Eigen/src/IterativeLinearSolvers/IncompleteCholesky.h - eigen - Git at Google

 // This file is part of Eigen, a lightweight C++ template library
 // for linear algebra.
 //
 // Copyright (C) 2012 Désiré Nuentsa-Wakam <desire.nuentsa_wakam@inria.fr>
 // Copyright (C) 2015 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_INCOMPLETE_CHOlESKY_H
 #define EIGEN_INCOMPLETE_CHOlESKY_H

 #include <vector>
 #include <list>

 namespace Eigen {
 /**
   * \brief Modified Incomplete Cholesky with dual threshold
   *
   * References : C-J. Lin and J. J. Moré, Incomplete Cholesky Factorizations with
   *              Limited memory, SIAM J. Sci. Comput.  21(1), pp. 24-45, 1999
   *
   * \tparam Scalar the scalar type of the input matrices
   * \tparam _UpLo The triangular part that will be used for the computations. It can be Lower
     *               or Upper. Default is Lower.
   * \tparam _OrderingType The ordering method to use, either AMDOrdering<> or NaturalOrdering<>. Default is AMDOrdering<int>,
   *                       unless EIGEN_MPL2_ONLY is defined, in which case the default is NaturalOrdering<int>.
   *
   * \implsparsesolverconcept
   *
   * It performs the following incomplete factorization: \f$ S P A P' S \approx L L' \f$
   * where L is a lower triangular factor, S is a diagonal scaling matrix, and P is a
   * fill-in reducing permutation as computed by the ordering method.
   *
   * \b Shifting \b strategy: Let \f$ B = S P A P' S \f$  be the scaled matrix on which the factorization is carried out,
   * and \f$ \beta \f$ be the minimum value of the diagonal. If \f$ \beta > 0 \f$ then, the factorization is directly performed
   * on the matrix B. Otherwise, the factorization is performed on the shifted matrix \f$ B + (\sigma+|\beta| I \f$ where
   * \f$ \sigma \f$ is the initial shift value as returned and set by setInitialShift() method. The default value is \f$ \sigma = 10^{-3} \f$.
   * If the factorization fails, then the shift in doubled until it succeed or a maximum of ten attempts. If it still fails, as returned by
   * the info() method, then you can either increase the initial shift, or better use another preconditioning technique.
   *
   */
 template <typename Scalar, int _UpLo = Lower, typename _OrderingType = AMDOrdering<int> >
 class IncompleteCholesky : public SparseSolverBase<IncompleteCholesky<Scalar,_UpLo,_OrderingType> >
 {
   protected:
     typedef SparseSolverBase<IncompleteCholesky<Scalar,_UpLo,_OrderingType> > Base;
     using Base::m_isInitialized;
   public:
     typedef typename NumTraits<Scalar>::Real RealScalar;
     typedef _OrderingType OrderingType;
     typedef typename OrderingType::PermutationType PermutationType;
     typedef typename PermutationType::StorageIndex StorageIndex;
     typedef SparseMatrix<Scalar,ColMajor,StorageIndex> FactorType;
     typedef Matrix<Scalar,Dynamic,1> VectorSx;
     typedef Matrix<RealScalar,Dynamic,1> VectorRx;
     typedef Matrix<StorageIndex,Dynamic, 1> VectorIx;
     typedef std::vector<std::list<StorageIndex> > VectorList;
     enum { UpLo = _UpLo };
     enum {
       ColsAtCompileTime = Dynamic,
       MaxColsAtCompileTime = Dynamic
     };
   public:

     /** Default constructor leaving the object in a partly non-initialized stage.
       *
       * You must call compute() or the pair analyzePattern()/factorize() to make it valid.
       *
       * \sa IncompleteCholesky(const MatrixType&)
       */
     IncompleteCholesky() : m_initialShift(1e-3),m_analysisIsOk(false),m_factorizationIsOk(false) {}

     /** Constructor computing the incomplete factorization for the given matrix \a matrix.
       */
     template<typename MatrixType>
     IncompleteCholesky(const MatrixType& matrix) : m_initialShift(1e-3),m_analysisIsOk(false),m_factorizationIsOk(false)
     {
       compute(matrix);
     }

