Eigen/src/LU/PartialPivLU.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>
 // Copyright (C) 2009 Gael Guennebaud <g.gael@free.fr>
 //
 // 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 PartialPivLU
   *
   * \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. Thus LAPACK's dgesv and dgesvx require the matrix to be square and invertible. The present class
   * does the same. It 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 FullPivLU.
   *
   * 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 FullPivLU.
   *
   * This LU decomposition is suitable to invert invertible matrices. It is what MatrixBase::inverse() uses
   * in the general case.
   * 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::partialPivLu(), MatrixBase::determinant(), MatrixBase::inverse(), MatrixBase::computeInverse(), class FullPivLU
   */
 template<typename _MatrixType> class PartialPivLU
 {
   public:

     typedef _MatrixType MatrixType;
     enum {
       RowsAtCompileTime = MatrixType::RowsAtCompileTime,
       ColsAtCompileTime = MatrixType::ColsAtCompileTime,
       Options = MatrixType::Options,
       MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
       MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
     };
     typedef typename MatrixType::Scalar Scalar;
     typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
     typedef typename ei_traits<MatrixType>::StorageKind StorageKind;
     typedef typename MatrixType::Index Index;
     typedef PermutationMatrix<RowsAtCompileTime, MaxRowsAtCompileTime> PermutationType;
     typedef Transpositions<RowsAtCompileTime, MaxRowsAtCompileTime> TranspositionType;


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

     /** \brief Default Constructor with memory preallocation
       *
       * Like the default constructor but with preallocation of the internal data
       * according to the specified problem \a size.
       * \sa PartialPivLU()
       */
     PartialPivLU(Index size);

     /** 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 FullPivLU instead.
       */
     PartialPivLU(const MatrixType& matrix);

     PartialPivLU& 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 FullPivLU).
       *
       * \sa matrixL(), matrixU()
       */
     inline const MatrixType& matrixLU() const
     {
       ei_assert(m_isInitialized && "PartialPivLU is not initialized.");
       return m_lu;
     }

     /** \returns the permutation matrix P.
       */
     inline const PermutationType& permutationP() const
     {
       ei_assert(m_isInitialized && "PartialPivLU is not initialized.");
       return m_p;
     }

     /** This method returns the solution x to the equation Ax=b, where A is the matrix of which
       * *this is the LU decomposition.
       *
       * \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.
       *
       * \returns the solution.
       *
       * Example: \include PartialPivLU_solve.cpp
       * Output: \verbinclude PartialPivLU_solve.out
       *
       * Since this PartialPivLU class assumes anyway that the matrix A is invertible, the solution
       * theoretically exists and is unique regardless of b.
       *
       * \sa TriangularView::solve(), inverse(), computeInverse()
       */
     template<typename Rhs>
     inline const ei_solve_retval<PartialPivLU, Rhs>
     solve(const MatrixBase<Rhs>& b) const
     {
       ei_assert(m_isInitialized && "PartialPivLU is not initialized.");
       return ei_solve_retval<PartialPivLU, Rhs>(*this, b.derived());
     }

     /** \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 FullPivLU instead.
       *
       * \sa MatrixBase::inverse(), LU::inverse()
       */
     inline const ei_solve_retval<PartialPivLU,typename MatrixType::IdentityReturnType> inverse() const
     {
       ei_assert(m_isInitialized && "PartialPivLU is not initialized.");
       return ei_solve_retval<PartialPivLU,typename MatrixType::IdentityReturnType>
                (*this, MatrixType::Identity(m_lu.rows(), m_lu.cols()));
     }

     /** \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;

     MatrixType reconstructedMatrix() const;

     inline Index rows() const { return m_lu.rows(); }
     inline Index cols() const { return m_lu.cols(); }

   protected:
     MatrixType m_lu;
     PermutationType m_p;
     TranspositionType m_rowsTranspositions;
     Index m_det_p;
     bool m_isInitialized;
 };

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

 template<typename MatrixType>
 PartialPivLU<MatrixType>::PartialPivLU(Index size)
   : m_lu(size, size),
     m_p(size),
     m_rowsTranspositions(size),
     m_det_p(0),
     m_isInitialized(false)
 {
 }

