename ResultType>
     void solve(const MatrixBase<OtherDerived>& b, ResultType *result) const;

     /** \returns the determinant of the matrix of which
       * *this is the LU decomposition. It has only linear complexity
       * (that is, O(n) where n is the dimension of the square matrix)
       * as the LU decomposition has already been computed.
       *
       * \note For fixed-size matrices of size up to 4, MatrixBase::determinant() offers
       *       optimized paths.
       *
       * \warning a determinant can be very big or small, so for matrices
       * of large enough dimension, there is a risk of overflow/underflow.
       *
       * \sa MatrixBase::determinant()
       */
     typename ei_traits<MatrixType>::Scalar determinant() const;

     /** Computes the inverse of the matrix of which *this is the LU decomposition.
       *
       * \param result a pointer to the matrix into which to store the inverse. Resized if needed.
       *
       * \warning The matrix being decomposed here is assumed to be invertible. If you need to check for
       *          invertibility, use class LU instead.
       *
       * \sa MatrixBase::computeInverse(), inverse()
       */
     inline void computeInverse(MatrixType *result) const
     {
       solve(MatrixType::Identity(m_lu.rows(), m_lu.cols()), result);
     }

     /** \returns the inverse of the matrix of which *this is the LU decomposition.
       *
       * \warning The matrix being decomposed here is assumed to be invertible. If you need to check for
       *          invertibility, use class LU instead.
       *
       * \sa computeInverse(), MatrixBase::inverse()
       */
     inline MatrixType inverse() const
     {
       MatrixType result;
       computeInverse(&result);
       return result;
     }

   protected:
     MatrixType m_lu;
     IntColVectorType m_p;
     int m_det_p;
     bool m_isInitialized;
 };

 template<typename MatrixType>
 PartialLU<MatrixType>::PartialLU()
   : m_lu(),
     m_p(),
     m_det_p(0),
     m_isInitialized(false)
 {
 }

 template<typename MatrixType>
 PartialLU<MatrixType>::PartialLU(const MatrixType& matrix)
   : m_lu(),
     m_p(),
     m_det_p(0),
     m_isInitialized(false)
 {
   compute(matrix);
 }


 /** This is the blocked version of ei_lu_unblocked() */
 template<typename Scalar, int StorageOrder>
 struct ei_partial_lu_impl
 {
   // FIXME add a stride to Map, so that the following mapping becomes easier,
   // another option would be to create an expression being able to automatically
   // warp any Map, Matrix, and Block expressions as a unique type, but since that's exactly
   // a Map + stride, why not adding a stride to Map, and convenient ctors from a Matrix,
   // and Block.
   typedef Map<Matrix<Scalar, Dynamic, Dynamic, StorageOrder> > MapLU;
   typedef Block<MapLU, Dynamic, Dynamic> MatrixType;
   typedef Block<MatrixType,Dynamic,Dynamic> BlockType;

   /** \internal performs the LU decomposition in-place of the matrix \a lu
     * using an unblocked algorithm.
     *
     * In addition, this function returns the row transpositions in the
     * vector \a row_transpositions which must have a size equal to the number
     * of columns of the matrix \a lu, and an integer \a nb_transpositions
     * which returns the actual number of transpositions.
     */
   static void unblocked_lu(MatrixType& lu, int* row_transpositions, int& nb_transpositions)
   {
     const int rows = lu.rows();
     const int size = std::min(lu.rows(),lu.cols());
     nb_transpositions = 0;
     for(int k = 0; k < size; ++k)
     {
       int row_of_biggest_in_col;
       lu.block(k,k,rows-k,1).cwise().abs().maxCoeff(&row_of_biggest_in_col);
       row_of_biggest_in_col += k;

       row_transpositions[k] = row_of_biggest_in_col;

       if(k != row_of_biggest_in_col)
       {
         lu.row(k).swap(lu.row(row_of_biggest_in_col));
         ++nb_transpositions;
       }

       if(k<rows-1)
       {
         int rrows = rows-k-1;
         int rsize = size-k-1;
         lu.col(k).end(rrows) /= lu.coeff(k,k);
         lu.corner(BottomRight,rrows,rsize).noalias() -= lu.col(k).end(rrows) * lu.row(k).end(rsize);
       }
     }
   }