     /** \returns number of rows of the factored matrix */
     Index rows() const { return m_L.rows(); }

     /** \returns number of columns of the factored matrix */
     Index cols() const { return m_L.cols(); }


     /** \brief Reports whether previous computation was successful.
       *
       * It triggers an assertion if \c *this has not been initialized through the respective constructor,
       * or a call to compute() or analyzePattern().
       *
       * \returns \c Success if computation was successful,
       *          \c NumericalIssue if the matrix appears to be negative.
       */
     ComputationInfo info() const
     {
       eigen_assert(m_isInitialized && "IncompleteCholesky is not initialized.");
       return m_info;
     }

     /** \brief Set the initial shift parameter \f$ \sigma \f$.
       */
     void setInitialShift(RealScalar shift) { m_initialShift = shift; }

     /** \brief Computes the fill reducing permutation vector using the sparsity pattern of \a mat
       */
     template<typename MatrixType>
     void analyzePattern(const MatrixType& mat)
     {
       OrderingType ord;
       PermutationType pinv;
       ord(mat.template selfadjointView<UpLo>(), pinv);
       if(pinv.size()>0) m_perm = pinv.inverse();
       else              m_perm.resize(0);
       m_L.resize(mat.rows(), mat.cols());
       m_analysisIsOk = true;
       m_isInitialized = true;
       m_info = Success;
     }

     /** \brief Performs the numerical factorization of the input matrix \a mat
       *
       * The method analyzePattern() or compute() must have been called beforehand
       * with a matrix having the same pattern.
       *
       * \sa compute(), analyzePattern()
       */
     template<typename MatrixType>
     void factorize(const MatrixType& mat);

     /** Computes or re-computes the incomplete Cholesky factorization of the input matrix \a mat
       *
       * It is a shortcut for a sequential call to the analyzePattern() and factorize() methods.
       *
       * \sa analyzePattern(), factorize()
       */
     template<typename MatrixType>
     void compute(const MatrixType& mat)
     {
       analyzePattern(mat);
       factorize(mat);
     }

     // internal
     template<typename Rhs, typename Dest>
     void _solve_impl(const Rhs& b, Dest& x) const
     {
       eigen_assert(m_factorizationIsOk && "factorize() should be called first");
       if (m_perm.rows() == b.rows())  x = m_perm * b;
       else                            x = b;
       x = m_scale.asDiagonal() * x;
       x = m_L.template triangularView<Lower>().solve(x);
       x = m_L.adjoint().template triangularView<Upper>().solve(x);
       x = m_scale.asDiagonal() * x;
       if (m_perm.rows() == b.rows())
         x = m_perm.inverse() * x;
     }

     /** \returns the sparse lower triangular factor L */
     const FactorType& matrixL() const { eigen_assert("m_factorizationIsOk"); return m_L; }

     /** \returns a vector representing the scaling factor S */
     const VectorRx& scalingS() const { eigen_assert("m_factorizationIsOk"); return m_scale; }

     /** \returns the fill-in reducing permutation P (can be empty for a natural ordering) */
     const PermutationType& permutationP() const { eigen_assert("m_analysisIsOk"); return m_perm; }

   protected:
     FactorType m_L;              // The lower part stored in CSC
     VectorRx m_scale;            // The vector for scaling the matrix
     RealScalar m_initialShift;   // The initial shift parameter
     bool m_analysisIsOk;
     bool m_factorizationIsOk;
     ComputationInfo m_info;
     PermutationType m_perm;

   private:
     inline void updateList(Ref<const VectorIx> colPtr, Ref<VectorIx> rowIdx, Ref<VectorSx> vals, const Index& col, const Index& jk, VectorIx& firstElt, VectorList& listCol);
 };

