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 <gael.guennebaud@inria.fr>
 //
 // This Source Code Form is subject to the terms of the Mozilla
 // Public License v. 2.0. If a copy of the MPL was not distributed
 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

 #ifndef EIGEN_PARTIALLU_H
 #define EIGEN_PARTIALLU_H

 #include "./InternalHeaderCheck.h"

 namespace Eigen {

 namespace internal {
 template<typename MatrixType_> struct traits<PartialPivLU<MatrixType_> >
  : traits<MatrixType_>
 {
   typedef MatrixXpr XprKind;
   typedef SolverStorage StorageKind;
   typedef int StorageIndex;
   typedef traits<MatrixType_> BaseTraits;
   enum {
     Flags = BaseTraits::Flags & RowMajorBit,
     CoeffReadCost = Dynamic
   };
 };

 template<typename T,typename Derived>
 struct enable_if_ref;
 // {
 //   typedef Derived type;
 // };

 template<typename T,typename Derived>
 struct enable_if_ref<Ref<T>,Derived> {
   typedef Derived type;
 };

 } // end namespace internal

 /** \ingroup LU_Module
   *
   * \class PartialPivLU
   *
   * \brief LU decomposition of a matrix with partial pivoting, and related features
   *
   * \tparam 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().
   *
   * This class supports the \link InplaceDecomposition inplace decomposition \endlink mechanism.
   *
   * \sa MatrixBase::partialPivLu(), MatrixBase::determinant(), MatrixBase::inverse(), MatrixBase::computeInverse(), class FullPivLU
   */
 template<typename MatrixType_> class PartialPivLU
   : public SolverBase<PartialPivLU<MatrixType_> >
 {
   public:

     typedef MatrixType_ MatrixType;
     typedef SolverBase<PartialPivLU> Base;
     friend class SolverBase<PartialPivLU>;

     EIGEN_GENERIC_PUBLIC_INTERFACE(PartialPivLU)
     enum {
       MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
       MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
     };
     typedef PermutationMatrix<RowsAtCompileTime, MaxRowsAtCompileTime> PermutationType;
     typedef Transpositions<RowsAtCompileTime, MaxRowsAtCompileTime> TranspositionType;
     typedef typename MatrixType::PlainObject PlainObject;

     /**
       * \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()
       */
     explicit 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.
       */
     template<typename InputType>
     explicit PartialPivLU(const EigenBase<InputType>& matrix);

     /** Constructor for \link InplaceDecomposition inplace decomposition \endlink
       *
       * \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.
       */
     template<typename InputType>
     explicit PartialPivLU(EigenBase<InputType>& matrix);

     template<typename InputType>
     PartialPivLU& compute(const EigenBase<InputType>& matrix) {
       m_lu = matrix.derived();
       compute();
       return *this;
     }

     /** \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
     {
       eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
       return m_lu;
     }

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

     #ifdef EIGEN_PARSED_BY_DOXYGEN
     /** 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 Solve<PartialPivLU, Rhs>
     solve(const MatrixBase<Rhs>& b) const;
     #endif

     /** \returns an estimate of the reciprocal condition number of the matrix of which \c *this is
         the LU decomposition.
       */
     inline RealScalar rcond() const
     {
       eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
       return internal::rcond_estimate_helper(m_l1_norm, *this);
     }

     /** \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 Inverse<PartialPivLU> inverse() const
     {
       eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
       return Inverse<PartialPivLU>(*this);
     }

     /** \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()
       */
     Scalar determinant() const;

     MatrixType reconstructedMatrix() const;

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

     #ifndef EIGEN_PARSED_BY_DOXYGEN
     template<typename RhsType, typename DstType>
     EIGEN_DEVICE_FUNC
     void _solve_impl(const RhsType &rhs, DstType &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.
       */

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

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

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

     template<bool Conjugate, typename RhsType, typename DstType>
     EIGEN_DEVICE_FUNC
     void _solve_impl_transposed(const RhsType &rhs, DstType &dst) const {
      /* The decomposition PA = LU can be rewritten as A^T = U^T L^T P.
       * So we proceed as follows:
       * Step 1: compute c as the solution to L^T c = b
       * Step 2: replace c by the solution x to U^T x = c.
       * Step 3: update  c = P^-1 c.
       */

       eigen_assert(rhs.rows() == m_lu.cols());