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

 /** \internal This is the blocked version of ei_fullpivlu_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;
   typedef typename MatrixType::RealScalar RealScalar;
   typedef typename MatrixType::Index Index;

   /** \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.
     *
     * \returns false if some pivot is exactly zero, in which case the matrix is left with
     *          undefined coefficients (to avoid generating inf/nan values). Returns true
     *          otherwise.
     */
   static bool unblocked_lu(MatrixType& lu, Index* row_transpositions, Index& nb_transpositions)
   {
     const Index rows = lu.rows();
     const Index size = std::min(lu.rows(),lu.cols());
     nb_transpositions = 0;
     for(Index k = 0; k < size; ++k)
     {
       Index row_of_biggest_in_col;
       RealScalar biggest_in_corner
         = lu.col(k).tail(rows-k).cwiseAbs().maxCoeff(&row_of_biggest_in_col);
       row_of_biggest_in_col += k;

       if(biggest_in_corner == 0) // the pivot is exactly zero: the matrix is singular
       {
         // end quickly, avoid generating inf/nan values. Although in this unblocked_lu case
         // the result is still valid, there's no need to boast about it because
         // the blocked_lu code can't guarantee the same.
         // before exiting, make sure to initialize the still uninitialized row_transpositions
         // in a sane state without destroying what we already have.
         for(Index i = k; i < size; i++)
           row_transpositions[i] = i;
         return false;
       }

       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)
       {
         Index rrows = rows-k-1;
         Index rsize = size-k-1;
         lu.col(k).tail(rrows) /= lu.coeff(k,k);
         lu.bottomRightCorner(rrows,rsize).noalias() -= lu.col(k).tail(rrows) * lu.row(k).tail(rsize);
       }
     }
     return true;
   }

   /** \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.
     *
     * \returns false if some pivot is exactly zero, in which case the matrix is left with
     *          undefined coefficients (to avoid generating inf/nan values). Returns true
     *          otherwise.
     *
     * \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 bool blocked_lu(Index rows, Index cols, Scalar* lu_data, Index luStride, Index* row_transpositions, Index& nb_transpositions, Index maxBlockSize=256)
   {
     MapLU lu1(lu_data,StorageOrder==RowMajor?rows:luStride,StorageOrder==RowMajor?luStride:cols);
     MatrixType lu(lu1,0,0,rows,cols);

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

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

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

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

       // partition the matrix:
       //                          A00 | A01 | A02
       // lu  = A_0 | A_1 | A_2 =  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);

       Index nb_transpositions_in_panel;
       // recursively call the blocked LU algorithm on [A11^T A21^T]^T
       // with a very small blocking size:
       if(!blocked_lu(trows+bs, bs, &lu.coeffRef(k,k), luStride,
                    row_transpositions+k, nb_transpositions_in_panel, 16))
       {
         // end quickly with undefined coefficients, just avoid generating inf/nan values.
         // before exiting, make sure to initialize the still uninitialized row_transpositions
         // in a sane state without destroying what we already have.
         for(Index i=k; i<size; ++i)
           row_transpositions[i] = i;
         return false;
       }
       nb_transpositions += nb_transpositions_in_panel;

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

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

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

         A22.noalias() -= A21 * A12;
       }
     }
     return true;
   }
 };

 /** \internal performs the LU decomposition with partial pivoting in-place.
   */
 template<typename MatrixType, typename TranspositionType>
 void ei_partial_lu_inplace(MatrixType& lu, TranspositionType& row_transpositions, typename MatrixType::Index& 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.outerStride(), &row_transpositions.coeffRef(0), nb_transpositions);
 }

 template<typename MatrixType>
 PartialPivLU<MatrixType>& PartialPivLU<MatrixType>::compute(const MatrixType& matrix)
 {
   m_lu = matrix;

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

   m_rowsTranspositions.resize(size);

   Index nb_transpositions;
   ei_partial_lu_inplace(m_lu, m_rowsTranspositions, nb_transpositions);
   m_det_p = (nb_transpositions%2) ? -1 : 1;

   m_p = m_rowsTranspositions;

   m_isInitialized = true;
   return *this;
 }

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

 /** \returns the matrix represented by the decomposition,
  * i.e., it returns the product: P^{-1} L U.
  * This function is provided for debug purpose. */
 template<typename MatrixType>
 MatrixType PartialPivLU<MatrixType>::reconstructedMatrix() const
 {
   ei_assert(m_isInitialized && "LU is not initialized.");
   // LU
   MatrixType res = m_lu.template triangularView<UnitLower>().toDenseMatrix()
                  * m_lu.template triangularView<Upper>();