 // Based on the following paper:
 //   C-J. Lin and J. J. Moré, Incomplete Cholesky Factorizations with
 //   Limited memory, SIAM J. Sci. Comput.  21(1), pp. 24-45, 1999
 //   http://ftp.mcs.anl.gov/pub/tech_reports/reports/P682.pdf
 template<typename Scalar, int _UpLo, typename OrderingType>
 template<typename _MatrixType>
 void IncompleteCholesky<Scalar,_UpLo, OrderingType>::factorize(const _MatrixType& mat)
 {
   using std::sqrt;
   eigen_assert(m_analysisIsOk && "analyzePattern() should be called first");

   // Dropping strategy : Keep only the p largest elements per column, where p is the number of elements in the column of the original matrix. Other strategies will be added

   // Apply the fill-reducing permutation computed in analyzePattern()
   if (m_perm.rows() == mat.rows() ) // To detect the null permutation
   {
     // The temporary is needed to make sure that the diagonal entry is properly sorted
     FactorType tmp(mat.rows(), mat.cols());
     tmp = mat.template selfadjointView<_UpLo>().twistedBy(m_perm);
     m_L.template selfadjointView<Lower>() = tmp.template selfadjointView<Lower>();
   }
   else
   {
     m_L.template selfadjointView<Lower>() = mat.template selfadjointView<_UpLo>();
   }

   Index n = m_L.cols();
   Index nnz = m_L.nonZeros();
   Map<VectorSx> vals(m_L.valuePtr(), nnz);         //values
   Map<VectorIx> rowIdx(m_L.innerIndexPtr(), nnz);  //Row indices
   Map<VectorIx> colPtr( m_L.outerIndexPtr(), n+1); // Pointer to the beginning of each row
   VectorIx firstElt(n-1); // for each j, points to the next entry in vals that will be used in the factorization
   VectorList listCol(n);  // listCol(j) is a linked list of columns to update column j
   VectorSx col_vals(n);   // Store a  nonzero values in each column
   VectorIx col_irow(n);   // Row indices of nonzero elements in each column
   VectorIx col_pattern(n);
   col_pattern.fill(-1);
   StorageIndex col_nnz;


   // Computes the scaling factors
   m_scale.resize(n);
   m_scale.setZero();
   for (Index j = 0; j < n; j++)
     for (Index k = colPtr[j]; k < colPtr[j+1]; k++)
     {
       m_scale(j) += numext::abs2(vals(k));
       if(rowIdx[k]!=j)
         m_scale(rowIdx[k]) += numext::abs2(vals(k));
     }

   m_scale = m_scale.cwiseSqrt().cwiseSqrt();

   for (Index j = 0; j < n; ++j)
     if(m_scale(j)>(std::numeric_limits<RealScalar>::min)())
       m_scale(j) = RealScalar(1)/m_scale(j);
     else
       m_scale(j) = 1;

   // TODO disable scaling if not needed, i.e., if it is roughly uniform? (this will make solve() faster)

   // Scale and compute the shift for the matrix
   RealScalar mindiag = NumTraits<RealScalar>::highest();
   for (Index j = 0; j < n; j++)
   {
     for (Index k = colPtr[j]; k < colPtr[j+1]; k++)
       vals[k] *= (m_scale(j)*m_scale(rowIdx[k]));
     eigen_internal_assert(rowIdx[colPtr[j]]==j && "IncompleteCholesky: only the lower triangular part must be stored");
     mindiag = numext::mini(numext::real(vals[colPtr[j]]), mindiag);
   }

   FactorType L_save = m_L;

   RealScalar shift = 0;
   if(mindiag <= RealScalar(0.))
     shift = m_initialShift - mindiag;

   m_info = NumericalIssue;

   // Try to perform the incomplete factorization using the current shift
   int iter = 0;
   do
   {
     // Apply the shift to the diagonal elements of the matrix
     for (Index j = 0; j < n; j++)
       vals[colPtr[j]] += shift;