   /** \internal performs the LU decomposition in-place of the matrix represented
     * by the variables \a rows, \a cols, \a lu_data, and \a lu_stride using a
     * recursive, blocked algorithm.
     *
     * In addition, this function returns the row transpositions in the
     * vector \a row_transpositions which must have a size equal to the number
     * of columns of the matrix \a lu, and an integer \a nb_transpositions
     * which returns the actual number of transpositions.
     *
     * \note This very low level interface using pointers, etc. is to:
     *   1 - reduce the number of instanciations to the strict minimum
     *   2 - avoid infinite recursion of the instanciations with Block<Block<Block<...> > >
     */
   static void blocked_lu(int rows, int cols, Scalar* lu_data, int luStride, int* row_transpositions, int& nb_transpositions, int maxBlockSize=256)
   {
     MapLU lu1(lu_data,StorageOrder==RowMajor?rows:luStride,StorageOrder==RowMajor?luStride:cols);
     MatrixType lu(lu1,0,0,rows,cols);

     const int size = std::min(rows,cols);

     // if the matrix is too small, no blocking:
     if(size<=16)
     {
       unblocked_lu(lu, row_transpositions, nb_transpositions);
       return;
     }

     // automatically adjust the number of subdivisions to the size
     // of the matrix so that there is enough sub blocks:
     int blockSize;
     {
       blockSize = size/8;
       blockSize = (blockSize/16)*16;
       blockSize = std::min(std::max(blockSize,8), maxBlockSize);
     }

     nb_transpositions = 0;
     for(int k = 0; k < size; k+=blockSize)
     {
       int bs = std::min(size-k,blockSize); // actual size of the block
       int trows = rows - k - bs; // trailing rows
       int tsize = size - k - bs; // trailing size

       // partition the matrix:
       //        A00 | A01 | A02
       // lu  =  A10 | A11 | A12
       //        A20 | A21 | A22
       BlockType A_0(lu,0,0,rows,k);
       BlockType A_2(lu,0,k+bs,rows,tsize);
       BlockType A11(lu,k,k,bs,bs);
       BlockType A12(lu,k,k+bs,bs,tsize);
       BlockType A21(lu,k+bs,k,trows,bs);
       BlockType A22(lu,k+bs,k+bs,trows,tsize);

       int nb_transpositions_in_panel;
       // recursively calls the blocked LU algorithm with a very small
       // blocking size:
       blocked_lu(trows+bs, bs, &lu.coeffRef(k,k), luStride,
                  row_transpositions+k, nb_transpositions_in_panel, 16);
       nb_transpositions += nb_transpositions_in_panel;

       // update permutations and apply them to A10
       for(int i=k;i<k+bs; ++i)
       {
         int piv = (row_transpositions[i] += k);
         A_0.row(i).swap(A_0.row(piv));
       }

       if(trows)
       {
         // apply permutations to A_2
         for(int i=k;i<k+bs; ++i)
           A_2.row(i).swap(A_2.row(row_transpositions[i]));

         // A12 = A11^-1 A12
         A11.template triangularView<UnitLowerTriangular>().solveInPlace(A12);

         A22 -= A21 * A12;
       }
     }
   }
 };

