src/SparseCholesky/SimplicialCholesky.h

// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2008-2010 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/.

/*

NOTE: the _symbolic, and _numeric functions has been adapted from
      the LDL library:

LDL Copyright (c) 2005 by Timothy A. Davis.  All Rights Reserved.

LDL License:

    Your use or distribution of LDL or any modified version of
    LDL implies that you agree to this License.

    This library 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 2.1 of the License, or (at your option) any later version.

    This library 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 for more details.

    You should have received a copy of the GNU Lesser General Public
    License along with this library; if not, write to the Free Software
    Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA  02110-1301
    USA

    Permission is hereby granted to use or copy this program under the
    terms of the GNU LGPL, provided that the Copyright, this License,
    and the Availability of the original version is retained on all copies.
    User documentation of any code that uses this code or any modified
    version of this code must cite the Copyright, this License, the
    Availability note, and "Used by permission." Permission to modify
    the code and to distribute modified code is granted, provided the
    Copyright, this License, and the Availability note are retained,
    and a notice that the code was modified is included.
 */

#include "../Core/util/NonMPL2.h"

#ifndef EIGEN_SIMPLICIAL_CHOLESKY_H
#define EIGEN_SIMPLICIAL_CHOLESKY_H

namespace Eigen {

enum SimplicialCholeskyMode {
  SimplicialCholeskyLLT,
  SimplicialCholeskyLDLT
};

/** \ingroup SparseCholesky_Module
  * \brief A direct sparse Cholesky factorizations
  *
  * These classes provide LL^T and LDL^T Cholesky factorizations of sparse matrices that are
  * selfadjoint and positive definite. The factorization allows for solving A.X = B where
  * X and B can be either dense or sparse.
  *
  * In order to reduce the fill-in, a symmetric permutation P is applied prior to the factorization
  * such that the factorized matrix is P A P^-1.
  *
  * \tparam _MatrixType the type of the sparse matrix A, it must be a SparseMatrix<>
  * \tparam _UpLo the triangular part that will be used for the computations. It can be Lower
  *               or Upper. Default is Lower.
  *
  */
template<typename Derived>
class SimplicialCholeskyBase : internal::noncopyable
{
  public:
    typedef typename internal::traits<Derived>::MatrixType MatrixType;
    enum { UpLo = internal::traits<Derived>::UpLo };
    typedef typename MatrixType::Scalar Scalar;
    typedef typename MatrixType::RealScalar RealScalar;
    typedef typename MatrixType::Index Index;
    typedef SparseMatrix<Scalar,ColMajor,Index> CholMatrixType;
    typedef Matrix<Scalar,Dynamic,1> VectorType;

  public:

    /** Default constructor */
    SimplicialCholeskyBase()
      : m_info(Success), m_isInitialized(false), m_shiftOffset(0), m_shiftScale(1)
    {}

    SimplicialCholeskyBase(const MatrixType& matrix)
      : m_info(Success), m_isInitialized(false), m_shiftOffset(0), m_shiftScale(1)
    {
      derived().compute(matrix);
    }

    ~SimplicialCholeskyBase()
    {
    }

    Derived& derived() { return *static_cast<Derived*>(this); }
    const Derived& derived() const { return *static_cast<const Derived*>(this); }

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

    /** \brief Reports whether previous computation was successful.
      *
      * \returns \c Success if computation was succesful,
      *          \c NumericalIssue if the matrix.appears to be negative.
      */
    ComputationInfo info() const
    {
      eigen_assert(m_isInitialized && "Decomposition is not initialized.");
      return m_info;
    }

    /** \returns the solution x of \f$ A x = b \f$ using the current decomposition of A.
      *
      * \sa compute()
      */
    template<typename Rhs>
    inline const internal::solve_retval<SimplicialCholeskyBase, Rhs>
    solve(const MatrixBase<Rhs>& b) const
    {
      eigen_assert(m_isInitialized && "Simplicial LLT or LDLT is not initialized.");
      eigen_assert(rows()==b.rows()
                && "SimplicialCholeskyBase::solve(): invalid number of rows of the right hand side matrix b");
      return internal::solve_retval<SimplicialCholeskyBase, Rhs>(*this, b.derived());
    }

