cart-elc

IncompleteLUT.h (14940B)

---

 // 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) 2014 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_LUT_H
 #define EIGEN_INCOMPLETE_LUT_H
 
 
 namespace Eigen {
 
 namespace internal {
 
 /** \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>
 Index QuickSplit(VectorV &row, VectorI &ind, Index ncut)
 {
   typedef typename VectorV::RealScalar RealScalar;
   using std::swap;
   using std::abs;
   Index mid;
   Index n = row.size(); /* length of the vector */
   Index 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 = abs(row(mid));
     for (Index j = first + 1; j <= last; j++) {
       if ( 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
 
 /** \ingroup IterativeLinearSolvers_Module
   * \class IncompleteLUT
   * \brief Incomplete LU factorization with dual-threshold strategy
   *
   * \implsparsesolverconcept
   *
   * During the numerical factorization, two dropping rules are used :
   *  1) any element whose magnitude is less than some tolerance is dropped.
   *    This tolerance is obtained by multiplying the input tolerance @p droptol
   *    by the average magnitude of all the original elements in the current row.
   *  2) After the elimination of the row, only the @p fill largest elements in
   *    the L part and the @p fill largest elements in the U part are kept
   *    (in addition to the diagonal element ). Note that @p fill is computed from
   *    the input parameter @p fillfactor which is used the ratio to control the fill_in
   *    relatively to the initial number of nonzero elements.
   *
   * The two extreme cases are when @p droptol=0 (to keep all the @p fill*2 largest elements)
   * and when @p fill=n/2 with @p droptol being different to zero.
   *
   * References : Yousef Saad, ILUT: A dual threshold incomplete LU factorization,
   *              Numerical Linear Algebra with Applications, 1(4), pp 387-402, 1994.
   *
   * NOTE : The following implementation is derived from the ILUT implementation
   * in the SPARSKIT package, Copyright (C) 2005, the Regents of the University of Minnesota
   *  released under the terms of the GNU LGPL:
   *    http://www-users.cs.umn.edu/~saad/software/SPARSKIT/README
   * However, Yousef Saad gave us permission to relicense his ILUT code to MPL2.
   * See the Eigen mailing list archive, thread: ILUT, date: July 8, 2012:
   *   http://listengine.tuxfamily.org/lists.tuxfamily.org/eigen/2012/07/msg00064.html
   * alternatively, on GMANE:
   *   http://comments.gmane.org/gmane.comp.lib.eigen/3302
   */
 template <typename _Scalar, typename _StorageIndex = int>
 class IncompleteLUT : public SparseSolverBase<IncompleteLUT<_Scalar, _StorageIndex> >
 {
   protected:
     typedef SparseSolverBase<IncompleteLUT> Base;
     using Base::m_isInitialized;
   public:
     typedef _Scalar Scalar;
     typedef _StorageIndex StorageIndex;
     typedef typename NumTraits<Scalar>::Real RealScalar;
     typedef Matrix<Scalar,Dynamic,1> Vector;
     typedef Matrix<StorageIndex,Dynamic,1> VectorI;
     typedef SparseMatrix<Scalar,RowMajor,StorageIndex> FactorType;
 
     enum {
       ColsAtCompileTime = Dynamic,
       MaxColsAtCompileTime = Dynamic
     };
 
   public:
 
     IncompleteLUT()
       : m_droptol(NumTraits<Scalar>::dummy_precision()), m_fillfactor(10),
         m_analysisIsOk(false), m_factorizationIsOk(false)
     {}
 
     template<typename MatrixType>
     explicit IncompleteLUT(const MatrixType& mat, const RealScalar& droptol=NumTraits<Scalar>::dummy_precision(), int fillfactor = 10)
       : m_droptol(droptol),m_fillfactor(fillfactor),
         m_analysisIsOk(false),m_factorizationIsOk(false)
     {
       eigen_assert(fillfactor != 0);
       compute(mat);
     }
 
     EIGEN_CONSTEXPR Index rows() const EIGEN_NOEXCEPT { return m_lu.rows(); }
 
     EIGEN_CONSTEXPR Index cols() const EIGEN_NOEXCEPT { return m_lu.cols(); }
 
     /** \brief Reports whether previous computation was successful.
       *
       * \returns \c Success if computation was successful,
       *          \c NumericalIssue if the matrix.appears to be negative.
       */
     ComputationInfo info() const
     {
       eigen_assert(m_isInitialized && "IncompleteLUT is not initialized.");
       return m_info;
     }
 
     template<typename MatrixType>
     void analyzePattern(const MatrixType& amat);
 
     template<typename MatrixType>
     void factorize(const MatrixType& amat);
 
     /**
       * Compute an incomplete LU factorization with dual threshold on the matrix mat
       * No pivoting is done in this version
       *
       **/
     template<typename MatrixType>
     IncompleteLUT& compute(const MatrixType& amat)
     {
       analyzePattern(amat);
       factorize(amat);
       return *this;
     }
 
     void setDroptol(const RealScalar& droptol);
     void setFillfactor(int fillfactor);
 
     template<typename Rhs, typename Dest>
     void _solve_impl(const Rhs& b, Dest& x) const
     {
       x = m_Pinv * b;
       x = m_lu.template triangularView<UnitLower>().solve(x);
       x = m_lu.template triangularView<Upper>().solve(x);
       x = m_P * x;
     }
 
 protected:
 
