// This file is part of Eigen, a lightweight C++ template library // for linear algebra. // // Copyright (C) 2008-2010 Gael Guennebaud // // Eigen is free software; you can redistribute it and/or // modify it under the terms of the GNU Lesser General Public // License as published by the Free Software Foundation; either // version 3 of the License, or (at your option) any later version. // // Alternatively, you can redistribute it and/or // modify it under the terms of the GNU General Public License as // published by the Free Software Foundation; either version 2 of // the License, or (at your option) any later version. // // Eigen is distributed in the hope that it will be useful, but WITHOUT ANY // WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS // FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the // GNU General Public License for more details. // // You should have received a copy of the GNU Lesser General Public // License and a copy of the GNU General Public License along with // Eigen. If not, see . /* NOTE: the _symbolic, and _numeric functions has been adapted from the LDL library: LDL Copyright (c) 2005 by Timothy A. Davis. All Rights Reserved. LDL License: Your use or distribution of LDL or any modified version of LDL implies that you agree to this License. This library is free software; you can redistribute it and/or modify it under the terms of the GNU Lesser General Public License as published by the Free Software Foundation; either version 2.1 of the License, or (at your option) any later version. This library is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for more details. You should have received a copy of the GNU Lesser General Public License along with this library; if not, write to the Free Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA Permission is hereby granted to use or copy this program under the terms of the GNU LGPL, provided that the Copyright, this License, and the Availability of the original version is retained on all copies. User documentation of any code that uses this code or any modified version of this code must cite the Copyright, this License, the Availability note, and "Used by permission." Permission to modify the code and to distribute modified code is granted, provided the Copyright, this License, and the Availability note are retained, and a notice that the code was modified is included. */ #ifndef EIGEN_SIMPLICIAL_CHOLESKY_H #define EIGEN_SIMPLICIAL_CHOLESKY_H enum SimplicialCholeskyMode { SimplicialCholeskyLLt, SimplicialCholeskyLDLt }; /** \brief A direct sparse Cholesky factorizations * * These classes provide LL^T and LDL^T Cholesky factorizations of sparse matrices that are * selfadjoint and positive definite. The factorization allows for solving A.X = B where * X and B can be either dense or sparse. * * \tparam _MatrixType the type of the sparse matrix A, it must be a SparseMatrix<> * \tparam _UpLo the triangular part that will be used for the computations. It can be Lower * or Upper. Default is Lower. * */ template class SimplicialCholeskyBase { public: typedef typename internal::traits::MatrixType MatrixType; enum { UpLo = internal::traits::UpLo }; typedef typename MatrixType::Scalar Scalar; typedef typename MatrixType::RealScalar RealScalar; typedef typename MatrixType::Index Index; typedef SparseMatrix CholMatrixType; typedef Matrix VectorType; public: SimplicialCholeskyBase() : m_info(Success), m_isInitialized(false) {} SimplicialCholeskyBase(const MatrixType& matrix) : m_info(Success), m_isInitialized(false) { compute(matrix); } ~SimplicialCholeskyBase() { } Derived& derived() { return *static_cast(this); } const Derived& derived() const { return *static_cast(this); } inline Index cols() const { return m_matrix.cols(); } inline Index rows() const { return m_matrix.rows(); } /** \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 && "Decomposition is not initialized."); return m_info; } /** Computes the sparse Cholesky decomposition of \a matrix */ Derived& compute(const MatrixType& matrix) { derived().analyzePattern(matrix); derived().factorize(matrix); return derived(); } /** \returns the