    /** \returns the solution x of \f$ A x = b \f$ using the current decomposition of A.
      *
      * \sa compute()
      */
    template<typename Rhs>
    inline const internal::sparse_solve_retval<SimplicialCholeskyBase, Rhs>
    solve(const SparseMatrixBase<Rhs>& b) const
    {
      eigen_assert(m_isInitialized && "Simplicial LLT or LDLT is not initialized.");
      eigen_assert(rows()==b.rows()
                && "SimplicialCholesky::solve(): invalid number of rows of the right hand side matrix b");
      return internal::sparse_solve_retval<SimplicialCholeskyBase, Rhs>(*this, b.derived());
    }

    /** \returns the permutation P
      * \sa permutationPinv() */
    const PermutationMatrix<Dynamic,Dynamic,Index>& permutationP() const
    { return m_P; }

    /** \returns the inverse P^-1 of the permutation P
      * \sa permutationP() */
    const PermutationMatrix<Dynamic,Dynamic,Index>& permutationPinv() const
    { return m_Pinv; }

    /** Sets the shift parameters that will be used to adjust the diagonal coefficients during the numerical factorization.
      *
      * During the numerical factorization, the diagonal coefficients are transformed by the following linear model:\n
      * \c d_ii = \a offset + \a scale * \c d_ii
      *
      * The default is the identity transformation with \a offset=0, and \a scale=1.
      *
      * \returns a reference to \c *this.
      */
    Derived& setShift(const RealScalar& offset, const RealScalar& scale = 1)
    {
      m_shiftOffset = offset;
      m_shiftScale = scale;
      return derived();
    }

#ifndef EIGEN_PARSED_BY_DOXYGEN
    /** \internal */
    template<typename Stream>
    void dumpMemory(Stream& s)
    {
      int total = 0;
      s << "  L:        " << ((total+=(m_matrix.cols()+1) * sizeof(int) + m_matrix.nonZeros()*(sizeof(int)+sizeof(Scalar))) >> 20) << "Mb" << "\n";
      s << "  diag:     " << ((total+=m_diag.size() * sizeof(Scalar)) >> 20) << "Mb" << "\n";
      s << "  tree:     " << ((total+=m_parent.size() * sizeof(int)) >> 20) << "Mb" << "\n";
      s << "  nonzeros: " << ((total+=m_nonZerosPerCol.size() * sizeof(int)) >> 20) << "Mb" << "\n";
      s << "  perm:     " << ((total+=m_P.size() * sizeof(int)) >> 20) << "Mb" << "\n";
      s << "  perm^-1:  " << ((total+=m_Pinv.size() * sizeof(int)) >> 20) << "Mb" << "\n";
      s << "  TOTAL:    " << (total>> 20) << "Mb" << "\n";
    }

    /** \internal */
    template<typename Rhs,typename Dest>
    void _solve(const MatrixBase<Rhs> &b, MatrixBase<Dest> &dest) const
    {
      eigen_assert(m_factorizationIsOk && "The decomposition is not in a valid state for solving, you must first call either compute() or symbolic()/numeric()");
      eigen_assert(m_matrix.rows()==b.rows());

      if(m_info!=Success)
        return;

      if(m_P.size()>0)
        dest = m_P * b;
      else
        dest = b;

      if(m_matrix.nonZeros()>0) // otherwise L==I
        derived().matrixL().solveInPlace(dest);

      if(m_diag.size()>0)
        dest = m_diag.asDiagonal().inverse() * dest;

      if (m_matrix.nonZeros()>0) // otherwise U==I
        derived().matrixU().solveInPlace(dest);

      if(m_P.size()>0)
        dest = m_Pinv * dest;
    }

    /** \internal */
    template<typename Rhs, typename DestScalar, int DestOptions, typename DestIndex>
    void _solve_sparse(const Rhs& b, SparseMatrix<DestScalar,DestOptions,DestIndex> &dest) const
    {
      eigen_assert(m_factorizationIsOk && "The decomposition is not in a valid state for solving, you must first call either compute() or symbolic()/numeric()");
      eigen_assert(m_matrix.rows()==b.rows());