 /** \internal performs the LU decomposition with partial pivoting in-place.
   */
 template<typename MatrixType, typename IntVector>
 void ei_partial_lu_inplace(MatrixType& lu, IntVector& row_transpositions, int& nb_transpositions)
 {
   ei_assert(lu.cols() == row_transpositions.size());
   ei_assert((&row_transpositions.coeffRef(1)-&row_transpositions.coeffRef(0)) == 1);

   ei_partial_lu_impl
     <typename MatrixType::Scalar, MatrixType::Flags&RowMajorBit?RowMajor:ColMajor>
     ::blocked_lu(lu.rows(), lu.cols(), &lu.coeffRef(0,0), lu.stride(), &row_transpositions.coeffRef(0), nb_transpositions);
 }

 template<typename MatrixType>
 PartialLU<MatrixType>& PartialLU<MatrixType>::compute(const MatrixType& matrix)
 {
   m_lu = matrix;
   m_p.resize(matrix.rows());

   ei_assert(matrix.rows() == matrix.cols() && "PartialLU is only for square (and moreover invertible) matrices");
   const int size = matrix.rows();

   IntColVectorType rows_transpositions(size);

   int nb_transpositions;
   ei_partial_lu_inplace(m_lu, rows_transpositions, nb_transpositions);
   m_det_p = (nb_transpositions%2) ? -1 : 1;

   for(int k = 0; k < size; ++k) m_p.coeffRef(k) = k;
   for(int k = size-1; k >= 0; --k)
     std::swap(m_p.coeffRef(k), m_p.coeffRef(rows_transpositions.coeff(k)));

   m_isInitialized = true;
   return *this;
 }

 template<typename MatrixType>
 typename ei_traits<MatrixType>::Scalar PartialLU<MatrixType>::determinant() const
 {
   ei_assert(m_isInitialized && "PartialLU is not initialized.");
   return Scalar(m_det_p) * m_lu.diagonal().prod();
 }

 template<typename MatrixType>
 template<typename OtherDerived, typename ResultType>
 void PartialLU<MatrixType>::solve(
   const MatrixBase<OtherDerived>& b,
   ResultType *result
 ) const
 {
   ei_assert(m_isInitialized && "PartialLU is not initialized.");

   /* The decomposition PA = LU can be rewritten as A = P^{-1} L U.
    * So we proceed as follows:
    * Step 1: compute c = Pb.
    * Step 2: replace c by the solution x to Lx = c.
    * Step 3: replace c by the solution x to Ux = c.
    */

   const int size = m_lu.rows();
   ei_assert(b.rows() == size);

   result->resize(size, b.cols());

   // Step 1
   for(int i = 0; i < size; ++i) result->row(m_p.coeff(i)) = b.row(i);

   // Step 2
   m_lu.template triangularView<UnitLowerTriangular>().solveInPlace(*result);

   // Step 3
   m_lu.template triangularView<UpperTriangular>().solveInPlace(*result);
 }

 /** \lu_module
   *
   * \return the LU decomposition of \c *this.
   *
   * \sa class LU
   */
 template<typename Derived>
 inline const PartialLU<typename MatrixBase<Derived>::PlainMatrixType>
 MatrixBase<Derived>::partialLu() const
 {
   return PartialLU<PlainMatrixType>(eval());
 }

 #endif // EIGEN_PARTIALLU_H
	// This file is part of Eigen, a lightweight C++ template library
	// for linear algebra.
	//
	// Copyright (C) 2006-2009 Benoit Jacob <jacob.benoit.1@gmail.com>
	//
	// Eigen is free software; you can redistribute it and/or
	// modify it under the terms of the GNU Lesser General Public
	// License as published by the Free Software Foundation; either
	// version 3 of the License, or (at your option) any later version.
	//
	// Alternatively, you can redistribute it and/or
	// modify it under the terms of the GNU General Public License as
	// published by the Free Software Foundation; either version 2 of
	// the License, or (at your option) any later version.
	//
	// Eigen is distributed in the hope that it will be useful, but WITHOUT ANY
	// WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
	// FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the
	// GNU General Public License for more details.
	//
	// You should have received a copy of the GNU Lesser General Public
	// License and a copy of the GNU General Public License along with
	// Eigen. If not, see <http://www.gnu.org/licenses/>.

