Incomplete Cholesky preconditioner... not yet stable

Eigen/src/IterativeLinearSolvers/IncompleteLUT.h

@@ -10,8 +10,56 @@
 #ifndef EIGEN_INCOMPLETE_LUT_H
 #define EIGEN_INCOMPLETE_LUT_H
 
+
 namespace Eigen { 
 
+namespace internal {
+    
+/**
+ * Compute a quick-sort split of a vector 
+ * On output, the vector row is permuted such that its elements satisfy
+ * abs(row(i)) >= abs(row(ncut)) if i<ncut
+ * abs(row(i)) <= abs(row(ncut)) if i>ncut 
+ * \param row The vector of values
+ * \param ind The array of index for the elements in @p row
+ * \param ncut  The number of largest elements to keep
+ **/ 
+template <typename VectorV, typename VectorI>
+int QuickSplit(VectorV &row, VectorI &ind, int ncut)
+{
+  typedef typename VectorV::RealScalar RealScalar;
+  using std::swap;
+  int mid;
+  int n = row.size(); /* length of the vector */
+  int first, last ; 
+  
+  ncut--; /* to fit the zero-based indices */
+  first = 0; 
+  last = n-1; 
+  if (ncut < first || ncut > last ) return 0;
+  
+  do {
+    mid = first; 
+    RealScalar abskey = std::abs(row(mid)); 
+    for (int j = first + 1; j <= last; j++) {
+      if ( std::abs(row(j)) > abskey) {
+        ++mid;
+        swap(row(mid), row(j));
+        swap(ind(mid), ind(j));
+      }
+    }
+    /* Interchange for the pivot element */
+    swap(row(mid), row(first));
+    swap(ind(mid), ind(first));
+    
+    if (mid > ncut) last = mid - 1;
+    else if (mid < ncut ) first = mid + 1; 
+  } while (mid != ncut );
+  
+  return 0; /* mid is equal to ncut */ 
+}
+
+}// end namespace internal
 /**
  * \brief Incomplete LU factorization with dual-threshold strategy
  * During the numerical factorization, two dropping rules are used :
@@ -126,10 +174,6 @@ class IncompleteLUT : internal::noncopyable
 
 protected:
 
-    template <typename VectorV, typename VectorI>
-    int QuickSplit(VectorV &row, VectorI &ind, int ncut);
-
-
     /** keeps off-diagonal entries; drops diagonal entries */
     struct keep_diag {
       inline bool operator() (const Index& row, const Index& col, const Scalar&) const
@@ -171,51 +215,6 @@ void IncompleteLUT<Scalar>::setFillfactor(int fillfactor)
   this->m_fillfactor = fillfactor;   
 }
 
-
-/**
- * Compute a quick-sort split of a vector 
- * On output, the vector row is permuted such that its elements satisfy
- * abs(row(i)) >= abs(row(ncut)) if i<ncut
- * abs(row(i)) <= abs(row(ncut)) if i>ncut 
- * \param row The vector of values
- * \param ind The array of index for the elements in @p row
- * \param ncut  The number of largest elements to keep
- **/ 
-template <typename Scalar>
-template <typename VectorV, typename VectorI>
-int IncompleteLUT<Scalar>::QuickSplit(VectorV &row, VectorI &ind, int ncut)
-{
-  using std::swap;
-  int mid;
-  int n = row.size(); /* length of the vector */
-  int first, last ; 
-  
-  ncut--; /* to fit the zero-based indices */
-  first = 0; 
-  last = n-1; 
-  if (ncut < first || ncut > last ) return 0;
-  
-  do {
-    mid = first; 
-    RealScalar abskey = std::abs(row(mid)); 
-    for (int j = first + 1; j <= last; j++) {
-      if ( std::abs(row(j)) > abskey) {
-        ++mid;
-        swap(row(mid), row(j));
-        swap(ind(mid), ind(j));
-      }
-    }
-    /* Interchange for the pivot element */
-    swap(row(mid), row(first));
-    swap(ind(mid), ind(first));
-    
-    if (mid > ncut) last = mid - 1;
-    else if (mid < ncut ) first = mid + 1; 
-  } while (mid != ncut );
-  
-  return 0; /* mid is equal to ncut */ 
-}
-
 template <typename Scalar>
 template<typename _MatrixType>
 void IncompleteLUT<Scalar>::analyzePattern(const _MatrixType& amat)
@@ -400,7 +399,7 @@ void IncompleteLUT<Scalar>::factorize(const _MatrixType& amat)
     len = (std::min)(sizel, nnzL);
     typename Vector::SegmentReturnType ul(u.segment(0, sizel));
     typename VectorXi::SegmentReturnType jul(ju.segment(0, sizel));
-    QuickSplit(ul, jul, len);
+    internal::QuickSplit(ul, jul, len);
 