       // Step 1
       dst = m_lu.template triangularView<Upper>().transpose()
                 .template conjugateIf<Conjugate>().solve(rhs);
       // Step 2
       m_lu.template triangularView<UnitLower>().transpose()
           .template conjugateIf<Conjugate>().solveInPlace(dst);
       // Step 3
       dst = permutationP().transpose() * dst;
     }
     #endif

   protected:

     EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar)

     void compute();

     MatrixType m_lu;
     PermutationType m_p;
     TranspositionType m_rowsTranspositions;
     RealScalar m_l1_norm;
     signed char m_det_p;
     bool m_isInitialized;
 };

 template<typename MatrixType>
 PartialPivLU<MatrixType>::PartialPivLU()
   : m_lu(),
     m_p(),
     m_rowsTranspositions(),
     m_l1_norm(0),
     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_l1_norm(0),
     m_det_p(0),
     m_isInitialized(false)
 {
 }

 template<typename MatrixType>
 template<typename InputType>
 PartialPivLU<MatrixType>::PartialPivLU(const EigenBase<InputType>& matrix)
   : m_lu(matrix.rows(),matrix.cols()),
     m_p(matrix.rows()),
     m_rowsTranspositions(matrix.rows()),
     m_l1_norm(0),
     m_det_p(0),
     m_isInitialized(false)
 {
   compute(matrix.derived());
 }

 template<typename MatrixType>
 template<typename InputType>
 PartialPivLU<MatrixType>::PartialPivLU(EigenBase<InputType>& matrix)
   : m_lu(matrix.derived()),
     m_p(matrix.rows()),
     m_rowsTranspositions(matrix.rows()),
     m_l1_norm(0),
     m_det_p(0),
     m_isInitialized(false)
 {
   compute();
 }

 namespace internal {

 /** \internal This is the blocked version of fullpivlu_unblocked() */
 template<typename Scalar, int StorageOrder, typename PivIndex, int SizeAtCompileTime=Dynamic>
 struct partial_lu_impl
 {
   static const int UnBlockedBound = 16;
   static const bool UnBlockedAtCompileTime = SizeAtCompileTime!=Dynamic && SizeAtCompileTime<=UnBlockedBound;
   static const int ActualSizeAtCompileTime = UnBlockedAtCompileTime ? SizeAtCompileTime : Dynamic;
   // Remaining rows and columns at compile-time:
   static const int RRows = SizeAtCompileTime==2 ? 1 : Dynamic;
   static const int RCols = SizeAtCompileTime==2 ? 1 : Dynamic;
   typedef Matrix<Scalar, ActualSizeAtCompileTime, ActualSizeAtCompileTime, StorageOrder> MatrixType;
   typedef Ref<MatrixType> MatrixTypeRef;
   typedef Ref<Matrix<Scalar, Dynamic, Dynamic, StorageOrder> > BlockType;
   typedef typename MatrixType::RealScalar RealScalar;

   /** \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 The index of the first pivot which is exactly zero if any, or a negative number otherwise.
     */
   static Index unblocked_lu(MatrixTypeRef& lu, PivIndex* row_transpositions, PivIndex& nb_transpositions)
   {
     typedef scalar_score_coeff_op<Scalar> Scoring;
     typedef typename Scoring::result_type Score;
     const Index rows = lu.rows();
     const Index cols = lu.cols();
     const Index size = (std::min)(rows,cols);
     // For small compile-time matrices it is worth processing the last row separately:
     //  speedup: +100% for 2x2, +10% for others.
     const Index endk = UnBlockedAtCompileTime ? size-1 : size;
     nb_transpositions = 0;
     Index first_zero_pivot = -1;
     for(Index k = 0; k < endk; ++k)
     {
       int rrows = internal::convert_index<int>(rows-k-1);
       int rcols = internal::convert_index<int>(cols-k-1);