	#ifndef EIGEN_PARTIALLU_H
	#define EIGEN_PARTIALLU_H

	/** \ingroup LU_Module
	*
	* \class PartialLU
	*
	* \brief LU decomposition of a matrix with partial pivoting, and related features
	*
	* \param MatrixType the type of the matrix of which we are computing the LU decomposition
	*
	* This class represents a LU decomposition of a \b square \b invertible matrix, with partial pivoting: the matrix A
	* is decomposed as A = PLU where L is unit-lower-triangular, U is upper-triangular, and P
	* is a permutation matrix.
	*
	* Typically, partial pivoting LU decomposition is only considered numerically stable for square invertible matrices.
	* So in this class, we plainly require that and take advantage of that to do some simplifications and optimizations.
	* This class will assert that the matrix is square, but it won't (actually it can't) check that the matrix is invertible:
	* it is your task to check that you only use this decomposition on invertible matrices.
	*
	* The guaranteed safe alternative, working for all matrices, is the full pivoting LU decomposition, provided by class LU.
	*
	* This is \b not a rank-revealing LU decomposition. Many features are intentionally absent from this class,
	* such as rank computation. If you need these features, use class LU.
	*
	* This LU decomposition is suitable to invert invertible matrices. It is what MatrixBase::inverse() uses. On the other hand,
	* it is \b not suitable to determine whether a given matrix is invertible.
	*
	* The data of the LU decomposition can be directly accessed through the methods matrixLU(), permutationP().
	*
	* \sa MatrixBase::partialLu(), MatrixBase::determinant(), MatrixBase::inverse(), MatrixBase::computeInverse(), class LU
	*/
	template<typename MatrixType> class PartialLU
	{
	public:

	typedef typename MatrixType::Scalar Scalar;
	typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
	typedef Matrix<int, 1, MatrixType::ColsAtCompileTime> IntRowVectorType;
	typedef Matrix<int, MatrixType::RowsAtCompileTime, 1> IntColVectorType;
	typedef Matrix<Scalar, 1, MatrixType::ColsAtCompileTime> RowVectorType;
	typedef Matrix<Scalar, MatrixType::RowsAtCompileTime, 1> ColVectorType;

	enum { MaxSmallDimAtCompileTime = EIGEN_ENUM_MIN(
	MatrixType::MaxColsAtCompileTime,
	MatrixType::MaxRowsAtCompileTime)
	};

	/**
	* \brief Default Constructor.
	*
	* The default constructor is useful in cases in which the user intends to
	* perform decompositions via PartialLU::compute(const MatrixType&).
	*/
	PartialLU();

	/** Constructor.
	*
	* \param matrix the matrix of which to compute the LU decomposition.
	*
	* \warning The matrix should have full rank (e.g. if it's square, it should be invertible).
	* If you need to deal with non-full rank, use class LU instead.
	*/
	PartialLU(const MatrixType& matrix);

	PartialLU& compute(const MatrixType& matrix);

	/** \returns the LU decomposition matrix: the upper-triangular part is U, the
	* unit-lower-triangular part is L (at least for square matrices; in the non-square
	* case, special care is needed, see the documentation of class LU).
	*
	* \sa matrixL(), matrixU()
	*/
	inline const MatrixType& matrixLU() const
	{
	ei_assert(m_isInitialized && "PartialLU is not initialized.");
	return m_lu;
	}

	/** \returns a vector of integers, whose size is the number of rows of the matrix being decomposed,
	* representing the P permutation i.e. the permutation of the rows. For its precise meaning,
	* see the examples given in the documentation of class LU.
	*/
	inline const IntColVectorType& permutationP() const
	{
	ei_assert(m_isInitialized && "PartialLU is not initialized.");
	return m_p;
	}