     // store the largest m_fill elements of the L part
     m_lu.startVec(ii);
@@ -429,7 +428,7 @@ void IncompleteLUT<Scalar>::factorize(const _MatrixType& amat)
     len = (std::min)(sizeu, nnzU);
     typename Vector::SegmentReturnType uu(u.segment(ii+1, sizeu-1));
     typename VectorXi::SegmentReturnType juu(ju.segment(ii+1, sizeu-1));
-    QuickSplit(uu, juu, len);
+    internal::QuickSplit(uu, juu, len);
 
     // store the largest elements of the U part
     for(int k = ii + 1; k < ii + len; k++)

bench/spbench/sp_solver.cpp

@@ -13,7 +13,7 @@
 #include <Eigen/SuperLUSupport>
 // #include <unsupported/Eigen/src/IterativeSolvers/Scaling.h>
 #include <bench/BenchTimer.h>
-
+#include <unsupported/Eigen/IterativeSolvers>
 using namespace std;
 using namespace Eigen;
 
@@ -26,7 +26,8 @@ int main(int argc, char **args)
   VectorXd b, x, tmp;
   BenchTimer timer,totaltime; 
   //SparseLU<SparseMatrix<double, ColMajor> >   solver;
-  SuperLU<SparseMatrix<double, ColMajor> >   solver;
+//   SuperLU<SparseMatrix<double, ColMajor> >   solver;
+  ConjugateGradient<SparseMatrix<double, ColMajor>, Lower,IncompleteCholesky<double,Lower> > solver; 
   ifstream matrix_file; 
   string line;
   int  n;

unsupported/Eigen/IterativeSolvers

@@ -33,6 +33,7 @@
 #include "../../Eigen/Jacobi"
 #include "../../Eigen/Householder"
 #include "src/IterativeSolvers/GMRES.h"
+#include "src/IterativeSolvers/IncompleteCholesky.h"
 //#include "src/IterativeSolvers/SSORPreconditioner.h"
 