       Index row_of_biggest_in_col;
       Score biggest_in_corner
         = lu.col(k).tail(rows-k).unaryExpr(Scoring()).maxCoeff(&row_of_biggest_in_col);
       row_of_biggest_in_col += k;

       row_transpositions[k] = PivIndex(row_of_biggest_in_col);

       if(biggest_in_corner != Score(0))
       {
         if(k != row_of_biggest_in_col)
         {
           lu.row(k).swap(lu.row(row_of_biggest_in_col));
           ++nb_transpositions;
         }

         lu.col(k).tail(fix<RRows>(rrows)) /= lu.coeff(k,k);
       }
       else if(first_zero_pivot==-1)
       {
         // the pivot is exactly zero, we record the index of the first pivot which is exactly 0,
         // and continue the factorization such we still have A = PLU
         first_zero_pivot = k;
       }

       if(k<rows-1)
         lu.bottomRightCorner(fix<RRows>(rrows),fix<RCols>(rcols)).noalias() -= lu.col(k).tail(fix<RRows>(rrows)) * lu.row(k).tail(fix<RCols>(rcols));
     }

     // special handling of the last entry
     if(UnBlockedAtCompileTime)
     {
       Index k = endk;
       row_transpositions[k] = PivIndex(k);
       if (Scoring()(lu(k, k)) == Score(0) && first_zero_pivot == -1)
         first_zero_pivot = k;
     }

     return first_zero_pivot;
   }

   /** \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 The index of the first pivot which is exactly zero if any, or a negative number otherwise.
     *
     * \note This very low level interface using pointers, etc. is to:
     *   1 - reduce the number of instantiations to the strict minimum
     *   2 - avoid infinite recursion of the instantiations with Block<Block<Block<...> > >
     */
   static Index blocked_lu(Index rows, Index cols, Scalar* lu_data, Index luStride, PivIndex* row_transpositions, PivIndex& nb_transpositions, Index maxBlockSize=256)
   {
     MatrixTypeRef lu = MatrixType::Map(lu_data,rows, cols, OuterStride<>(luStride));

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

     // if the matrix is too small, no blocking:
     if(UnBlockedAtCompileTime || size<=UnBlockedBound)
     {
       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;
     Index first_zero_pivot = -1;
     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.block(0,0,rows,k);
       BlockType A_2 = lu.block(0,k+bs,rows,tsize);
       BlockType A11 = lu.block(k,k,bs,bs);
       BlockType A12 = lu.block(k,k+bs,bs,tsize);
       BlockType A21 = lu.block(k+bs,k,trows,bs);
       BlockType A22 = lu.block(k+bs,k+bs,trows,tsize);

       PivIndex nb_transpositions_in_panel;
       // recursively call the blocked LU algorithm on [A11^T A21^T]^T
       // with a very small blocking size:
       Index ret = blocked_lu(trows+bs, bs, &lu.coeffRef(k,k), luStride,
                    row_transpositions+k, nb_transpositions_in_panel, 16);
       if(ret>=0 && first_zero_pivot==-1)
         first_zero_pivot = k+ret;

       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] += internal::convert_index<PivIndex>(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 first_zero_pivot;
   }
 };

 /** \internal performs the LU decomposition with partial pivoting in-place.
   */
 template<typename MatrixType, typename TranspositionType>
 void partial_lu_inplace(MatrixType& lu, TranspositionType& row_transpositions, typename TranspositionType::StorageIndex& nb_transpositions)
 {
   // Special-case of zero matrix.
   if (lu.rows() == 0 || lu.cols() == 0) {
     nb_transpositions = 0;
     return;
   }
   eigen_assert(lu.cols() == row_transpositions.size());
   eigen_assert(row_transpositions.size() < 2 || (&row_transpositions.coeffRef(1)-&row_transpositions.coeffRef(0)) == 1);

   partial_lu_impl
     < typename MatrixType::Scalar, MatrixType::Flags&RowMajorBit?RowMajor:ColMajor,
       typename TranspositionType::StorageIndex,
       internal::min_size_prefer_fixed(MatrixType::RowsAtCompileTime, MatrixType::ColsAtCompileTime)>
     ::blocked_lu(lu.rows(), lu.cols(), &lu.coeffRef(0,0), lu.outerStride(), &row_transpositions.coeffRef(0), nb_transpositions);
 }