solution x of \f$ A x = b \f$ using the current decomposition of A. * * \sa compute() */ template inline const internal::solve_retval solve(const MatrixBase& b) const { eigen_assert(m_isInitialized && "Simplicial LLt or LDLt is not initialized."); eigen_assert(rows()==b.rows() && "SimplicialCholeskyBase::solve(): invalid number of rows of the right hand side matrix b"); return internal::solve_retval(*this, b.derived()); } /** \returns the solution x of \f$ A x = b \f$ using the current decomposition of A. * * \sa compute() */ template inline const internal::sparse_solve_retval solve(const SparseMatrixBase& b) const { eigen_assert(m_isInitialized && "Simplicial LLt or LDLt is not initialized."); eigen_assert(rows()==b.rows() && "SimplicialCholesky::solve(): invalid number of rows of the right hand side matrix b"); return internal::sparse_solve_retval(*this, b.derived()); } /** \returns the permutation P * \sa permutationPinv() */ const PermutationMatrix& permutationP() const { return m_P; } /** \returns the inverse P^-1 of the permutation P * \sa permutationP() */ const PermutationMatrix& permutationPinv() const { return m_Pinv; } #ifndef EIGEN_PARSED_BY_DOXYGEN /** \internal */ template void dumpMemory(Stream& s) { int total = 0; s << " L: " << ((total+=(m_matrix.cols()+1) * sizeof(int) + m_matrix.nonZeros()*(sizeof(int)+sizeof(Scalar))) >> 20) << "Mb" << "\n"; s << " diag: " << ((total+=m_diag.size() * sizeof(Scalar)) >> 20) << "Mb" << "\n"; s << " tree: " << ((total+=m_parent.size() * sizeof(int)) >> 20) << "Mb" << "\n"; s << " nonzeros: " << ((total+=m_nonZerosPerCol.size() * sizeof(int)) >> 20) << "Mb" << "\n"; s << " perm: " << ((total+=m_P.size() * sizeof(int)) >> 20) << "Mb" << "\n"; s << " perm^-1: " << ((total+=m_Pinv.size() * sizeof(int)) >> 20) << "Mb" << "\n"; s << " TOTAL: " << (total>> 20) << "Mb" << "\n"; } /** \internal */ template void _solve(const MatrixBase &b, MatrixBase &dest) const { eigen_assert(m_factorizationIsOk && "The decomposition is not in a valid state for solving, you must first call either compute() or symbolic()/numeric()"); eigen_assert(m_matrix.rows()==b.rows()); if(m_info!=Success) return; if(m_P.size()>0) dest = m_Pinv * b; else dest = b; if(m_matrix.nonZeros()>0) // otherwise L==I derived().matrixL().solveInPlace(dest); if(m_diag.size()>0) dest = m_diag.asDiagonal().inverse() * dest; if (m_matrix.nonZeros()>0) // otherwise I==I derived().matrixU().solveInPlace(dest); if(m_P.size()>0) dest = m_P * dest; } /** \internal */ template void _solve_sparse(const Rhs& b, SparseMatrix &dest) const { eigen_assert(m_factorizationIsOk && "The decomposition is not in a valid state for solving, you must first call either compute() or symbolic()/numeric()"); eigen_assert(m_matrix.rows()==b.rows()); // we process the sparse rhs per block of NbColsAtOnce columns temporarily stored into a dense matrix. static const int NbColsAtOnce = 4; int rhsCols = b.cols(); int size = b.rows(); Eigen::Matrix tmp(size,rhsCols); for(int k=0; k(rhsCols-k, NbColsAtOnce); tmp.leftCols(actualCols) = b.middleCols(k,actualCols); tmp.leftCols(actualCols) = derived().solve(tmp.leftCols(actualCols)); dest.middleCols(k,actualCols) = tmp.leftCols(actualCols).sparseView(); } } #endif // EIGEN_PARSED_BY_DOXYGEN protected: template void factorize(const MatrixType& a); void analyzePattern(const MatrixType& a, bool doLDLt); /** 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; } }; mutable ComputationInfo m_info; bool m_isInitialized; bool m_factorizationIsOk; bool m_analysisIsOk; CholMatrixType m_matrix; VectorType m_diag; // the diagonal coefficients (LDLt mode) VectorXi m_parent; // elimination tree VectorXi m_nonZerosPerCol; PermutationMatrix m_P; // the permutation PermutationMatrix m_Pinv; // the inverse permutation }; template class SimplicialLLt; template class SimplicialLDLt; template class SimplicialCholesky; namespace internal { template struct traits > { typedef _MatrixType MatrixType; enum { UpLo = _UpLo }; typedef typename MatrixType::Scalar Scalar; typedef typename MatrixType::Index Index; typedef SparseMatrix CholMatrixType; typedef SparseTriangularView MatrixL; typedef