   // P^{-1}(LU)
   res = m_p.inverse() * res;

   return res;
 }

 /***** Implementation of solve() *****************************************************/

 template<typename _MatrixType, typename Rhs>
 struct ei_solve_retval<PartialPivLU<_MatrixType>, Rhs>
   : ei_solve_retval_base<PartialPivLU<_MatrixType>, Rhs>
 {
   EIGEN_MAKE_SOLVE_HELPERS(PartialPivLU<_MatrixType>,Rhs)

   template<typename Dest> void evalTo(Dest& dst) const
   {
     /* 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.
     */

     ei_assert(rhs().rows() == dec().matrixLU().rows());

     // Step 1
     dst = dec().permutationP() * rhs();

     // Step 2
     dec().matrixLU().template triangularView<UnitLower>().solveInPlace(dst);

     // Step 3
     dec().matrixLU().template triangularView<Upper>().solveInPlace(dst);
   }
 };

 /******** MatrixBase methods *******/

 /** \lu_module
   *
   * \return the partial-pivoting LU decomposition of \c *this.
   *
   * \sa class PartialPivLU
   */
 template<typename Derived>
 inline const PartialPivLU<typename MatrixBase<Derived>::PlainObject>
 MatrixBase<Derived>::partialPivLu() const
 {
   return PartialPivLU<PlainObject>(eval());
 }

 /** \lu_module
   *
   * Synonym of partialPivLu().
   *
   * \return the partial-pivoting LU decomposition of \c *this.
   *
   * \sa class PartialPivLU
   */
 template<typename Derived>
 inline const PartialPivLU<typename MatrixBase<Derived>::PlainObject>
 MatrixBase<Derived>::lu() const
 {
   return PartialPivLU<PlainObject>(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>
	// Copyright (C) 2009 Gael Guennebaud <g.gael@free.fr>
	//
	// 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 PartialPivLU
	*
	* \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. Thus LAPACK's dgesv and dgesvx require the matrix to be square and invertible. The present class
	* does the same. It 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 FullPivLU.
	*
	* 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 FullPivLU.
	*
	* This LU decomposition is suitable to invert invertible matrices. It is what MatrixBase::inverse() uses
	* in the general case.
	* 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::partialPivLu(), MatrixBase::determinant(), MatrixBase::inverse(), MatrixBase::computeInverse(), class FullPivLU
	*/
	template<typename _MatrixType> class PartialPivLU
	{
	public:

	typedef _MatrixType MatrixType;
	enum {
	RowsAtCompileTime = MatrixType::RowsAtCompileTime,
	ColsAtCompileTime = MatrixType::ColsAtCompileTime,
	Options = MatrixType::Options,
	MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
	MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
	};
	typedef typename MatrixType::Scalar Scalar;
	typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
	typedef typename ei_traits<MatrixType>::StorageKind StorageKind;
	typedef typename MatrixType::Index Index;
	typedef PermutationMatrix<RowsAtCompileTime, MaxRowsAtCompileTime> PermutationType;
	typedef Transpositions<RowsAtCompileTime, MaxRowsAtCompileTime> TranspositionType;


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

	/** \brief Default Constructor with memory preallocation
	*
	* Like the default constructor but with preallocation of the internal data
	* according to the specified problem \a size.
	* \sa PartialPivLU()
	*/
	PartialPivLU(Index size);

	/** 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 FullPivLU instead.
	*/
	PartialPivLU(const MatrixType& matrix);

	PartialPivLU& 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 FullPivLU).
	*
	* \sa matrixL(), matrixU()
	*/
	inline const MatrixType& matrixLU() const
	{
	ei_assert(m_isInitialized && "PartialPivLU is not initialized.");
	return m_lu;
	}

	/** \returns the permutation matrix P.
	*/
	inline const PermutationType& permutationP() const
	{
	ei_assert(m_isInitialized && "PartialPivLU is not initialized.");
	return m_p;
	}