 } // end namespace internal

 template<typename MatrixType>
 void PartialPivLU<MatrixType>::compute()
 {
   // the row permutation is stored as int indices, so just to be sure:
   eigen_assert(m_lu.rows()<NumTraits<int>::highest());

   if(m_lu.cols()>0)
     m_l1_norm = m_lu.cwiseAbs().colwise().sum().maxCoeff();
   else
     m_l1_norm = RealScalar(0);

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

   m_rowsTranspositions.resize(size);

   typename TranspositionType::StorageIndex nb_transpositions;
   internal::partial_lu_inplace(m_lu, m_rowsTranspositions, nb_transpositions);
   m_det_p = (nb_transpositions%2) ? -1 : 1;

   m_p = m_rowsTranspositions;

   m_isInitialized = true;
 }

 template<typename MatrixType>
 typename PartialPivLU<MatrixType>::Scalar PartialPivLU<MatrixType>::determinant() const
 {
   eigen_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
 {
   eigen_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 details *****************************************************/

 namespace internal {

 /***** Implementation of inverse() *****************************************************/
 template<typename DstXprType, typename MatrixType>
 struct Assignment<DstXprType, Inverse<PartialPivLU<MatrixType> >, internal::assign_op<typename DstXprType::Scalar,typename PartialPivLU<MatrixType>::Scalar>, Dense2Dense>
 {
   typedef PartialPivLU<MatrixType> LuType;
   typedef Inverse<LuType> SrcXprType;
   static void run(DstXprType &dst, const SrcXprType &src, const internal::assign_op<typename DstXprType::Scalar,typename LuType::Scalar> &)
   {
     dst = src.nestedExpression().solve(MatrixType::Identity(src.rows(), src.cols()));
   }
 };
 } // end namespace internal

 /******** 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());
 }

 } // end namespace Eigen

 #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 <gael.guennebaud@inria.fr>
	//
	// This Source Code Form is subject to the terms of the Mozilla
	// Public License v. 2.0. If a copy of the MPL was not distributed
	// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

	#ifndef EIGEN_PARTIALLU_H
	#define EIGEN_PARTIALLU_H

	#include "./InternalHeaderCheck.h"

	namespace Eigen {

	namespace internal {
	template<typename MatrixType_> struct traits<PartialPivLU<MatrixType_> >
	: traits<MatrixType_>
	{
	typedef MatrixXpr XprKind;
	typedef SolverStorage StorageKind;
	typedef int StorageIndex;
	typedef traits<MatrixType_> BaseTraits;
	enum {
	Flags = BaseTraits::Flags & RowMajorBit,
	CoeffReadCost = Dynamic
	};
	};

	template<typename T,typename Derived>
	struct enable_if_ref;
	// {
	// typedef Derived type;
	// };

	template<typename T,typename Derived>
	struct enable_if_ref<Ref<T>,Derived> {
	typedef Derived type;
	};

	} // end namespace internal

	/** \ingroup LU_Module
	*
	* \class PartialPivLU
	*
	* \brief LU decomposition of a matrix with partial pivoting, and related features
	*
	* \tparam 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().
	*
	* This class supports the \link InplaceDecomposition inplace decomposition \endlink mechanism.
	*
	* \sa MatrixBase::partialPivLu(), MatrixBase::determinant(), MatrixBase::inverse(), MatrixBase::computeInverse(), class FullPivLU
	*/
	template<typename MatrixType_> class PartialPivLU
	: public SolverBase<PartialPivLU<MatrixType_> >
	{
	public:

	typedef MatrixType_ MatrixType;
	typedef SolverBase<PartialPivLU> Base;
	friend class SolverBase<PartialPivLU>;

	EIGEN_GENERIC_PUBLIC_INTERFACE(PartialPivLU)
	enum {
	MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
	MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
	};
	typedef PermutationMatrix<RowsAtCompileTime, MaxRowsAtCompileTime> PermutationType;
	typedef Transpositions<RowsAtCompileTime, MaxRowsAtCompileTime> TranspositionType;
	typedef typename MatrixType::PlainObject PlainObject;