      // we process the sparse rhs per block of NbColsAtOnce columns temporarily stored into a dense matrix.
      static const int NbColsAtOnce = 4;
      int rhsCols = b.cols();
      int size = b.rows();
      Eigen::Matrix<DestScalar,Dynamic,Dynamic> tmp(size,rhsCols);
      for(int k=0; k<rhsCols; k+=NbColsAtOnce)
      {
        int actualCols = std::min<int>(rhsCols-k, NbColsAtOnce);
        tmp.leftCols(actualCols) = b.middleCols(k,actualCols);
        tmp.leftCols(actualCols) = derived().solve(tmp.leftCols(actualCols));
        dest.middleCols(k,actualCols) = tmp.leftCols(actualCols).sparseView();
      }
    }

#endif // EIGEN_PARSED_BY_DOXYGEN

  protected:

    /** Computes the sparse Cholesky decomposition of \a matrix */
    template<bool DoLDLT>
    void compute(const MatrixType& matrix)
    {
      eigen_assert(matrix.rows()==matrix.cols());
      Index size = matrix.cols();
      CholMatrixType ap(size,size);
      ordering(matrix, ap);
      analyzePattern_preordered(ap, DoLDLT);
      factorize_preordered<DoLDLT>(ap);
    }

    template<bool DoLDLT>
    void factorize(const MatrixType& a)
    {
      eigen_assert(a.rows()==a.cols());
      int size = a.cols();
      CholMatrixType ap(size,size);
      ap.template selfadjointView<Upper>() = a.template selfadjointView<UpLo>().twistedBy(m_P);
      factorize_preordered<DoLDLT>(ap);
    }

    template<bool DoLDLT>
    void factorize_preordered(const CholMatrixType& a);

    void analyzePattern(const MatrixType& a, bool doLDLT)
    {
      eigen_assert(a.rows()==a.cols());
      int size = a.cols();
      CholMatrixType ap(size,size);
      ordering(a, ap);
      analyzePattern_preordered(ap,doLDLT);
    }
    void analyzePattern_preordered(const CholMatrixType& a, bool doLDLT);

    void ordering(const MatrixType& a, CholMatrixType& ap);

    /** keeps off-diagonal entries; drops diagonal entries */
    struct keep_diag {
      inline bool operator() (const Index& row, const Index& col, const Scalar&) const
      {
        return row!=col;
      }
    };

    mutable ComputationInfo m_info;
    bool m_isInitialized;
    bool m_factorizationIsOk;
    bool m_analysisIsOk;

    CholMatrixType m_matrix;
    VectorType m_diag;                                // the diagonal coefficients (LDLT mode)
    VectorXi m_parent;                                // elimination tree
    VectorXi m_nonZerosPerCol;
    PermutationMatrix<Dynamic,Dynamic,Index> m_P;     // the permutation
    PermutationMatrix<Dynamic,Dynamic,Index> m_Pinv;  // the inverse permutation

    RealScalar m_shiftOffset;
    RealScalar m_shiftScale;
};

template<typename _MatrixType, int _UpLo = Lower> class SimplicialLLT;
template<typename _MatrixType, int _UpLo = Lower> class SimplicialLDLT;
template<typename _MatrixType, int _UpLo = Lower> class SimplicialCholesky;

namespace internal {

template<typename _MatrixType, int _UpLo> struct traits<SimplicialLLT<_MatrixType,_UpLo> >
{
  typedef _MatrixType MatrixType;
  enum { UpLo = _UpLo };
  typedef typename MatrixType::Scalar                         Scalar;
  typedef typename MatrixType::Index                          Index;
  typedef SparseMatrix<Scalar, ColMajor, Index>               CholMatrixType;
  typedef SparseTriangularView<CholMatrixType, Eigen::Lower>  MatrixL;
  typedef SparseTriangularView<typename CholMatrixType::AdjointReturnType, Eigen::Upper>   MatrixU;
  static inline MatrixL getL(const MatrixType& m) { return m; }
  static inline MatrixU getU(const MatrixType& m) { return m.adjoint(); }
};