     // jki version of the Cholesky factorization
     Index j=0;
     for (; j < n; ++j)
     {
       // Left-looking factorization of the j-th column
       // First, load the j-th column into col_vals
       Scalar diag = vals[colPtr[j]];  // It is assumed that only the lower part is stored
       col_nnz = 0;
       for (Index i = colPtr[j] + 1; i < colPtr[j+1]; i++)
       {
         StorageIndex l = rowIdx[i];
         col_vals(col_nnz) = vals[i];
         col_irow(col_nnz) = l;
         col_pattern(l) = col_nnz;
         col_nnz++;
       }
       {
         typename std::list<StorageIndex>::iterator k;
         // Browse all previous columns that will update column j
         for(k = listCol[j].begin(); k != listCol[j].end(); k++)
         {
           Index jk = firstElt(*k); // First element to use in the column
           eigen_internal_assert(rowIdx[jk]==j);
           Scalar v_j_jk = numext::conj(vals[jk]);

           jk += 1;
           for (Index i = jk; i < colPtr[*k+1]; i++)
           {
             StorageIndex l = rowIdx[i];
             if(col_pattern[l]<0)
             {
               col_vals(col_nnz) = vals[i] * v_j_jk;
               col_irow[col_nnz] = l;
               col_pattern(l) = col_nnz;
               col_nnz++;
             }
             else
               col_vals(col_pattern[l]) -= vals[i] * v_j_jk;
           }
           updateList(colPtr,rowIdx,vals, *k, jk, firstElt, listCol);
         }
       }

       // Scale the current column
       if(numext::real(diag) <= 0)
       {
         if(++iter>=10)
           return;

         // increase shift
         shift = numext::maxi(m_initialShift,RealScalar(2)*shift);
         // restore m_L, col_pattern, and listCol
         vals = Map<const VectorSx>(L_save.valuePtr(), nnz);
         rowIdx = Map<const VectorIx>(L_save.innerIndexPtr(), nnz);
         colPtr = Map<const VectorIx>(L_save.outerIndexPtr(), n+1);
         col_pattern.fill(-1);
         for(Index i=0; i<n; ++i)
           listCol[i].clear();

         break;
       }

       RealScalar rdiag = sqrt(numext::real(diag));
       vals[colPtr[j]] = rdiag;
       for (Index k = 0; k<col_nnz; ++k)
       {
         Index i = col_irow[k];
         //Scale
         col_vals(k) /= rdiag;
         //Update the remaining diagonals with col_vals
         vals[colPtr[i]] -= numext::abs2(col_vals(k));
       }
       // Select the largest p elements
       // p is the original number of elements in the column (without the diagonal)
       Index p = colPtr[j+1] - colPtr[j] - 1 ;
       Ref<VectorSx> cvals = col_vals.head(col_nnz);
       Ref<VectorIx> cirow = col_irow.head(col_nnz);
       internal::QuickSplit(cvals,cirow, p);
       // Insert the largest p elements in the matrix
       Index cpt = 0;
       for (Index i = colPtr[j]+1; i < colPtr[j+1]; i++)
       {
         vals[i] = col_vals(cpt);
         rowIdx[i] = col_irow(cpt);
         // restore col_pattern:
         col_pattern(col_irow(cpt)) = -1;
         cpt++;
       }
       // Get the first smallest row index and put it after the diagonal element
       Index jk = colPtr(j)+1;
       updateList(colPtr,rowIdx,vals,j,jk,firstElt,listCol);
     }

     if(j==n)
     {
       m_factorizationIsOk = true;
       m_info = Success;
     }
   } while(m_info!=Success);
 }

 template<typename Scalar, int _UpLo, typename OrderingType>
 inline void IncompleteCholesky<Scalar,_UpLo, OrderingType>::updateList(Ref<const VectorIx> colPtr, Ref<VectorIx> rowIdx, Ref<VectorSx> vals, const Index& col, const Index& jk, VectorIx& firstElt, VectorList& listCol)
 {
   if (jk < colPtr(col+1) )
   {
     Index p = colPtr(col+1) - jk;
     Index minpos;
     rowIdx.segment(jk,p).minCoeff(&minpos);
     minpos += jk;
     if (rowIdx(minpos) != rowIdx(jk))
     {
       //Swap
       std::swap(rowIdx(jk),rowIdx(minpos));
       std::swap(vals(jk),vals(minpos));
     }
     firstElt(col) = internal::convert_index<StorageIndex,Index>(jk);
     listCol[rowIdx(jk)].push_back(internal::convert_index<StorageIndex,Index>(col));
   }
 }