SparseTriangularView MatrixU; inline static MatrixL getL(const MatrixType& m) { return m; } inline static MatrixU getU(const MatrixType& m) { return m.adjoint(); } }; //template struct traits > //{ // typedef _MatrixType MatrixType; // enum { UpLo = Upper }; // typedef typename MatrixType::Scalar Scalar; // typedef typename MatrixType::Index Index; // typedef SparseMatrix CholMatrixType; // typedef TriangularView MatrixL; // typedef TriangularView MatrixU; // inline static MatrixL getL(const MatrixType& m) { return m.adjoint(); } // inline static MatrixU getU(const MatrixType& m) { return m; } //}; template struct traits > { typedef _MatrixType MatrixType; enum { UpLo = _UpLo }; typedef typename MatrixType::Scalar Scalar; typedef typename MatrixType::Index Index; typedef SparseMatrix CholMatrixType; typedef SparseTriangularView MatrixL; typedef SparseTriangularView MatrixU; inline static MatrixL getL(const MatrixType& m) { return m; } inline static MatrixU getU(const MatrixType& m) { return m.adjoint(); } }; //template struct traits > //{ // typedef _MatrixType MatrixType; // enum { UpLo = Upper }; // typedef typename MatrixType::Scalar Scalar; // typedef typename MatrixType::Index Index; // typedef SparseMatrix CholMatrixType; // typedef TriangularView MatrixL; // typedef TriangularView MatrixU; // inline static MatrixL getL(const MatrixType& m) { return m.adjoint(); } // inline static MatrixU getU(const MatrixType& m) { return m; } //}; template struct traits > { typedef _MatrixType MatrixType; enum { UpLo = _UpLo }; }; } /** \class SimplicialLLt * \brief A direct sparse LLt Cholesky factorizations * * This class provides a LL^T Cholesky factorizations of sparse matrices that are * selfadjoint and positive definite. The factorization allows for solving A.X = B where * X and B can be either dense or sparse. * * \tparam _MatrixType the type of the sparse matrix A, it must be a SparseMatrix<> * \tparam _UpLo the triangular part that will be used for the computations. It can be Lower * or Upper. Default is Lower. * * \sa class SimplicialLDLt */ template class SimplicialLLt : public SimplicialCholeskyBase > { public: typedef _MatrixType MatrixType; enum { UpLo = _UpLo }; typedef SimplicialCholeskyBase Base; typedef typename MatrixType::Scalar Scalar; typedef typename MatrixType::RealScalar RealScalar; typedef typename MatrixType::Index Index; typedef SparseMatrix CholMatrixType; typedef Matrix VectorType; typedef internal::traits Traits; typedef typename Traits::MatrixL MatrixL; typedef typename Traits::MatrixU MatrixU; public: SimplicialLLt() : Base() {} SimplicialLLt(const MatrixType& matrix) : Base(matrix) {} inline const MatrixL matrixL() const { eigen_assert(Base::m_factorizationIsOk && "Simplicial LLt not factorized"); return Traits::getL(Base::m_matrix); } inline const MatrixU matrixU() const { eigen_assert(Base::m_factorizationIsOk && "Simplicial LLt not factorized"); return Traits::getU(Base::m_matrix); } /** Performs a symbolic decomposition on the sparcity of \a matrix. * * This function is particularly useful when solving for several problems having the same structure. * * \sa factorize() */ void analyzePattern(const MatrixType& a) { Base::analyzePattern(a, false); } /** Performs a numeric decomposition of \a matrix * * The given matrix must has the same sparcity than the matrix on which the symbolic decomposition has been performed. * * \sa analyzePattern() */ void factorize(const MatrixType& a) { Base::template factorize(a); } Scalar determinant() const { Scalar detL = Diagonal(Base::m_matrix).prod(); return internal::abs2(detL); } }; /** \class SimplicialLDLt * \brief A direct sparse LDLt Cholesky factorizations without square root. * * This class provides a LDL^T Cholesky factorizations without square root of sparse matrices that are * selfadjoint and positive definite. The factorization allows for solving A.X = B where * X and B can be either dense or sparse. * * \tparam _MatrixType the type of the sparse matrix A, it must be a SparseMatrix<> * \tparam _UpLo the triangular part that will be used for the computations. It can be Lower * or Upper. Default is Lower. * * \sa class