	/**
	* \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()
	*/
	explicit 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.
	*/
	template<typename InputType>
	explicit PartialPivLU(const EigenBase<InputType>& matrix);

	/** Constructor for \link InplaceDecomposition inplace decomposition \endlink
	*
	* \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.
	*/
	template<typename InputType>
	explicit PartialPivLU(EigenBase<InputType>& matrix);

	template<typename InputType>
	PartialPivLU& compute(const EigenBase<InputType>& matrix) {
	m_lu = matrix.derived();
	compute();
	return *this;
	}

	/** \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
	{
	eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
	return m_lu;
	}

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

	#ifdef EIGEN_PARSED_BY_DOXYGEN
	/** 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 Solve<PartialPivLU, Rhs>
	solve(const MatrixBase<Rhs>& b) const;
	#endif

	/** \returns an estimate of the reciprocal condition number of the matrix of which \c *this is
	the LU decomposition.
	*/
	inline RealScalar rcond() const
	{
	eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
	return internal::rcond_estimate_helper(m_l1_norm, *this);
	}

	/** \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 Inverse<PartialPivLU> inverse() const
	{
	eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
	return Inverse<PartialPivLU>(*this);
	}

	/** \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()
	*/
	Scalar determinant() const;

	MatrixType reconstructedMatrix() const;

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

	#ifndef EIGEN_PARSED_BY_DOXYGEN
	template<typename RhsType, typename DstType>
	EIGEN_DEVICE_FUNC
	void _solve_impl(const RhsType &rhs, DstType &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.
	*/

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

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

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

	template<bool Conjugate, typename RhsType, typename DstType>
	EIGEN_DEVICE_FUNC
	void _solve_impl_transposed(const RhsType &rhs, DstType &dst) const {
	/* The decomposition PA = LU can be rewritten as A^T = U^T L^T P.
	* So we proceed as follows:
	* Step 1: compute c as the solution to L^T c = b
	* Step 2: replace c by the solution x to U^T x = c.
	* Step 3: update c = P^-1 c.
	*/

	eigen_assert(rhs.rows() == m_lu.cols());

	// Step 1
	dst = m_lu.template triangularView<Upper>().transpose()
	.template conjugateIf<Conjugate>().solve(rhs);
	// Step 2
	m_lu.template triangularView<UnitLower>().transpose()
	.template conjugateIf<Conjugate>().solveInPlace(dst);
	// Step 3
	dst = permutationP().transpose() * dst;
	}
	#endif

	protected:

	EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar)

	void compute();

	MatrixType m_lu;
	PermutationType m_p;
	TranspositionType m_rowsTranspositions;
	RealScalar m_l1_norm;
	signed char m_det_p;
	bool m_isInitialized;
	};

	template<typename MatrixType>
	PartialPivLU<MatrixType>::PartialPivLU()
	: m_lu(),
	m_p(),
	m_rowsTranspositions(),
	m_l1_norm(0),
	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_l1_norm(0),
	m_det_p(0),
	m_isInitialized(false)
	{
	}

	template<typename MatrixType>
	template<typename InputType>
	PartialPivLU<MatrixType>::PartialPivLU(const EigenBase<InputType>& matrix)
	: m_lu(matrix.rows(),matrix.cols()),
	m_p(matrix.rows()),
	m_rowsTranspositions(matrix.rows()),
	m_l1_norm(0),
	m_det_p(0),
	m_isInitialized(false)
	{
	compute(matrix.derived());
	}

	template<typename MatrixType>
	template<typename InputType>
	PartialPivLU<MatrixType>::PartialPivLU(EigenBase<InputType>& matrix)
	: m_lu(matrix.derived()),
	m_p(matrix.rows()),
	m_rowsTranspositions(matrix.rows()),
	m_l1_norm(0),
	m_det_p(0),
	m_isInitialized(false)
	{
	compute();
	}