	/** This method finds the solution x to the equation Ax=b, where A is the matrix of which
	* *this is the LU decomposition. Since if this partial pivoting decomposition the matrix is assumed
	* to have full rank, such a solution is assumed to exist and to be unique.
	*
	* \warning Again, if your matrix may not have full rank, use class LU instead. See LU::solve().
	*
	* \param b the right-hand-side of the equation to solve. Can be a vector or a matrix,
	* the only requirement in order for the equation to make sense is that
	* b.rows()==A.rows(), where A is the matrix of which *this is the LU decomposition.
	* \param result a pointer to the vector or matrix in which to store the solution, if any exists.
	* Resized if necessary, so that result->rows()==A.cols() and result->cols()==b.cols().
	* If no solution exists, *result is left with undefined coefficients.
	*
	* Example: \include PartialLU_solve.cpp
	* Output: \verbinclude PartialLU_solve.out
	*
	* \sa TriangularView::solve(), inverse(), computeInverse()
	*/
	template<typename OtherDerived, typename ResultType>
	void solve(const MatrixBase<OtherDerived>& b, ResultType *result) const;

	/** \returns the determinant of the matrix of which
	* *this is the LU decomposition. It has only linear complexity
	* (that is, O(n) where n is the dimension of the square matrix)
	* as the LU decomposition has already been computed.
	*
	* \note For fixed-size matrices of size up to 4, MatrixBase::determinant() offers
	* optimized paths.
	*
	* \warning a determinant can be very big or small, so for matrices
	* of large enough dimension, there is a risk of overflow/underflow.
	*
	* \sa MatrixBase::determinant()
	*/
	typename ei_traits<MatrixType>::Scalar determinant() const;

	/** Computes the inverse of the matrix of which *this is the LU decomposition.
	*
	* \param result a pointer to the matrix into which to store the inverse. Resized if needed.
	*
	* \warning The matrix being decomposed here is assumed to be invertible. If you need to check for
	* invertibility, use class LU instead.
	*
	* \sa MatrixBase::computeInverse(), inverse()
	*/
	inline void computeInverse(MatrixType *result) const
	{
	solve(MatrixType::Identity(m_lu.rows(), m_lu.cols()), result);
	}

	/** \returns the inverse of the matrix of which *this is the LU decomposition.
	*
	* \warning The matrix being decomposed here is assumed to be invertible. If you need to check for
	* invertibility, use class LU instead.
	*
	* \sa computeInverse(), MatrixBase::inverse()
	*/
	inline MatrixType inverse() const
	{
	MatrixType result;
	computeInverse(&result);
	return result;
	}

	protected:
	MatrixType m_lu;
	IntColVectorType m_p;
	int m_det_p;
	bool m_isInitialized;
	};

	template<typename MatrixType>
	PartialLU<MatrixType>::PartialLU()
	: m_lu(),
	m_p(),
	m_det_p(0),
	m_isInitialized(false)
	{
	}

	template<typename MatrixType>
	PartialLU<MatrixType>::PartialLU(const MatrixType& matrix)
	: m_lu(),
	m_p(),
	m_det_p(0),
	m_isInitialized(false)
	{
	compute(matrix);
	}



	/** This is the blocked version of ei_lu_unblocked() */
	template<typename Scalar, int StorageOrder>
	struct ei_partial_lu_impl
	{
	// FIXME add a stride to Map, so that the following mapping becomes easier,
	// another option would be to create an expression being able to automatically
	// warp any Map, Matrix, and Block expressions as a unique type, but since that's exactly
	// a Map + stride, why not adding a stride to Map, and convenient ctors from a Matrix,
	// and Block.
	typedef Map<Matrix<Scalar, Dynamic, Dynamic, StorageOrder> > MapLU;
	typedef Block<MapLU, Dynamic, Dynamic> MatrixType;
	typedef Block<MatrixType,Dynamic,Dynamic> BlockType;