	/** This method returns the solution x to the equation Ax=b, where A is the matrix of which
	* *this is the LU decomposition.
	*
	* \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.
	*
	* \returns the solution.
	*
	* Example: \include PartialPivLU_solve.cpp
	* Output: \verbinclude PartialPivLU_solve.out
	*
	* Since this PartialPivLU class assumes anyway that the matrix A is invertible, the solution
	* theoretically exists and is unique regardless of b.
	*
	* \sa TriangularView::solve(), inverse(), computeInverse()
	*/
	template<typename Rhs>
	inline const ei_solve_retval<PartialPivLU, Rhs>
	solve(const MatrixBase<Rhs>& b) const
	{
	ei_assert(m_isInitialized && "PartialPivLU is not initialized.");
	return ei_solve_retval<PartialPivLU, Rhs>(*this, b.derived());
	}

	/** \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 FullPivLU instead.
	*
	* \sa MatrixBase::inverse(), LU::inverse()
	*/
	inline const ei_solve_retval<PartialPivLU,typename MatrixType::IdentityReturnType> inverse() const
	{
	ei_assert(m_isInitialized && "PartialPivLU is not initialized.");
	return ei_solve_retval<PartialPivLU,typename MatrixType::IdentityReturnType>
	(*this, MatrixType::Identity(m_lu.rows(), m_lu.cols()));
	}

	/** \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;

	MatrixType reconstructedMatrix() const;

	inline Index rows() const { return m_lu.rows(); }
	inline Index cols() const { return m_lu.cols(); }

	protected:
	MatrixType m_lu;
	PermutationType m_p;
	TranspositionType m_rowsTranspositions;
	Index m_det_p;
	bool m_isInitialized;
	};

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

	template<typename MatrixType>
	PartialPivLU<MatrixType>::PartialPivLU(Index size)
	: m_lu(size, size),
	m_p(size),
	m_rowsTranspositions(size),
	m_det_p(0),
	m_isInitialized(false)
	{
	}

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

	/** \internal This is the blocked version of ei_fullpivlu_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;
	typedef typename MatrixType::RealScalar RealScalar;
	typedef typename MatrixType::Index Index;

	/** \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.
	*
	* \returns false if some pivot is exactly zero, in which case the matrix is left with
	* undefined coefficients (to avoid generating inf/nan values). Returns true
	* otherwise.
	*/
	static bool unblocked_lu(MatrixType& lu, Index* row_transpositions, Index& nb_transpositions)
	{
	const Index rows = lu.rows();
	const Index size = std::min(lu.rows(),lu.cols());
	nb_transpositions = 0;
	for(Index k = 0; k < size; ++k)
	{
	Index row_of_biggest_in_col;
	RealScalar biggest_in_corner
	= lu.col(k).tail(rows-k).cwiseAbs().maxCoeff(&row_of_biggest_in_col);
	row_of_biggest_in_col += k;

	if(biggest_in_corner == 0) // the pivot is exactly zero: the matrix is singular
	{
	// end quickly, avoid generating inf/nan values. Although in this unblocked_lu case
	// the result is still valid, there's no need to boast about it because
	// the blocked_lu code can't guarantee the same.
	// before exiting, make sure to initialize the still uninitialized row_transpositions
	// in a sane state without destroying what we already have.
	for(Index i = k; i < size; i++)
	row_transpositions[i] = i;
	return false;
	}

	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)
	{
	Index rrows = rows-k-1;
	Index rsize = size-k-1;
	lu.col(k).tail(rrows) /= lu.coeff(k,k);
	lu.bottomRightCorner(rrows,rsize).noalias() -= lu.col(k).tail(rrows) * lu.row(k).tail(rsize);
	}
	}
	return true;
	}

	/** \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.
	*
	* \returns false if some pivot is exactly zero, in which case the matrix is left with
	* undefined coefficients (to avoid generating inf/nan values). Returns true
	* otherwise.
	*
	* \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 bool blocked_lu(Index rows, Index cols, Scalar* lu_data, Index luStride, Index* row_transpositions, Index& nb_transpositions, Index maxBlockSize=256)
	{
	MapLU lu1(lu_data,StorageOrder==RowMajor?rows:luStride,StorageOrder==RowMajor?luStride:cols);
	MatrixType lu(lu1,0,0,rows,cols);