	namespace internal {

	/** \internal This is the blocked version of fullpivlu_unblocked() */
	template<typename Scalar, int StorageOrder, typename PivIndex, int SizeAtCompileTime=Dynamic>
	struct partial_lu_impl
	{
	static const int UnBlockedBound = 16;
	static const bool UnBlockedAtCompileTime = SizeAtCompileTime!=Dynamic && SizeAtCompileTime<=UnBlockedBound;
	static const int ActualSizeAtCompileTime = UnBlockedAtCompileTime ? SizeAtCompileTime : Dynamic;
	// Remaining rows and columns at compile-time:
	static const int RRows = SizeAtCompileTime==2 ? 1 : Dynamic;
	static const int RCols = SizeAtCompileTime==2 ? 1 : Dynamic;
	typedef Matrix<Scalar, ActualSizeAtCompileTime, ActualSizeAtCompileTime, StorageOrder> MatrixType;
	typedef Ref<MatrixType> MatrixTypeRef;
	typedef Ref<Matrix<Scalar, Dynamic, Dynamic, StorageOrder> > BlockType;
	typedef typename MatrixType::RealScalar RealScalar;

	/** \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 The index of the first pivot which is exactly zero if any, or a negative number otherwise.
	*/
	static Index unblocked_lu(MatrixTypeRef& lu, PivIndex* row_transpositions, PivIndex& nb_transpositions)
	{
	typedef scalar_score_coeff_op<Scalar> Scoring;
	typedef typename Scoring::result_type Score;
	const Index rows = lu.rows();
	const Index cols = lu.cols();
	const Index size = (std::min)(rows,cols);
	// For small compile-time matrices it is worth processing the last row separately:
	// speedup: +100% for 2x2, +10% for others.
	const Index endk = UnBlockedAtCompileTime ? size-1 : size;
	nb_transpositions = 0;
	Index first_zero_pivot = -1;
	for(Index k = 0; k < endk; ++k)
	{
	int rrows = internal::convert_index<int>(rows-k-1);
	int rcols = internal::convert_index<int>(cols-k-1);

	Index row_of_biggest_in_col;
	Score biggest_in_corner
	= lu.col(k).tail(rows-k).unaryExpr(Scoring()).maxCoeff(&row_of_biggest_in_col);
	row_of_biggest_in_col += k;

	row_transpositions[k] = PivIndex(row_of_biggest_in_col);

	if(biggest_in_corner != Score(0))
	{
	if(k != row_of_biggest_in_col)
	{
	lu.row(k).swap(lu.row(row_of_biggest_in_col));
	++nb_transpositions;
	}

	lu.col(k).tail(fix<RRows>(rrows)) /= lu.coeff(k,k);
	}
	else if(first_zero_pivot==-1)
	{
	// the pivot is exactly zero, we record the index of the first pivot which is exactly 0,
	// and continue the factorization such we still have A = PLU
	first_zero_pivot = k;
	}

	if(k<rows-1)
	lu.bottomRightCorner(fix<RRows>(rrows),fix<RCols>(rcols)).noalias() -= lu.col(k).tail(fix<RRows>(rrows)) * lu.row(k).tail(fix<RCols>(rcols));
	}

	// special handling of the last entry
	if(UnBlockedAtCompileTime)
	{
	Index k = endk;
	row_transpositions[k] = PivIndex(k);
	if (Scoring()(lu(k, k)) == Score(0) && first_zero_pivot == -1)
	first_zero_pivot = k;
	}

	return first_zero_pivot;
	}

	/** \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 The index of the first pivot which is exactly zero if any, or a negative number otherwise.
	*
	* \note This very low level interface using pointers, etc. is to:
	* 1 - reduce the number of instantiations to the strict minimum
	* 2 - avoid infinite recursion of the instantiations with Block<Block<Block<...> > >
	*/
	static Index blocked_lu(Index rows, Index cols, Scalar* lu_data, Index luStride, PivIndex* row_transpositions, PivIndex& nb_transpositions, Index maxBlockSize=256)
	{
	MatrixTypeRef lu = MatrixType::Map(lu_data,rows, cols, OuterStride<>(luStride));

