Quelle IncompleteLUT.h Sprache: C

// 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

quality94%

¤ Dauer der Verarbeitung: 0.8 Sekunden ¤

Wurzel

Suchen

Beweissystem der NASA

Beweissystem Isabelle

NIST Cobol Testsuite

Cephes Mathematical Library

Wiener Entwicklungsmethode

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.

Bemerkung:

Die farbliche Syntaxdarstellung ist noch experimentell.