template<typename _MatrixType,int _UpLo> struct traits<SimplicialLDLT<_MatrixType,_UpLo> >
{
  typedef _MatrixType MatrixType;
  enum { UpLo = _UpLo };
  typedef typename MatrixType::Scalar                             Scalar;
  typedef typename MatrixType::Index                              Index;
  typedef SparseMatrix<Scalar, ColMajor, Index>                   CholMatrixType;
  typedef SparseTriangularView<CholMatrixType, Eigen::UnitLower>  MatrixL;
  typedef SparseTriangularView<typename CholMatrixType::AdjointReturnType, Eigen::UnitUpper> MatrixU;
  static inline MatrixL getL(const MatrixType& m) { return m; }
  static inline MatrixU getU(const MatrixType& m) { return m.adjoint(); }
};

template<typename _MatrixType, int _UpLo> struct traits<SimplicialCholesky<_MatrixType,_UpLo> >
{
  typedef _MatrixType MatrixType;
  enum { UpLo = _UpLo };
};

}

/** \ingroup SparseCholesky_Module
  * \class SimplicialLLT
  * \brief A direct sparse LLT Cholesky factorizations
  *
  * This class provides a LL^T Cholesky factorizations of sparse matrices that are
  * selfadjoint and positive definite. The factorization allows for solving A.X = B where
  * X and B can be either dense or sparse.
  *
  * In order to reduce the fill-in, a symmetric permutation P is applied prior to the factorization
  * such that the factorized matrix is P A P^-1.
  *
  * \tparam _MatrixType the type of the sparse matrix A, it must be a SparseMatrix<>
  * \tparam _UpLo the triangular part that will be used for the computations. It can be Lower
  *               or Upper. Default is Lower.
  *
  * \sa class SimplicialLDLT
  */
template<typename _MatrixType, int _UpLo>
    class SimplicialLLT : public SimplicialCholeskyBase<SimplicialLLT<_MatrixType,_UpLo> >
{
public:
    typedef _MatrixType MatrixType;
    enum { UpLo = _UpLo };
    typedef SimplicialCholeskyBase<SimplicialLLT> Base;
    typedef typename MatrixType::Scalar Scalar;
    typedef typename MatrixType::RealScalar RealScalar;
    typedef typename MatrixType::Index Index;
    typedef SparseMatrix<Scalar,ColMajor,Index> CholMatrixType;
    typedef Matrix<Scalar,Dynamic,1> VectorType;
    typedef internal::traits<SimplicialLLT> Traits;
    typedef typename Traits::MatrixL  MatrixL;
    typedef typename Traits::MatrixU  MatrixU;
public:
    /** Default constructor */
    SimplicialLLT() : Base() {}
    /** Constructs and performs the LLT factorization of \a matrix */
    SimplicialLLT(const MatrixType& matrix)
        : Base(matrix) {}

    /** \returns an expression of the factor L */
    inline const MatrixL matrixL() const {
        eigen_assert(Base::m_factorizationIsOk && "Simplicial LLT not factorized");
        return Traits::getL(Base::m_matrix);
    }

    /** \returns an expression of the factor U (= L^*) */
    inline const MatrixU matrixU() const {
        eigen_assert(Base::m_factorizationIsOk && "Simplicial LLT not factorized");
        return Traits::getU(Base::m_matrix);
    }

    /** Computes the sparse Cholesky decomposition of \a matrix */
    SimplicialLLT& compute(const MatrixType& matrix)
    {
      Base::template compute<false>(matrix);
      return *this;
    }

    /** Performs a symbolic decomposition on the sparcity of \a matrix.
      *
      * This function is particularly useful when solving for several problems having the same structure.
      *
      * \sa factorize()
      */
    void analyzePattern(const MatrixType& a)
    {
      Base::analyzePattern(a, false);
    }