	/** \internal performs the LU decomposition in-place of the matrix \a lu
	* using an unblocked algorithm.
	*
	* In addition, this function returns the row transpositions in the
	* vector \a row_transpositions which must have a size equal to the number
	* of columns of the matrix \a lu, and an integer \a nb_transpositions
	* which returns the actual number of transpositions.
	*/
	static void unblocked_lu(MatrixType& lu, int* row_transpositions, int& nb_transpositions)
	{
	const int rows = lu.rows();
	const int size = std::min(lu.rows(),lu.cols());
	nb_transpositions = 0;
	for(int k = 0; k < size; ++k)
	{
	int row_of_biggest_in_col;
	lu.block(k,k,rows-k,1).cwise().abs().maxCoeff(&row_of_biggest_in_col);
	row_of_biggest_in_col += k;

	row_transpositions[k] = row_of_biggest_in_col;

	if(k != row_of_biggest_in_col)
	{
	lu.row(k).swap(lu.row(row_of_biggest_in_col));
	++nb_transpositions;
	}

	if(k<rows-1)
	{
	int rrows = rows-k-1;
	int rsize = size-k-1;
	lu.col(k).end(rrows) /= lu.coeff(k,k);
	lu.corner(BottomRight,rrows,rsize).noalias() -= lu.col(k).end(rrows) * lu.row(k).end(rsize);
	}
	}
	}

	/** \internal performs the LU decomposition in-place of the matrix represented
	* by the variables \a rows, \a cols, \a lu_data, and \a lu_stride using a
	* recursive, blocked algorithm.
	*
	* In addition, this function returns the row transpositions in the
	* vector \a row_transpositions which must have a size equal to the number
	* of columns of the matrix \a lu, and an integer \a nb_transpositions
	* which returns the actual number of transpositions.
	*
	* \note This very low level interface using pointers, etc. is to:
	* 1 - reduce the number of instanciations to the strict minimum
	* 2 - avoid infinite recursion of the instanciations with Block<Block<Block<...> > >
	*/
	static void blocked_lu(int rows, int cols, Scalar* lu_data, int luStride, int* row_transpositions, int& nb_transpositions, int maxBlockSize=256)
	{
	MapLU lu1(lu_data,StorageOrder==RowMajor?rows:luStride,StorageOrder==RowMajor?luStride:cols);
	MatrixType lu(lu1,0,0,rows,cols);

	const int size = std::min(rows,cols);

	// if the matrix is too small, no blocking:
	if(size<=16)
	{
	unblocked_lu(lu, row_transpositions, nb_transpositions);
	return;
	}

	// automatically adjust the number of subdivisions to the size
	// of the matrix so that there is enough sub blocks:
	int blockSize;
	{
	blockSize = size/8;
	blockSize = (blockSize/16)*16;
	blockSize = std::min(std::max(blockSize,8), maxBlockSize);
	}

	nb_transpositions = 0;
	for(int k = 0; k < size; k+=blockSize)
	{
	int bs = std::min(size-k,blockSize); // actual size of the block
	int trows = rows - k - bs; // trailing rows
	int tsize = size - k - bs; // trailing size

	// partition the matrix:
	// A00 \| A01 \| A02
	// lu = A10 \| A11 \| A12
	// A20 \| A21 \| A22
	BlockType A_0(lu,0,0,rows,k);
	BlockType A_2(lu,0,k+bs,rows,tsize);
	BlockType A11(lu,k,k,bs,bs);
	BlockType A12(lu,k,k+bs,bs,tsize);
	BlockType A21(lu,k+bs,k,trows,bs);
	BlockType A22(lu,k+bs,k+bs,trows,tsize);

	int nb_transpositions_in_panel;
	// recursively calls the blocked LU algorithm with a very small
	// blocking size:
	blocked_lu(trows+bs, bs, &lu.coeffRef(k,k), luStride,
	row_transpositions+k, nb_transpositions_in_panel, 16);
	nb_transpositions += nb_transpositions_in_panel;