 } // end namespace Eigen

 #endif
	// This file is part of Eigen, a lightweight C++ template library
	// for linear algebra.
	//
	// Copyright (C) 2012 Désiré Nuentsa-Wakam <desire.nuentsa_wakam@inria.fr>
	// Copyright (C) 2015 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_INCOMPLETE_CHOlESKY_H
	#define EIGEN_INCOMPLETE_CHOlESKY_H

	#include <vector>
	#include <list>

	namespace Eigen {
	/**
	* \brief Modified Incomplete Cholesky with dual threshold
	*
	* References : C-J. Lin and J. J. Moré, Incomplete Cholesky Factorizations with
	* Limited memory, SIAM J. Sci. Comput. 21(1), pp. 24-45, 1999
	*
	* \tparam Scalar the scalar type of the input matrices
	* \tparam _UpLo The triangular part that will be used for the computations. It can be Lower
	* or Upper. Default is Lower.
	* \tparam _OrderingType The ordering method to use, either AMDOrdering<> or NaturalOrdering<>. Default is AMDOrdering<int>,
	* unless EIGEN_MPL2_ONLY is defined, in which case the default is NaturalOrdering<int>.
	*
	* \implsparsesolverconcept
	*
	* It performs the following incomplete factorization: \f$ S P A P' S \approx L L' \f$
	* where L is a lower triangular factor, S is a diagonal scaling matrix, and P is a
	* fill-in reducing permutation as computed by the ordering method.
	*
	* \b Shifting \b strategy: Let \f$ B = S P A P' S \f$ be the scaled matrix on which the factorization is carried out,
	* and \f$ \beta \f$ be the minimum value of the diagonal. If \f$ \beta > 0 \f$ then, the factorization is directly performed
	* on the matrix B. Otherwise, the factorization is performed on the shifted matrix \f$ B + (\sigma+\|\beta\| I \f$ where
	* \f$ \sigma \f$ is the initial shift value as returned and set by setInitialShift() method. The default value is \f$ \sigma = 10^{-3} \f$.
	* If the factorization fails, then the shift in doubled until it succeed or a maximum of ten attempts. If it still fails, as returned by
	* the info() method, then you can either increase the initial shift, or better use another preconditioning technique.
	*
	*/
	template <typename Scalar, int _UpLo = Lower, typename _OrderingType = AMDOrdering<int> >
	class IncompleteCholesky : public SparseSolverBase<IncompleteCholesky<Scalar,_UpLo,_OrderingType> >
	{
	protected:
	typedef SparseSolverBase<IncompleteCholesky<Scalar,_UpLo,_OrderingType> > Base;
	using Base::m_isInitialized;
	public:
	typedef typename NumTraits<Scalar>::Real RealScalar;
	typedef _OrderingType OrderingType;
	typedef typename OrderingType::PermutationType PermutationType;
	typedef typename PermutationType::StorageIndex StorageIndex;
	typedef SparseMatrix<Scalar,ColMajor,StorageIndex> FactorType;
	typedef Matrix<Scalar,Dynamic,1> VectorSx;
	typedef Matrix<RealScalar,Dynamic,1> VectorRx;
	typedef Matrix<StorageIndex,Dynamic, 1> VectorIx;
	typedef std::vector<std::list<StorageIndex> > VectorList;
	enum { UpLo = _UpLo };
	enum {
	ColsAtCompileTime = Dynamic,
	MaxColsAtCompileTime = Dynamic
	};
	public:

	/** Default constructor leaving the object in a partly non-initialized stage.
	*
	* You must call compute() or the pair analyzePattern()/factorize() to make it valid.
	*
	* \sa IncompleteCholesky(const MatrixType&)
	*/
	IncompleteCholesky() : m_initialShift(1e-3),m_analysisIsOk(false),m_factorizationIsOk(false) {}

	/** Constructor computing the incomplete factorization for the given matrix \a matrix.
	*/
	template<typename MatrixType>
	IncompleteCholesky(const MatrixType& matrix) : m_initialShift(1e-3),m_analysisIsOk(false),m_factorizationIsOk(false)
	{
	compute(matrix);
	}

