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