	// update permutations and apply them to A10
	for(int i=k;i<k+bs; ++i)
	{
	int piv = (row_transpositions[i] += k);
	A_0.row(i).swap(A_0.row(piv));
	}

	if(trows)
	{
	// apply permutations to A_2
	for(int i=k;i<k+bs; ++i)
	A_2.row(i).swap(A_2.row(row_transpositions[i]));

	// A12 = A11^-1 A12
	A11.template triangularView<UnitLowerTriangular>().solveInPlace(A12);

	A22 -= A21 * A12;
	}
	}
	}
	};

	/** \internal performs the LU decomposition with partial pivoting in-place.
	*/
	template<typename MatrixType, typename IntVector>
	void ei_partial_lu_inplace(MatrixType& lu, IntVector& row_transpositions, int& nb_transpositions)
	{
	ei_assert(lu.cols() == row_transpositions.size());
	ei_assert((&row_transpositions.coeffRef(1)-&row_transpositions.coeffRef(0)) == 1);

	ei_partial_lu_impl
	<typename MatrixType::Scalar, MatrixType::Flags&RowMajorBit?RowMajor:ColMajor>
	::blocked_lu(lu.rows(), lu.cols(), &lu.coeffRef(0,0), lu.stride(), &row_transpositions.coeffRef(0), nb_transpositions);
	}

	template<typename MatrixType>
	PartialLU<MatrixType>& PartialLU<MatrixType>::compute(const MatrixType& matrix)
	{
	m_lu = matrix;
	m_p.resize(matrix.rows());

	ei_assert(matrix.rows() == matrix.cols() && "PartialLU is only for square (and moreover invertible) matrices");
	const int size = matrix.rows();

	IntColVectorType rows_transpositions(size);

	int nb_transpositions;
	ei_partial_lu_inplace(m_lu, rows_transpositions, nb_transpositions);
	m_det_p = (nb_transpositions%2) ? -1 : 1;

	for(int k = 0; k < size; ++k) m_p.coeffRef(k) = k;
	for(int k = size-1; k >= 0; --k)
	std::swap(m_p.coeffRef(k), m_p.coeffRef(rows_transpositions.coeff(k)));

	m_isInitialized = true;
	return *this;
	}

	template<typename MatrixType>
	typename ei_traits<MatrixType>::Scalar PartialLU<MatrixType>::determinant() const
	{
	ei_assert(m_isInitialized && "PartialLU is not initialized.");
	return Scalar(m_det_p) * m_lu.diagonal().prod();
	}

	template<typename MatrixType>
	template<typename OtherDerived, typename ResultType>
	void PartialLU<MatrixType>::solve(
	const MatrixBase<OtherDerived>& b,
	ResultType *result
	) const
	{
	ei_assert(m_isInitialized && "PartialLU is not initialized.");

	/* The decomposition PA = LU can be rewritten as A = P^{-1} L U.
	* So we proceed as follows:
	* Step 1: compute c = Pb.
	* Step 2: replace c by the solution x to Lx = c.
	* Step 3: replace c by the solution x to Ux = c.
	*/

	const int size = m_lu.rows();
	ei_assert(b.rows() == size);

	result->resize(size, b.cols());

	// Step 1
	for(int i = 0; i < size; ++i) result->row(m_p.coeff(i)) = b.row(i);

	// Step 2
	m_lu.template triangularView<UnitLowerTriangular>().solveInPlace(*result);

	// Step 3
	m_lu.template triangularView<UpperTriangular>().solveInPlace(*result);
	}

	/** \lu_module
	*
	* \return the LU decomposition of \c *this.
	*
	* \sa class LU
	*/
	template<typename Derived>
	inline const PartialLU<typename MatrixBase<Derived>::PlainMatrixType>
	MatrixBase<Derived>::partialLu() const
	{
	return PartialLU<PlainMatrixType>(eval());
	}

	#endif // EIGEN_PARTIALLU_H