	/** \returns number of rows of the factored matrix */
	Index rows() const { return m_L.rows(); }

	/** \returns number of columns of the factored matrix */
	Index cols() const { return m_L.cols(); }


	/** \brief Reports whether previous computation was successful.
	*
	* It triggers an assertion if \c *this has not been initialized through the respective constructor,
	* or a call to compute() or analyzePattern().
	*
	* \returns \c Success if computation was successful,
	* \c NumericalIssue if the matrix appears to be negative.
	*/
	ComputationInfo info() const
	{
	eigen_assert(m_isInitialized && "IncompleteCholesky is not initialized.");
	return m_info;
	}

	/** \brief Set the initial shift parameter \f$ \sigma \f$.
	*/
	void setInitialShift(RealScalar shift) { m_initialShift = shift; }

	/** \brief Computes the fill reducing permutation vector using the sparsity pattern of \a mat
	*/
	template<typename MatrixType>
	void analyzePattern(const MatrixType& mat)
	{
	OrderingType ord;
	PermutationType pinv;
	ord(mat.template selfadjointView<UpLo>(), pinv);
	if(pinv.size()>0) m_perm = pinv.inverse();
	else m_perm.resize(0);
	m_L.resize(mat.rows(), mat.cols());
	m_analysisIsOk = true;
	m_isInitialized = true;
	m_info = Success;
	}

	/** \brief Performs the numerical factorization of the input matrix \a mat
	*
	* The method analyzePattern() or compute() must have been called beforehand
	* with a matrix having the same pattern.
	*
	* \sa compute(), analyzePattern()
	*/
	template<typename MatrixType>
	void factorize(const MatrixType& mat);

	/** Computes or re-computes the incomplete Cholesky factorization of the input matrix \a mat
	*
	* It is a shortcut for a sequential call to the analyzePattern() and factorize() methods.
	*
	* \sa analyzePattern(), factorize()
	*/
	template<typename MatrixType>
	void compute(const MatrixType& mat)
	{
	analyzePattern(mat);
	factorize(mat);
	}

	// internal
	template<typename Rhs, typename Dest>
	void _solve_impl(const Rhs& b, Dest& x) const
	{
	eigen_assert(m_factorizationIsOk && "factorize() should be called first");
	if (m_perm.rows() == b.rows()) x = m_perm * b;
	else x = b;
	x = m_scale.asDiagonal() * x;
	x = m_L.template triangularView<Lower>().solve(x);
	x = m_L.adjoint().template triangularView<Upper>().solve(x);
	x = m_scale.asDiagonal() * x;
	if (m_perm.rows() == b.rows())
	x = m_perm.inverse() * x;
	}

	/** \returns the sparse lower triangular factor L */
	const FactorType& matrixL() const { eigen_assert("m_factorizationIsOk"); return m_L; }

	/** \returns a vector representing the scaling factor S */
	const VectorRx& scalingS() const { eigen_assert("m_factorizationIsOk"); return m_scale; }

	/** \returns the fill-in reducing permutation P (can be empty for a natural ordering) */
	const PermutationType& permutationP() const { eigen_assert("m_analysisIsOk"); return m_perm; }

	protected:
	FactorType m_L; // The lower part stored in CSC
	VectorRx m_scale; // The vector for scaling the matrix
	RealScalar m_initialShift; // The initial shift parameter
	bool m_analysisIsOk;
	bool m_factorizationIsOk;
	ComputationInfo m_info;
	PermutationType m_perm;

	private:
	inline void updateList(Ref<const VectorIx> colPtr, Ref<VectorIx> rowIdx, Ref<VectorSx> vals, const Index& col, const Index& jk, VectorIx& firstElt, VectorList& listCol);
	};

	// Based on the following paper:
	// C-J. Lin and J. J. Moré, Incomplete Cholesky Factorizations with
	// Limited memory, SIAM J. Sci. Comput. 21(1), pp. 24-45, 1999
	// http://ftp.mcs.anl.gov/pub/tech_reports/reports/P682.pdf
	template<typename Scalar, int _UpLo, typename OrderingType>
	template<typename _MatrixType>
	void IncompleteCholesky<Scalar,_UpLo, OrderingType>::factorize(const _MatrixType& mat)
	{
	using std::sqrt;
	eigen_assert(m_analysisIsOk && "analyzePattern() should be called first");