SimplicialLLt */ template class SimplicialLDLt : public SimplicialCholeskyBase > { public: typedef _MatrixType MatrixType; enum { UpLo = _UpLo }; typedef SimplicialCholeskyBase Base; typedef typename MatrixType::Scalar Scalar; typedef typename MatrixType::RealScalar RealScalar; typedef typename MatrixType::Index Index; typedef SparseMatrix CholMatrixType; typedef Matrix VectorType; typedef internal::traits Traits; typedef typename Traits::MatrixL MatrixL; typedef typename Traits::MatrixU MatrixU; public: SimplicialLDLt() : Base() {} SimplicialLDLt(const MatrixType& matrix) : Base(matrix) {} inline const VectorType vectorD() const { eigen_assert(Base::m_factorizationIsOk && "Simplicial LDLt not factorized"); return Base::m_diag; } inline const MatrixL matrixL() const { eigen_assert(Base::m_factorizationIsOk && "Simplicial LDLt not factorized"); return Traits::getL(Base::m_matrix); } inline const MatrixU matrixU() const { eigen_assert(Base::m_factorizationIsOk && "Simplicial LDLt not factorized"); return Traits::getU(Base::m_matrix); } /** Performs a symbolic decomposition on the sparcity of \a matrix. * * This function is particularly useful when solving for several problems having the same structure. * * \sa factorize() */ void analyzePattern(const MatrixType& a) { Base::analyzePattern(a, true); } /** Performs a numeric decomposition of \a matrix * * The given matrix must has the same sparcity than the matrix on which the symbolic decomposition has been performed. * * \sa analyzePattern() */ void factorize(const MatrixType& a) { Base::template factorize(a); } Scalar determinant() const { return Base::m_diag.prod(); } }; /** \class SimplicialCholesky * \deprecated * \sa class SimplicialLDLt, class SimplicialLLt */ template class SimplicialCholesky : public SimplicialCholeskyBase > { public: typedef _MatrixType MatrixType; enum { UpLo = _UpLo }; typedef SimplicialCholeskyBase Base; typedef typename MatrixType::Scalar Scalar; typedef typename MatrixType::RealScalar RealScalar; typedef typename MatrixType::Index Index; typedef SparseMatrix CholMatrixType; typedef Matrix VectorType; typedef internal::traits Traits; typedef internal::traits > LDLtTraits; typedef internal::traits > LLtTraits; public: SimplicialCholesky() : Base(), m_LDLt(true) {} SimplicialCholesky(const MatrixType& matrix) : Base(), m_LDLt(true) { Base::compute(matrix); } SimplicialCholesky& setMode(SimplicialCholeskyMode mode) { switch(mode) { case SimplicialCholeskyLLt: m_LDLt = false; break; case SimplicialCholeskyLDLt: m_LDLt = true; break; default: break; } return *this; } inline const VectorType vectorD() const { eigen_assert(Base::m_factorizationIsOk && "Simplicial Cholesky not factorized"); return Base::m_diag; } inline const CholMatrixType rawMatrix() const { eigen_assert(Base::m_factorizationIsOk && "Simplicial Cholesky not factorized"); return Base::m_matrix; } /** Performs a symbolic decomposition on the sparcity of \a matrix. * * This function is particularly useful when solving for several problems having the same structure. * * \sa factorize() */ void analyzePattern(const MatrixType& a) { Base::analyzePattern(a, m_LDLt); } /** Performs a numeric decomposition of \a matrix * * The given matrix must has the same sparcity than the matrix on which the symbolic decomposition has been performed. * * \sa analyzePattern() */ void factorize(const MatrixType& a) { if(m_LDLt) Base::template factorize(a); else Base::template factorize(a); } /** \internal */ template void _solve(const MatrixBase &b, MatrixBase &dest) const { eigen_assert(Base::m_factorizationIsOk && "The decomposition is not in a valid state for solving, you must first call either compute() or symbolic()/numeric()"); eigen_assert(Base::m_matrix.rows()==b.rows()); if(Base::m_info!