    /** Performs a numeric decomposition of \a matrix
      *
      * The given matrix must has the same sparcity than the matrix on which the symbolic decomposition has been performed.
      *
      * \sa analyzePattern()
      */
    void factorize(const MatrixType& a)
    {
      Base::template factorize<false>(a);
    }

    /** \returns the determinant of the underlying matrix from the current factorization */
    Scalar determinant() const
    {
      Scalar detL = Base::m_matrix.diagonal().prod();
      return internal::abs2(detL);
    }
};

/** \ingroup SparseCholesky_Module
  * \class SimplicialLDLT
  * \brief A direct sparse LDLT Cholesky factorizations without square root.
  *
  * This class provides a LDL^T Cholesky factorizations without square root of sparse matrices that are
  * selfadjoint and positive definite. The factorization allows for solving A.X = B where
  * X and B can be either dense or sparse.
  *
  * In order to reduce the fill-in, a symmetric permutation P is applied prior to the factorization
  * such that the factorized matrix is P A P^-1.
  *
  * \tparam _MatrixType the type of the sparse matrix A, it must be a SparseMatrix<>
  * \tparam _UpLo the triangular part that will be used for the computations. It can be Lower
  *               or Upper. Default is Lower.
  *
  * \sa class SimplicialLLT
  */
template<typename _MatrixType, int _UpLo>
    class SimplicialLDLT : public SimplicialCholeskyBase<SimplicialLDLT<_MatrixType,_UpLo> >
{
public:
    typedef _MatrixType MatrixType;
    enum { UpLo = _UpLo };
    typedef SimplicialCholeskyBase<SimplicialLDLT> Base;
    typedef typename MatrixType::Scalar Scalar;
    typedef typename MatrixType::RealScalar RealScalar;
    typedef typename MatrixType::Index Index;
    typedef SparseMatrix<Scalar,ColMajor,Index> CholMatrixType;
    typedef Matrix<Scalar,Dynamic,1> VectorType;
    typedef internal::traits<SimplicialLDLT> Traits;
    typedef typename Traits::MatrixL  MatrixL;
    typedef typename Traits::MatrixU  MatrixU;
public:
    /** Default constructor */
    SimplicialLDLT() : Base() {}

    /** Constructs and performs the LLT factorization of \a matrix */
    SimplicialLDLT(const MatrixType& matrix)
        : Base(matrix) {}

    /** \returns a vector expression of the diagonal D */
    inline const VectorType vectorD() const {
        eigen_assert(Base::m_factorizationIsOk && "Simplicial LDLT not factorized");
        return Base::m_diag;
    }
    /** \returns an expression of the factor L */
    inline const MatrixL matrixL() const {
        eigen_assert(Base::m_factorizationIsOk && "Simplicial LDLT not factorized");
        return Traits::getL(Base::m_matrix);
    }

    /** \returns an expression of the factor U (= L^*) */
    inline const MatrixU matrixU() const {
        eigen_assert(Base::m_factorizationIsOk && "Simplicial LDLT not factorized");
        return Traits::getU(Base::m_matrix);
    }

    /** Computes the sparse Cholesky decomposition of \a matrix */
    SimplicialLDLT& compute(const MatrixType& matrix)
    {
      Base::template compute<true>(matrix);
      return *this;
    }

    /** Performs a symbolic decomposition on the sparcity of \a matrix.
      *
      * This function is particularly useful when solving for several problems having the same structure.
      *
      * \sa factorize()
      */
    void analyzePattern(const MatrixType& a)
    {
      Base::analyzePattern(a, true);
    }

    /** Performs a numeric decomposition of \a matrix
      *
      * The given matrix must has the same sparcity than the matrix on which the symbolic decomposition has been performed.
      *
      * \sa analyzePattern()
      */
    void factorize(const MatrixType& a)
    {
      Base::template factorize<true>(a);
    }

    /** \returns the determinant of the underlying matrix from the current factorization */
    Scalar determinant() const
    {
      return Base::m_diag.prod();
    }
};