	// Dropping strategy : Keep only the p largest elements per column, where p is the number of elements in the column of the original matrix. Other strategies will be added

	// Apply the fill-reducing permutation computed in analyzePattern()
	if (m_perm.rows() == mat.rows() ) // To detect the null permutation
	{
	// The temporary is needed to make sure that the diagonal entry is properly sorted
	FactorType tmp(mat.rows(), mat.cols());
	tmp = mat.template selfadjointView<_UpLo>().twistedBy(m_perm);
	m_L.template selfadjointView<Lower>() = tmp.template selfadjointView<Lower>();
	}
	else
	{
	m_L.template selfadjointView<Lower>() = mat.template selfadjointView<_UpLo>();
	}

	Index n = m_L.cols();
	Index nnz = m_L.nonZeros();
	Map<VectorSx> vals(m_L.valuePtr(), nnz); //values
	Map<VectorIx> rowIdx(m_L.innerIndexPtr(), nnz); //Row indices
	Map<VectorIx> colPtr( m_L.outerIndexPtr(), n+1); // Pointer to the beginning of each row
	VectorIx firstElt(n-1); // for each j, points to the next entry in vals that will be used in the factorization
	VectorList listCol(n); // listCol(j) is a linked list of columns to update column j
	VectorSx col_vals(n); // Store a nonzero values in each column
	VectorIx col_irow(n); // Row indices of nonzero elements in each column
	VectorIx col_pattern(n);
	col_pattern.fill(-1);
	StorageIndex col_nnz;


	// Computes the scaling factors
	m_scale.resize(n);
	m_scale.setZero();
	for (Index j = 0; j < n; j++)
	for (Index k = colPtr[j]; k < colPtr[j+1]; k++)
	{
	m_scale(j) += numext::abs2(vals(k));
	if(rowIdx[k]!=j)
	m_scale(rowIdx[k]) += numext::abs2(vals(k));
	}

	m_scale = m_scale.cwiseSqrt().cwiseSqrt();

	for (Index j = 0; j < n; ++j)
	if(m_scale(j)>(std::numeric_limits<RealScalar>::min)())
	m_scale(j) = RealScalar(1)/m_scale(j);
	else
	m_scale(j) = 1;

	// TODO disable scaling if not needed, i.e., if it is roughly uniform? (this will make solve() faster)

	// Scale and compute the shift for the matrix
	RealScalar mindiag = NumTraits<RealScalar>::highest();
	for (Index j = 0; j < n; j++)
	{
	for (Index k = colPtr[j]; k < colPtr[j+1]; k++)
	vals[k] = (m_scale(j)m_scale(rowIdx[k]));
	eigen_internal_assert(rowIdx[colPtr[j]]==j && "IncompleteCholesky: only the lower triangular part must be stored");
	mindiag = numext::mini(numext::real(vals[colPtr[j]]), mindiag);
	}

	FactorType L_save = m_L;

	RealScalar shift = 0;
	if(mindiag <= RealScalar(0.))
	shift = m_initialShift - mindiag;

	m_info = NumericalIssue;

	// Try to perform the incomplete factorization using the current shift
	int iter = 0;
	do
	{
	// Apply the shift to the diagonal elements of the matrix
	for (Index j = 0; j < n; j++)
	vals[colPtr[j]] += shift;