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

	// if the matrix is too small, no blocking:
	if(UnBlockedAtCompileTime \|\| size<=UnBlockedBound)
	{
	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;
	Index first_zero_pivot = -1;
	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.block(0,0,rows,k);
	BlockType A_2 = lu.block(0,k+bs,rows,tsize);
	BlockType A11 = lu.block(k,k,bs,bs);
	BlockType A12 = lu.block(k,k+bs,bs,tsize);
	BlockType A21 = lu.block(k+bs,k,trows,bs);
	BlockType A22 = lu.block(k+bs,k+bs,trows,tsize);

	PivIndex nb_transpositions_in_panel;
	// recursively call the blocked LU algorithm on [A11^T A21^T]^T
	// with a very small blocking size:
	Index ret = blocked_lu(trows+bs, bs, &lu.coeffRef(k,k), luStride,
	row_transpositions+k, nb_transpositions_in_panel, 16);
	if(ret>=0 && first_zero_pivot==-1)
	first_zero_pivot = k+ret;

	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] += internal::convert_index<PivIndex>(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 first_zero_pivot;
	}
	};

	/** \internal performs the LU decomposition with partial pivoting in-place.
	*/
	template<typename MatrixType, typename TranspositionType>
	void partial_lu_inplace(MatrixType& lu, TranspositionType& row_transpositions, typename TranspositionType::StorageIndex& nb_transpositions)
	{
	// Special-case of zero matrix.
	if (lu.rows() == 0 \|\| lu.cols() == 0) {
	nb_transpositions = 0;
	return;
	}
	eigen_assert(lu.cols() == row_transpositions.size());
	eigen_assert(row_transpositions.size() < 2 \|\| (&row_transpositions.coeffRef(1)-&row_transpositions.coeffRef(0)) == 1);

	partial_lu_impl
	< typename MatrixType::Scalar, MatrixType::Flags&RowMajorBit?RowMajor:ColMajor,
	typename TranspositionType::StorageIndex,
	internal::min_size_prefer_fixed(MatrixType::RowsAtCompileTime, MatrixType::ColsAtCompileTime)>
	::blocked_lu(lu.rows(), lu.cols(), &lu.coeffRef(0,0), lu.outerStride(), &row_transpositions.coeffRef(0), nb_transpositions);
	}

	} // end namespace internal

	template<typename MatrixType>
	void PartialPivLU<MatrixType>::compute()
	{
	// the row permutation is stored as int indices, so just to be sure:
	eigen_assert(m_lu.rows()<NumTraits<int>::highest());

	if(m_lu.cols()>0)
	m_l1_norm = m_lu.cwiseAbs().colwise().sum().maxCoeff();
	else
	m_l1_norm = RealScalar(0);

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

	m_rowsTranspositions.resize(size);

	typename TranspositionType::StorageIndex nb_transpositions;
	internal::partial_lu_inplace(m_lu, m_rowsTranspositions, nb_transpositions);
	m_det_p = (nb_transpositions%2) ? -1 : 1;

	m_p = m_rowsTranspositions;

	m_isInitialized = true;
	}

	template<typename MatrixType>
	typename PartialPivLU<MatrixType>::Scalar PartialPivLU<MatrixType>::determinant() const
	{
	eigen_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
	{
	eigen_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 details ***************************************************/

	namespace internal {

	/*** Implementation of inverse() ***************************************************/
	template<typename DstXprType, typename MatrixType>
	struct Assignment<DstXprType, Inverse<PartialPivLU<MatrixType> >, internal::assign_op<typename DstXprType::Scalar,typename PartialPivLU<MatrixType>::Scalar>, Dense2Dense>
	{
	typedef PartialPivLU<MatrixType> LuType;
	typedef Inverse<LuType> SrcXprType;
	static void run(DstXprType &dst, const SrcXprType &src, const internal::assign_op<typename DstXprType::Scalar,typename LuType::Scalar> &)
	{
	dst = src.nestedExpression().solve(MatrixType::Identity(src.rows(), src.cols()));
	}
	};
	} // end namespace internal

	/****** 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());
	}

	} // end namespace Eigen

	#endif // EIGEN_PARTIALLU_H