/** \deprecated use SimplicialLDLT or class SimplicialLLT
  * \ingroup SparseCholesky_Module
  * \class SimplicialCholesky
  *
  * \sa class SimplicialLDLT, class SimplicialLLT
  */
template<typename _MatrixType, int _UpLo>
    class SimplicialCholesky : public SimplicialCholeskyBase<SimplicialCholesky<_MatrixType,_UpLo> >
{
public:
    typedef _MatrixType MatrixType;
    enum { UpLo = _UpLo };
    typedef SimplicialCholeskyBase<SimplicialCholesky> Base;
    typedef typename MatrixType::Scalar Scalar;
    typedef typename MatrixType::RealScalar RealScalar;
    typedef typename MatrixType::Index Index;
    typedef SparseMatrix<Scalar,ColMajor,Index> CholMatrixType;
    typedef Matrix<Scalar,Dynamic,1> VectorType;
    typedef internal::traits<SimplicialCholesky> Traits;
    typedef internal::traits<SimplicialLDLT<MatrixType,UpLo> > LDLTTraits;
    typedef internal::traits<SimplicialLLT<MatrixType,UpLo>  > LLTTraits;
  public:
    SimplicialCholesky() : Base(), m_LDLT(true) {}

    SimplicialCholesky(const MatrixType& matrix)
      : Base(), m_LDLT(true)
    {
      compute(matrix);
    }

    SimplicialCholesky& setMode(SimplicialCholeskyMode mode)
    {
      switch(mode)
      {
      case SimplicialCholeskyLLT:
        m_LDLT = false;
        break;
      case SimplicialCholeskyLDLT:
        m_LDLT = true;
        break;
      default:
        break;
      }

      return *this;
    }

    inline const VectorType vectorD() const {
        eigen_assert(Base::m_factorizationIsOk && "Simplicial Cholesky not factorized");
        return Base::m_diag;
    }
    inline const CholMatrixType rawMatrix() const {
        eigen_assert(Base::m_factorizationIsOk && "Simplicial Cholesky not factorized");
        return Base::m_matrix;
    }

    /** Computes the sparse Cholesky decomposition of \a matrix */
    SimplicialCholesky& compute(const MatrixType& matrix)
    {
      if(m_LDLT)
        Base::template compute<true>(matrix);
      else
        Base::template compute<false>(matrix);
      return *this;
    }

    /** Performs a symbolic decomposition on the sparcity of \a matrix.
      *
      * This function is particularly useful when solving for several problems having the same structure.
      *
      * \sa factorize()
      */
    void analyzePattern(const MatrixType& a)
    {
      Base::analyzePattern(a, m_LDLT);
    }

    /** Performs a numeric decomposition of \a matrix
      *
      * The given matrix must has the same sparcity than the matrix on which the symbolic decomposition has been performed.
      *
      * \sa analyzePattern()
      */
    void factorize(const MatrixType& a)
    {
      if(m_LDLT)
        Base::template factorize<true>(a);
      else
        Base::template factorize<false>(a);
    }

    /** \internal */
    template<typename Rhs,typename Dest>
    void _solve(const MatrixBase<Rhs> &b, MatrixBase<Dest> &dest) const
    {
      eigen_assert(Base::m_factorizationIsOk && "The decomposition is not in a valid state for solving, you must first call either compute() or symbolic()/numeric()");
      eigen_assert(Base::m_matrix.rows()==b.rows());

      if(Base::m_info!=Success)
        return;

      if(Base::m_P.size()>0)
        dest = Base::m_P * b;
      else
        dest = b;

      if(Base::m_matrix.nonZeros()>0) // otherwise L==I
      {
        if(m_LDLT)
          LDLTTraits::getL(Base::m_matrix).solveInPlace(dest);
        else
          LLTTraits::getL(Base::m_matrix).solveInPlace(dest);
      }

      if(Base::m_diag.size()>0)
        dest = Base::m_diag.asDiagonal().inverse() * dest;