 //@}

unsupported/Eigen/src/IterativeSolvers/IncompleteCholesky.h

@@ -0,0 +1,221 @@
+// 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>
+//
+// 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 "Eigen/src/IterativeLinearSolvers/IncompleteLUT.h" 
+#include <Eigen/OrderingMethods>
+#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 _MatrixType The type of the sparse matrix. It should be a symmetric 
+ *                     matrix. It is advised to give  a row-oriented sparse matrix 
+ * \tparam _UpLo The triangular part of the matrix to reference. 
+ * \tparam _OrderingType 
+ */
+
+template <typename Scalar, int _UpLo = Lower, typename _OrderingType = NaturalOrdering<int> >
+class IncompleteCholesky : internal::noncopyable
+{
+  public:
+    typedef SparseMatrix<Scalar,ColMajor> MatrixType;
+    typedef _OrderingType OrderingType;
+    typedef typename MatrixType::RealScalar RealScalar; 
+    typedef typename MatrixType::Index Index; 
+    typedef PermutationMatrix<Dynamic, Dynamic, Index> PermutationType;
+    typedef Matrix<Scalar,Dynamic,1> VectorType; 
+    typedef Matrix<Index,Dynamic, 1> IndexType; 
+
+  public:
+    IncompleteCholesky() {}
+    IncompleteCholesky(const MatrixType& matrix)
+    {
+      compute(matrix);
+    }
+    
+    Index rows() const { return m_L.rows(); }
+    
+    Index cols() const { return m_L.cols(); }
+    
+
+    /** \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 && "IncompleteLLT is not initialized.");
+      return m_info;
+    }
+    /**
+    * \brief Computes the fill reducing permutation vector. 
+    */
+    template<typename MatrixType>
+    void analyzePattern(const MatrixType& mat)
+    {
+      OrderingType ord; 
+      ord(mat, m_perm); 
+      m_analysisIsOk = true; 
+    }
+    
+    template<typename MatrixType>
+    void factorize(const MatrixType& amat);
+    
+    template<typename MatrixType>
+    void compute (const MatrixType& matrix)
+    {
+      analyzePattern(matrix); 
+      factorize(matrix);
+    }
+    
+    template<typename Rhs, typename Dest>
+    void _solve(const Rhs& b, Dest& x) const
+    {
+      eigen_assert(m_factorizationIsOk && "factorize() should be called first");
+      if (m_perm.rows() == b.rows())
+        x = m_perm.inverse() * b; 
+      else 
+        x = b; 
+      x = m_L.template triangularView<UnitLower>().solve(x); 
+      x = m_L.adjoint().template triangularView<Upper>().solve(x); 
+      if (m_perm.rows() == b.rows())
+        x = m_perm * x;
+    }
+    template<typename Rhs> inline const internal::solve_retval<IncompleteCholesky, Rhs>
+    solve(const MatrixBase<Rhs>& b) const
+    {
+      eigen_assert(m_isInitialized && "IncompleteLLT is not initialized.");
+      eigen_assert(cols()==b.rows()
+                && "IncompleteLLT::solve(): invalid number of rows of the right hand side matrix b");
+      return internal::solve_retval<IncompleteCholesky, Rhs>(*this, b.derived());
+    }
+  protected:
+    SparseMatrix<Scalar,ColMajor> m_L;  // The lower part stored in CSC
+    bool m_analysisIsOk; 
+    bool m_factorizationIsOk; 
+    bool m_isInitialized;
+    ComputationInfo m_info;
+    PermutationType m_perm; 
+    
+}; 
+
+template<typename Scalar, int _UpLo, typename OrderingType>
+template<typename _MatrixType>
+void IncompleteCholesky<Scalar,_UpLo, OrderingType>::factorize(const _MatrixType& mat)
+{
+  eigen_assert(m_analysisIsOk && "analyzePattern() should be called first"); 
+  
+  // FIXME Stability: We should probably compute the scaling factors and the shifts that are needed to ensure an efficient LLT preconditioner. 
+  
+  // Dropping strategies : 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() )
+    m_L.template selfadjointView<Lower>() = mat.template selfadjointView<_UpLo>().twistedBy(m_perm);
+  else
+    m_L.template selfadjointView<Lower>() = mat.template selfadjointView<_UpLo>();
+  
+  int n = mat.cols(); 
+  
+  Scalar *vals = m_L.valuePtr(); //Values 
+  Index *rowIdx = m_L.innerIndexPtr(); //Row indices 
+  Index *colPtr = m_L.outerIndexPtr(); // Pointer to the beginning of each row
+  VectorType firstElt(n-1); // for each j, points to the next entry in vals that will be used in the factorization
+  // Initialize firstElt; 
+  for (int j = 0; j < n-1; j++) firstElt(j) = colPtr[j]+1; 
+  std::vector<std::list<Index> > listCol(n); // listCol(j) is a linked list of columns to update column j
+  VectorType curCol(n); // Store a  nonzero values in each column
+  VectorType irow(n); // Row indices of nonzero elements in each column
+  // jki version of the Cholesky factorization 
+  for (int j=0; j < n; j++)
+  {
+     //Left-looking factorize the column j 
+     // First, load the jth column into curCol 
+     Scalar diag = vals[colPtr[j]];  // Lower diagonal matrix with 
+     curCol.setZero();
+     irow.setLinSpaced(n,0,n-1); 
+     for (int i = colPtr[j] + 1; i < colPtr[j+1]; i++)
+     {
+       curCol(rowIdx[i]) = vals[i]; 
+       irow(rowIdx[i]) = rowIdx[i]; 
+     }
+     
+     std::list<int>::iterator k; 
+     // Browse all previous columns that will update column j
+     for(k = listCol[j].begin(); k != listCol[j].end(); k++) 
+     {
+       int jk = firstElt(*k); // First element to use in the column 
+       Scalar a_jk = vals[jk]; 
+       diag -= a_jk * a_jk; 
+       jk += 1; 
+       for (int i = jk; i < colPtr[*k]; i++)
+       {
+         curCol(rowIdx[i]) -= vals[i] * a_jk ;
+       }
+       firstElt(*k) = jk; 
+       if (jk < colPtr[*k+1]) 
+       {
+         // Add this column to the updating columns list for column *k+1
+         listCol[rowIdx[jk]].push_back(*k); 
+       }
+     }
+     
+     // Select the largest p elements
+     //  p is the original number of elements in the column (without the diagonal)
+     int p = colPtr[j+1] - colPtr[j] - 2 ; 
+     internal::QuickSplit(curCol, irow, p); 
+     if(RealScalar(diag) <= 0)
+     {
+       m_info = NumericalIssue; 
+       return; 
+     }
+     RealScalar rdiag = internal::sqrt(RealScalar(diag));
+     Scalar scal = Scalar(1)/rdiag; 
+     vals[colPtr[j]] = rdiag;
+     // Insert the largest p elements in the matrix and scale them meanwhile  
+     int cpt = 0; 
+     for (int i = colPtr[j]+1; i < colPtr[j+1]; i++)
+     {
+       vals[i] = curCol(cpt) * scal; 
+       rowIdx[i] = irow(cpt); 
+       cpt ++; 
+     }
+  }
+  m_factorizationIsOk = true; 
+  m_isInitialized = true;
+  m_info = Success; 
+}
+
+namespace internal {
+
+template<typename _MatrixType, typename Rhs>
+struct solve_retval<IncompleteCholesky<_MatrixType>, Rhs>
+  : solve_retval_base<IncompleteCholesky<_MatrixType>, Rhs>
+{
+  typedef IncompleteCholesky<_MatrixType> Dec;
+  EIGEN_MAKE_SOLVE_HELPERS(Dec,Rhs)
+
+  template<typename Dest> void evalTo(Dest& dst) const
+  {
+    dec()._solve(rhs(),dst);
+  }
+};
+
+} // end namespace internal
+
+} // end namespace Eigen 
+
+#endif