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

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

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

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

	// partition the matrix:
	// A00 \| A01 \| A02
	// lu = A_0 \| A_1 \| A_2 = 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);

	Index nb_transpositions_in_panel;
	// recursively call the blocked LU algorithm on [A11^T A21^T]^T
	// with a very small blocking size:
	if(!blocked_lu(trows+bs, bs, &lu.coeffRef(k,k), luStride,
	row_transpositions+k, nb_transpositions_in_panel, 16))
	{
	// end quickly with undefined coefficients, just avoid generating inf/nan values.
	// before exiting, make sure to initialize the still uninitialized row_transpositions
	// in a sane state without destroying what we already have.
	for(Index i=k; i<size; ++i)
	row_transpositions[i] = i;
	return false;
	}
	nb_transpositions += nb_transpositions_in_panel;

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

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

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

	A22.noalias() -= A21 * A12;
	}
	}
	return true;
	}
	};

	/** \internal performs the LU decomposition with partial pivoting in-place.
	*/
	template<typename MatrixType, typename TranspositionType>
	void ei_partial_lu_inplace(MatrixType& lu, TranspositionType& row_transpositions, typename MatrixType::Index& 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.outerStride(), &row_transpositions.coeffRef(0), nb_transpositions);
	}

	template<typename MatrixType>
	PartialPivLU<MatrixType>& PartialPivLU<MatrixType>::compute(const MatrixType& matrix)
	{
	m_lu = matrix;

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

	m_rowsTranspositions.resize(size);

	Index nb_transpositions;
	ei_partial_lu_inplace(m_lu, m_rowsTranspositions, nb_transpositions);
	m_det_p = (nb_transpositions%2) ? -1 : 1;

	m_p = m_rowsTranspositions;

	m_isInitialized = true;
	return *this;
	}

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

	/** \returns the matrix represented by the decomposition,
	* i.e., it returns the product: P^{-1} L U.
	* This function is provided for debug purpose. */
	template<typename MatrixType>
	MatrixType PartialPivLU<MatrixType>::reconstructedMatrix() const
	{
	ei_assert(m_isInitialized && "LU is not initialized.");
	// LU
	MatrixType res = m_lu.template triangularView<UnitLower>().toDenseMatrix()
	* m_lu.template triangularView<Upper>();

	// P^{-1}(LU)
	res = m_p.inverse() * res;

	return res;
	}

	/*** Implementation of solve() ***************************************************/

	template<typename _MatrixType, typename Rhs>
	struct ei_solve_retval<PartialPivLU<_MatrixType>, Rhs>
	: ei_solve_retval_base<PartialPivLU<_MatrixType>, Rhs>
	{
	EIGEN_MAKE_SOLVE_HELPERS(PartialPivLU<_MatrixType>,Rhs)

	template<typename Dest> void evalTo(Dest& dst) const
	{
	/* 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.
	*/

	ei_assert(rhs().rows() == dec().matrixLU().rows());

	// Step 1
	dst = dec().permutationP() * rhs();

	// Step 2
	dec().matrixLU().template triangularView<UnitLower>().solveInPlace(dst);

	// Step 3
	dec().matrixLU().template triangularView<Upper>().solveInPlace(dst);
	}
	};

	/****** MatrixBase methods *****/

	/** \lu_module
	*
	* \return the partial-pivoting LU decomposition of \c *this.
	*
	* \sa class PartialPivLU
	*/
	template<typename Derived>
	inline const PartialPivLU<typename MatrixBase<Derived>::PlainObject>
	MatrixBase<Derived>::partialPivLu() const
	{
	return PartialPivLU<PlainObject>(eval());
	}

	/** \lu_module
	*
	* Synonym of partialPivLu().
	*
	* \return the partial-pivoting LU decomposition of \c *this.
	*
	* \sa class PartialPivLU
	*/
	template<typename Derived>
	inline const PartialPivLU<typename MatrixBase<Derived>::PlainObject>
	MatrixBase<Derived>::lu() const
	{
	return PartialPivLU<PlainObject>(eval());
	}

	#endif // EIGEN_PARTIALLU_H