      if (Base::m_matrix.nonZeros()>0) // otherwise I==I
      {
        if(m_LDLT)
          LDLTTraits::getU(Base::m_matrix).solveInPlace(dest);
        else
          LLTTraits::getU(Base::m_matrix).solveInPlace(dest);
      }

      if(Base::m_P.size()>0)
        dest = Base::m_Pinv * dest;
    }

    Scalar determinant() const
    {
      if(m_LDLT)
      {
        return Base::m_diag.prod();
      }
      else
      {
        Scalar detL = Diagonal<const CholMatrixType>(Base::m_matrix).prod();
        return internal::abs2(detL);
      }
    }

  protected:
    bool m_LDLT;
};

template<typename Derived>
void SimplicialCholeskyBase<Derived>::ordering(const MatrixType& a, CholMatrixType& ap)
{
  eigen_assert(a.rows()==a.cols());
  const Index size = a.rows();
  // TODO allows to configure the permutation
  // Note that amd compute the inverse permutation
  {
    CholMatrixType C;
    C = a.template selfadjointView<UpLo>();
    // remove diagonal entries:
    // seems not to be needed
    // C.prune(keep_diag());
    internal::minimum_degree_ordering(C, m_Pinv);
  }

  if(m_Pinv.size()>0)
    m_P = m_Pinv.inverse();
  else
    m_P.resize(0);

  ap.resize(size,size);
  ap.template selfadjointView<Upper>() = a.template selfadjointView<UpLo>().twistedBy(m_P);
}

template<typename Derived>
void SimplicialCholeskyBase<Derived>::analyzePattern_preordered(const CholMatrixType& ap, bool doLDLT)
{
  const Index size = ap.rows();
  m_matrix.resize(size, size);
  m_parent.resize(size);
  m_nonZerosPerCol.resize(size);

  ei_declare_aligned_stack_constructed_variable(Index, tags, size, 0);

  for(Index k = 0; k < size; ++k)
  {
    /* L(k,:) pattern: all nodes reachable in etree from nz in A(0:k-1,k) */
    m_parent[k] = -1;             /* parent of k is not yet known */
    tags[k] = k;                  /* mark node k as visited */
    m_nonZerosPerCol[k] = 0;      /* count of nonzeros in column k of L */
    for(typename CholMatrixType::InnerIterator it(ap,k); it; ++it)
    {
      Index i = it.index();
      if(i < k)
      {
        /* follow path from i to root of etree, stop at flagged node */
        for(; tags[i] != k; i = m_parent[i])
        {
          /* find parent of i if not yet determined */
          if (m_parent[i] == -1)
            m_parent[i] = k;
          m_nonZerosPerCol[i]++;        /* L (k,i) is nonzero */
          tags[i] = k;                  /* mark i as visited */
        }
      }
    }
  }

  /* construct Lp index array from m_nonZerosPerCol column counts */
  Index* Lp = m_matrix.outerIndexPtr();
  Lp[0] = 0;
  for(Index k = 0; k < size; ++k)
    Lp[k+1] = Lp[k] + m_nonZerosPerCol[k] + (doLDLT ? 0 : 1);

  m_matrix.resizeNonZeros(Lp[size]);

  m_isInitialized     = true;
  m_info              = Success;
  m_analysisIsOk      = true;
  m_factorizationIsOk = false;
}


template<typename Derived>
template<bool DoLDLT>
void SimplicialCholeskyBase<Derived>::factorize_preordered(const CholMatrixType& ap)
{
  eigen_assert(m_analysisIsOk && "You must first call analyzePattern()");
  eigen_assert(ap.rows()==ap.cols());
  const Index size = ap.rows();
  eigen_assert(m_parent.size()==size);
  eigen_assert(m_nonZerosPerCol.size()==size);

  const Index* Lp = m_matrix.outerIndexPtr();
  Index* Li = m_matrix.innerIndexPtr();
  Scalar* Lx = m_matrix.valuePtr();

  ei_declare_aligned_stack_constructed_variable(Scalar, y, size, 0);
  ei_declare_aligned_stack_constructed_variable(Index,  pattern, size, 0);
  ei_declare_aligned_stack_constructed_variable(Index,  tags, size, 0);