     /** 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;
       }
     };
 
 protected:
 
     FactorType m_lu;
     RealScalar m_droptol;
     int m_fillfactor;
     bool m_analysisIsOk;
     bool m_factorizationIsOk;
     ComputationInfo m_info;
     PermutationMatrix<Dynamic,Dynamic,StorageIndex> m_P;     // Fill-reducing permutation
     PermutationMatrix<Dynamic,Dynamic,StorageIndex> m_Pinv;  // Inverse permutation
 };
 
 /**
  * Set control parameter droptol
  *  \param droptol   Drop any element whose magnitude is less than this tolerance
  **/
 template<typename Scalar, typename StorageIndex>
 void IncompleteLUT<Scalar,StorageIndex>::setDroptol(const RealScalar& droptol)
 {
   this->m_droptol = droptol;
 }
 
 /**
  * Set control parameter fillfactor
  * \param fillfactor  This is used to compute the  number @p fill_in of largest elements to keep on each row.
  **/
 template<typename Scalar, typename StorageIndex>
 void IncompleteLUT<Scalar,StorageIndex>::setFillfactor(int fillfactor)
 {
   this->m_fillfactor = fillfactor;
 }
 
 template <typename Scalar, typename StorageIndex>
 template<typename _MatrixType>
 void IncompleteLUT<Scalar,StorageIndex>::analyzePattern(const _MatrixType& amat)
 {
   // Compute the Fill-reducing permutation
   // Since ILUT does not perform any numerical pivoting,
   // it is highly preferable to keep the diagonal through symmetric permutations.
   // To this end, let's symmetrize the pattern and perform AMD on it.
   SparseMatrix<Scalar,ColMajor, StorageIndex> mat1 = amat;
   SparseMatrix<Scalar,ColMajor, StorageIndex> mat2 = amat.transpose();
   // FIXME for a matrix with nearly symmetric pattern, mat2+mat1 is the appropriate choice.
   //       on the other hand for a really non-symmetric pattern, mat2*mat1 should be preferred...
   SparseMatrix<Scalar,ColMajor, StorageIndex> AtA = mat2 + mat1;
   AMDOrdering<StorageIndex> ordering;
   ordering(AtA,m_P);
   m_Pinv  = m_P.inverse(); // cache the inverse permutation
   m_analysisIsOk = true;
   m_factorizationIsOk = false;
   m_isInitialized = true;
 }
 
 template <typename Scalar, typename StorageIndex>
 template<typename _MatrixType>
 void IncompleteLUT<Scalar,StorageIndex>::factorize(const _MatrixType& amat)
 {
   using std::sqrt;
   using std::swap;
   using std::abs;
   using internal::convert_index;
 
   eigen_assert((amat.rows() == amat.cols()) && "The factorization should be done on a square matrix");
   Index n = amat.cols();  // Size of the matrix
   m_lu.resize(n,n);
   // Declare Working vectors and variables
   Vector u(n) ;     // real values of the row -- maximum size is n --
   VectorI ju(n);   // column position of the values in u -- maximum size  is n
   VectorI jr(n);   // Indicate the position of the nonzero elements in the vector u -- A zero location is indicated by -1
 
   // Apply the fill-reducing permutation
   eigen_assert(m_analysisIsOk && "You must first call analyzePattern()");
   SparseMatrix<Scalar,RowMajor, StorageIndex> mat;
   mat = amat.twistedBy(m_Pinv);
 
   // Initialization
   jr.fill(-1);
   ju.fill(0);
   u.fill(0);
 
   // number of largest elements to keep in each row:
   Index fill_in = (amat.nonZeros()*m_fillfactor)/n + 1;
   if (fill_in > n) fill_in = n;
 
   // number of largest nonzero elements to keep in the L and the U part of the current row:
   Index nnzL = fill_in/2;
   Index nnzU = nnzL;
   m_lu.reserve(n * (nnzL + nnzU + 1));
 
   // global loop over the rows of the sparse matrix
   for (Index ii = 0; ii < n; ii++)
   {
     // 1 - copy the lower and the upper part of the row i of mat in the working vector u
 
     Index sizeu = 1; // number of nonzero elements in the upper part of the current row
     Index sizel = 0; // number of nonzero elements in the lower part of the current row
     ju(ii)    = convert_index<StorageIndex>(ii);
     u(ii)     = 0;
     jr(ii)    = convert_index<StorageIndex>(ii);
     RealScalar rownorm = 0;
 
     typename FactorType::InnerIterator j_it(mat, ii); // Iterate through the current row ii
     for (; j_it; ++j_it)
     {
       Index k = j_it.index();
       if (k < ii)
       {
         // copy the lower part
         ju(sizel) = convert_index<StorageIndex>(k);
         u(sizel) = j_it.value();
         jr(k) = convert_index<StorageIndex>(sizel);
         ++sizel;
       }
       else if (k == ii)
       {
         u(ii) = j_it.value();
       }
       else
       {
         // copy the upper part
         Index jpos = ii + sizeu;
         ju(jpos) = convert_index<StorageIndex>(k);
         u(jpos) = j_it.value();
         jr(k) = convert_index<StorageIndex>(jpos);
         ++sizeu;
       }
       rownorm += numext::abs2(j_it.value());
     }
 