	// jki version of the Cholesky factorization
	Index j=0;
	for (; j < n; ++j)
	{
	// Left-looking factorization of the j-th column
	// First, load the j-th column into col_vals
	Scalar diag = vals[colPtr[j]]; // It is assumed that only the lower part is stored
	col_nnz = 0;
	for (Index i = colPtr[j] + 1; i < colPtr[j+1]; i++)
	{
	StorageIndex l = rowIdx[i];
	col_vals(col_nnz) = vals[i];
	col_irow(col_nnz) = l;
	col_pattern(l) = col_nnz;
	col_nnz++;
	}
	{
	typename std::list<StorageIndex>::iterator k;
	// Browse all previous columns that will update column j
	for(k = listCol[j].begin(); k != listCol[j].end(); k++)
	{
	Index jk = firstElt(*k); // First element to use in the column
	eigen_internal_assert(rowIdx[jk]==j);
	Scalar v_j_jk = numext::conj(vals[jk]);

	jk += 1;
	for (Index i = jk; i < colPtr[*k+1]; i++)
	{
	StorageIndex l = rowIdx[i];
	if(col_pattern[l]<0)
	{
	col_vals(col_nnz) = vals[i] * v_j_jk;
	col_irow[col_nnz] = l;
	col_pattern(l) = col_nnz;
	col_nnz++;
	}
	else
	col_vals(col_pattern[l]) -= vals[i] * v_j_jk;
	}
	updateList(colPtr,rowIdx,vals, *k, jk, firstElt, listCol);
	}
	}

	// Scale the current column
	if(numext::real(diag) <= 0)
	{
	if(++iter>=10)
	return;

	// increase shift
	shift = numext::maxi(m_initialShift,RealScalar(2)*shift);
	// restore m_L, col_pattern, and listCol
	vals = Map<const VectorSx>(L_save.valuePtr(), nnz);
	rowIdx = Map<const VectorIx>(L_save.innerIndexPtr(), nnz);
	colPtr = Map<const VectorIx>(L_save.outerIndexPtr(), n+1);
	col_pattern.fill(-1);
	for(Index i=0; i<n; ++i)
	listCol[i].clear();

	break;
	}

	RealScalar rdiag = sqrt(numext::real(diag));
	vals[colPtr[j]] = rdiag;
	for (Index k = 0; k<col_nnz; ++k)
	{
	Index i = col_irow[k];
	//Scale
	col_vals(k) /= rdiag;
	//Update the remaining diagonals with col_vals
	vals[colPtr[i]] -= numext::abs2(col_vals(k));
	}
	// Select the largest p elements
	// p is the original number of elements in the column (without the diagonal)
	Index p = colPtr[j+1] - colPtr[j] - 1 ;
	Ref<VectorSx> cvals = col_vals.head(col_nnz);
	Ref<VectorIx> cirow = col_irow.head(col_nnz);
	internal::QuickSplit(cvals,cirow, p);
	// Insert the largest p elements in the matrix
	Index cpt = 0;
	for (Index i = colPtr[j]+1; i < colPtr[j+1]; i++)
	{
	vals[i] = col_vals(cpt);
	rowIdx[i] = col_irow(cpt);
	// restore col_pattern:
	col_pattern(col_irow(cpt)) = -1;
	cpt++;
	}
	// Get the first smallest row index and put it after the diagonal element
	Index jk = colPtr(j)+1;
	updateList(colPtr,rowIdx,vals,j,jk,firstElt,listCol);
	}

	if(j==n)
	{
	m_factorizationIsOk = true;
	m_info = Success;
	}
	} while(m_info!=Success);
	}

	template<typename Scalar, int _UpLo, typename OrderingType>
	inline void IncompleteCholesky<Scalar,_UpLo, OrderingType>::updateList(Ref<const VectorIx> colPtr, Ref<VectorIx> rowIdx, Ref<VectorSx> vals, const Index& col, const Index& jk, VectorIx& firstElt, VectorList& listCol)
	{
	if (jk < colPtr(col+1) )
	{
	Index p = colPtr(col+1) - jk;
	Index minpos;
	rowIdx.segment(jk,p).minCoeff(&minpos);
	minpos += jk;
	if (rowIdx(minpos) != rowIdx(jk))
	{
	//Swap
	std::swap(rowIdx(jk),rowIdx(minpos));
	std::swap(vals(jk),vals(minpos));
	}
	firstElt(col) = internal::convert_index<StorageIndex,Index>(jk);
	listCol[rowIdx(jk)].push_back(internal::convert_index<StorageIndex,Index>(col));
	}
	}

	} // end namespace Eigen

	#endif