  bool ok = true;
  m_diag.resize(DoLDLT ? size : 0);

  for(Index k = 0; k < size; ++k)
  {
    // compute nonzero pattern of kth row of L, in topological order
    y[k] = 0.0;                     // Y(0:k) is now all zero
    Index top = size;               // stack for pattern is empty
    tags[k] = k;                    // mark node k as visited
    m_nonZerosPerCol[k] = 0;        // count of nonzeros in column k of L
    for(typename MatrixType::InnerIterator it(ap,k); it; ++it)
    {
      Index i = it.index();
      if(i <= k)
      {
        y[i] += internal::conj(it.value());            /* scatter A(i,k) into Y (sum duplicates) */
        Index len;
        for(len = 0; tags[i] != k; i = m_parent[i])
        {
          pattern[len++] = i;     /* L(k,i) is nonzero */
          tags[i] = k;            /* mark i as visited */
        }
        while(len > 0)
          pattern[--top] = pattern[--len];
      }
    }

    /* compute numerical values kth row of L (a sparse triangular solve) */

    RealScalar d = internal::real(y[k]) * m_shiftScale + m_shiftOffset;    // get D(k,k), apply the shift function, and clear Y(k)
    y[k] = 0.0;
    for(; top < size; ++top)
    {
      Index i = pattern[top];       /* pattern[top:n-1] is pattern of L(:,k) */
      Scalar yi = y[i];             /* get and clear Y(i) */
      y[i] = 0.0;

      /* the nonzero entry L(k,i) */
      Scalar l_ki;
      if(DoLDLT)
        l_ki = yi / m_diag[i];
      else
        yi = l_ki = yi / Lx[Lp[i]];

      Index p2 = Lp[i] + m_nonZerosPerCol[i];
      Index p;
      for(p = Lp[i] + (DoLDLT ? 0 : 1); p < p2; ++p)
        y[Li[p]] -= internal::conj(Lx[p]) * yi;
      d -= internal::real(l_ki * internal::conj(yi));
      Li[p] = k;                          /* store L(k,i) in column form of L */
      Lx[p] = l_ki;
      ++m_nonZerosPerCol[i];              /* increment count of nonzeros in col i */
    }
    if(DoLDLT)
    {
      m_diag[k] = d;
      if(d == RealScalar(0))
      {
        ok = false;                         /* failure, D(k,k) is zero */
        break;
      }
    }
    else
    {
      Index p = Lp[k] + m_nonZerosPerCol[k]++;
      Li[p] = k ;                /* store L(k,k) = sqrt (d) in column k */
      if(d <= RealScalar(0)) {
        ok = false;              /* failure, matrix is not positive definite */
        break;
      }
      Lx[p] = internal::sqrt(d) ;
    }
  }

  m_info = ok ? Success : NumericalIssue;
  m_factorizationIsOk = true;
}

namespace internal {

template<typename Derived, typename Rhs>
struct solve_retval<SimplicialCholeskyBase<Derived>, Rhs>
  : solve_retval_base<SimplicialCholeskyBase<Derived>, Rhs>
{
  typedef SimplicialCholeskyBase<Derived> Dec;
  EIGEN_MAKE_SOLVE_HELPERS(Dec,Rhs)

  template<typename Dest> void evalTo(Dest& dst) const
  {
    dec().derived()._solve(rhs(),dst);
  }
};

template<typename Derived, typename Rhs>
struct sparse_solve_retval<SimplicialCholeskyBase<Derived>, Rhs>
  : sparse_solve_retval_base<SimplicialCholeskyBase<Derived>, Rhs>
{
  typedef SimplicialCholeskyBase<Derived> Dec;
  EIGEN_MAKE_SPARSE_SOLVE_HELPERS(Dec,Rhs)

  template<typename Dest> void evalTo(Dest& dst) const
  {
    dec().derived()._solve_sparse(rhs(),dst);
  }
};

} // end namespace internal

} // end namespace Eigen

#endif // EIGEN_SIMPLICIAL_CHOLESKY_H