     // 2 - detect possible zero row
     if(rownorm==0)
     {
       m_info = NumericalIssue;
       return;
     }
     // Take the 2-norm of the current row as a relative tolerance
     rownorm = sqrt(rownorm);
 
     // 3 - eliminate the previous nonzero rows
     Index jj = 0;
     Index len = 0;
     while (jj < sizel)
     {
       // In order to eliminate in the correct order,
       // we must select first the smallest column index among  ju(jj:sizel)
       Index k;
       Index minrow = ju.segment(jj,sizel-jj).minCoeff(&k); // k is relative to the segment
       k += jj;
       if (minrow != ju(jj))
       {
         // swap the two locations
         Index j = ju(jj);
         swap(ju(jj), ju(k));
         jr(minrow) = convert_index<StorageIndex>(jj);
         jr(j) = convert_index<StorageIndex>(k);
         swap(u(jj), u(k));
       }
       // Reset this location
       jr(minrow) = -1;
 
       // Start elimination
       typename FactorType::InnerIterator ki_it(m_lu, minrow);
       while (ki_it && ki_it.index() < minrow) ++ki_it;
       eigen_internal_assert(ki_it && ki_it.col()==minrow);
       Scalar fact = u(jj) / ki_it.value();
 
       // drop too small elements
       if(abs(fact) <= m_droptol)
       {
         jj++;
         continue;
       }
 
       // linear combination of the current row ii and the row minrow
       ++ki_it;
       for (; ki_it; ++ki_it)
       {
         Scalar prod = fact * ki_it.value();
         Index j     = ki_it.index();
         Index jpos  = jr(j);
         if (jpos == -1) // fill-in element
         {
           Index newpos;
           if (j >= ii) // dealing with the upper part
           {
             newpos = ii + sizeu;
             sizeu++;
             eigen_internal_assert(sizeu<=n);
           }
           else // dealing with the lower part
           {
             newpos = sizel;
             sizel++;
             eigen_internal_assert(sizel<=ii);
           }
           ju(newpos) = convert_index<StorageIndex>(j);
           u(newpos) = -prod;
           jr(j) = convert_index<StorageIndex>(newpos);
         }
         else
           u(jpos) -= prod;
       }
       // store the pivot element
       u(len)  = fact;
       ju(len) = convert_index<StorageIndex>(minrow);
       ++len;
 
       jj++;
     } // end of the elimination on the row ii
 
     // reset the upper part of the pointer jr to zero
     for(Index k = 0; k <sizeu; k++) jr(ju(ii+k)) = -1;
 
     // 4 - partially sort and insert the elements in the m_lu matrix
 
     // sort the L-part of the row
     sizel = len;
     len = (std::min)(sizel, nnzL);
     typename Vector::SegmentReturnType ul(u.segment(0, sizel));
     typename VectorI::SegmentReturnType jul(ju.segment(0, sizel));
     internal::QuickSplit(ul, jul, len);
 
     // store the largest m_fill elements of the L part
     m_lu.startVec(ii);
     for(Index k = 0; k < len; k++)
       m_lu.insertBackByOuterInnerUnordered(ii,ju(k)) = u(k);
 
     // store the diagonal element
     // apply a shifting rule to avoid zero pivots (we are doing an incomplete factorization)
     if (u(ii) == Scalar(0))
       u(ii) = sqrt(m_droptol) * rownorm;
     m_lu.insertBackByOuterInnerUnordered(ii, ii) = u(ii);
 
     // sort the U-part of the row
     // apply the dropping rule first
     len = 0;
     for(Index k = 1; k < sizeu; k++)
     {
       if(abs(u(ii+k)) > m_droptol * rownorm )
       {
         ++len;
         u(ii + len)  = u(ii + k);
         ju(ii + len) = ju(ii + k);
       }
     }
     sizeu = len + 1; // +1 to take into account the diagonal element
     len = (std::min)(sizeu, nnzU);
     typename Vector::SegmentReturnType uu(u.segment(ii+1, sizeu-1));
     typename VectorI::SegmentReturnType juu(ju.segment(ii+1, sizeu-1));
     internal::QuickSplit(uu, juu, len);
 
     // store the largest elements of the U part
     for(Index k = ii + 1; k < ii + len; k++)
       m_lu.insertBackByOuterInnerUnordered(ii,ju(k)) = u(k);
   }
   m_lu.finalize();
   m_lu.makeCompressed();
 
   m_factorizationIsOk = true;
   m_info = Success;
 }
 
 } // end namespace Eigen
 
 #endif // EIGEN_INCOMPLETE_LUT_H