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diff --git a/eigen/Eigen/src/IterativeLinearSolvers/IncompleteLUT.h b/eigen/Eigen/src/IterativeLinearSolvers/IncompleteLUT.h new file mode 100644 index 0000000..d3f37fe --- /dev/null +++ b/eigen/Eigen/src/IterativeLinearSolvers/IncompleteLUT.h @@ -0,0 +1,478 @@ +// 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_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, typename Index> +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 + * + * 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> +class IncompleteLUT : internal::noncopyable +{ + typedef _Scalar Scalar; + typedef typename NumTraits<Scalar>::Real RealScalar; + typedef Matrix<Scalar,Dynamic,1> Vector; + typedef SparseMatrix<Scalar,RowMajor> FactorType; + typedef SparseMatrix<Scalar,ColMajor> PermutType; + typedef typename FactorType::Index Index; + + public: + typedef Matrix<Scalar,Dynamic,Dynamic> MatrixType; + + IncompleteLUT() + : m_droptol(NumTraits<Scalar>::dummy_precision()), m_fillfactor(10), + m_analysisIsOk(false), m_factorizationIsOk(false), m_isInitialized(false) + {} + + template<typename MatrixType> + 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),m_isInitialized(false) + { + eigen_assert(fillfactor != 0); + compute(mat); + } + + Index rows() const { return m_lu.rows(); } + + Index cols() const { return m_lu.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 && "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<Scalar>& 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(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; + } + + template<typename Rhs> inline const internal::solve_retval<IncompleteLUT, Rhs> + solve(const MatrixBase<Rhs>& b) const + { + eigen_assert(m_isInitialized && "IncompleteLUT is not initialized."); + eigen_assert(cols()==b.rows() + && "IncompleteLUT::solve(): invalid number of rows of the right hand side matrix b"); + return internal::solve_retval<IncompleteLUT, Rhs>(*this, b.derived()); + } + +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; + bool m_isInitialized; + ComputationInfo m_info; + PermutationMatrix<Dynamic,Dynamic,Index> m_P; // Fill-reducing permutation + PermutationMatrix<Dynamic,Dynamic,Index> m_Pinv; // Inverse permutation +}; + +/** + * Set control parameter droptol + * \param droptol Drop any element whose magnitude is less than this tolerance + **/ +template<typename Scalar> +void IncompleteLUT<Scalar>::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> +void IncompleteLUT<Scalar>::setFillfactor(int fillfactor) +{ + this->m_fillfactor = fillfactor; +} + +template <typename Scalar> +template<typename _MatrixType> +void IncompleteLUT<Scalar>::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. +#ifndef EIGEN_MPL2_ONLY + // To this end, let's symmetrize the pattern and perform AMD on it. + SparseMatrix<Scalar,ColMajor, Index> mat1 = amat; + SparseMatrix<Scalar,ColMajor, Index> 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 prefered... + SparseMatrix<Scalar,ColMajor, Index> AtA = mat2 + mat1; + AMDOrdering<Index> ordering; + ordering(AtA,m_P); + m_Pinv = m_P.inverse(); // cache the inverse permutation +#else + // If AMD is not available, (MPL2-only), then let's use the slower COLAMD routine. + SparseMatrix<Scalar,ColMajor, Index> mat1 = amat; + COLAMDOrdering<Index> ordering; + ordering(mat1,m_Pinv); + m_P = m_Pinv.inverse(); +#endif + + m_analysisIsOk = true; + m_factorizationIsOk = false; + m_isInitialized = false; +} + +template <typename Scalar> +template<typename _MatrixType> +void IncompleteLUT<Scalar>::factorize(const _MatrixType& amat) +{ + using std::sqrt; + using std::swap; + using std::abs; + + 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 -- + VectorXi ju(n); // column position of the values in u -- maximum size is n + VectorXi 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, Index> 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 = static_cast<Index> (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) = ii; + u(ii) = 0; + jr(ii) = 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) = k; + u(sizel) = j_it.value(); + jr(k) = sizel; + ++sizel; + } + else if (k == ii) + { + u(ii) = j_it.value(); + } + else + { + // copy the upper part + Index jpos = ii + sizeu; + ju(jpos) = k; + u(jpos) = j_it.value(); + jr(k) = 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) = jj; jr(j) = 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) = j; + u(newpos) = -prod; + jr(j) = newpos; + } + else + u(jpos) -= prod; + } + // store the pivot element + u(len) = fact; + ju(len) = 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 VectorXi::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 VectorXi::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_isInitialized = m_factorizationIsOk; + m_info = Success; +} + +namespace internal { + +template<typename _MatrixType, typename Rhs> +struct solve_retval<IncompleteLUT<_MatrixType>, Rhs> + : solve_retval_base<IncompleteLUT<_MatrixType>, Rhs> +{ + typedef IncompleteLUT<_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 // EIGEN_INCOMPLETE_LUT_H |