=Success) return; if(Base::m_P.size()>0) dest = Base::m_Pinv * b; else dest = b; if(Base::m_matrix.nonZeros()>0) // otherwise L==I { if(m_LDLt) LDLtTraits::getL(Base::m_matrix).solveInPlace(dest); else LLtTraits::getL(Base::m_matrix).solveInPlace(dest); } if(Base::m_diag.size()>0) dest = Base::m_diag.asDiagonal().inverse() * dest; if (Base::m_matrix.nonZeros()>0) // otherwise I==I { if(m_LDLt) LDLtTraits::getU(Base::m_matrix).solveInPlace(dest); else LLtTraits::getU(Base::m_matrix).solveInPlace(dest); } if(Base::m_P.size()>0) dest = Base::m_P * dest; } Scalar determinant() const { if(m_LDLt) { return Base::m_diag.prod(); } else { Scalar detL = Diagonal(Base::m_matrix).prod(); return internal::abs2(detL); } } protected: bool m_LDLt; }; template void SimplicialCholeskyBase::analyzePattern(const MatrixType& a, bool doLDLt) { eigen_assert(a.rows()==a.cols()); const Index size = a.rows(); m_matrix.resize(size, size); m_parent.resize(size); m_nonZerosPerCol.resize(size); ei_declare_aligned_stack_constructed_variable(Index, tags, size, 0); // TODO allows to configure the permutation { CholMatrixType C; C = a.template selfadjointView(); // remove diagonal entries: C.prune(keep_diag()); internal::minimum_degree_ordering(C, m_P); } if(m_P.size()>0) m_Pinv = m_P.inverse(); else m_Pinv.resize(0); SparseMatrix ap(size,size); ap.template selfadjointView() = a.template selfadjointView().twistedBy(m_Pinv); for(Index k = 0; k < size; ++k) { /* L(k,:) pattern: all nodes reachable in etree from nz in A(0:k-1,k) */ m_parent[k] = -1; /* parent of k is not yet known */ tags[k] = k; /* mark node k as visited */ m_nonZerosPerCol[k] = 0; /* count of nonzeros in column k of L */ for(typename CholMatrixType::InnerIterator it(ap,k); it; ++it) { Index i = it.index(); if(i < k) { /* follow path from i to root of etree, stop at flagged node */ for(; tags[i] != k; i = m_parent[i]) { /* find parent of i if not yet determined */ if (m_parent[i] == -1) m_parent[i] = k; m_nonZerosPerCol[i]++; /* L (k,i) is nonzero */ tags[i] = k; /* mark i as visited */ } } } } /* construct Lp index array from m_nonZerosPerCol column counts */ Index* Lp = m_matrix._outerIndexPtr(); Lp[0] = 0; for(Index k = 0; k < size; ++k) Lp[k+1] = Lp[k] + m_nonZerosPerCol[k] + (doLDLt ? 0 : 1); m_matrix.resizeNonZeros(Lp[size]); m_isInitialized = true; m_info = Success; m_analysisIsOk = true; m_factorizationIsOk = false; } template template void SimplicialCholeskyBase::factorize(const MatrixType& a) { eigen_assert(m_analysisIsOk && "You must first call analyzePattern()"); eigen_assert(a.rows()==a.cols()); const Index size = a.rows(); eigen_assert(m_parent.size()==size); eigen_assert(m_nonZerosPerCol.size()==size); const Index* Lp = m_matrix._outerIndexPtr(); Index* Li = m_matrix._innerIndexPtr(); Scalar* Lx = m_matrix._valuePtr(); ei_declare_aligned_stack_constructed_variable(Scalar, y, size, 0); ei_declare_aligned_stack_constructed_variable(Index, pattern, size, 0); ei_declare_aligned_stack_constructed_variable(Index, tags, size, 0); SparseMatrix ap(size,size); ap.template selfadjointView() = a.template selfadjointView().twistedBy(m_Pinv); bool ok = true; m_diag.resize(DoLDLt ? size : 0); for(Index k = 0; k < size; ++k) { // compute nonzero pattern of kth row of L, in topological order y[k] = 0.0; // Y(0:k) is now all zero Index top = size; // stack for pattern is empty tags[k] = k; // mark node k as visited m_nonZerosPerCol[k] = 0; // count of nonzeros in column k of L for(typename MatrixType::InnerIterator it(ap,k); it; ++it) { Index i = it.index(); if(i <= k) { y[i] += internal::conj(it.value()); /* scatter A(i,k) into Y (sum duplicates) */ Index len; for(len = 0; tags[i] != k; i = m_parent[i]) { pattern[len++] = i; /* L(k,i) is nonzero */ tags[i] = k; /* mark i as visited */ } while(len > 0) pattern[--top] = pattern[--len]; } } /* compute numerical values kth row of L (a sparse triangular solve) */ Scalar d = y[k]; // get D(k,k) and clear Y(k) y[k] = 0.0; for(; top < size; ++top) { Index i = pattern[top]; /* pattern[top:n-1] is pattern of L(:,k) */ Scalar yi = y[i]; /* get and clear Y(i) */ y[i] = 0.0; /* the nonzero entry L(k,i) */ Scalar l_ki; if(DoLDLt) l_ki = yi / m_diag[i]; else yi = l_ki = yi / Lx[Lp[i]]; Index p2 = Lp[i] + m_nonZerosPerCol[i]; Index p; for(p = Lp[i] + (DoLDLt ? 0 : 1); p < p2; ++p) y[Li[p]] -= internal::conj(Lx[p]) * yi; d -= l_ki * internal::conj(yi); Li[p] = k; /* store L(k,i) in column form of L */ Lx[p] = l_ki; ++m_nonZerosPerCol[i]; /* increment count of nonzeros in col i */ } if(DoLDLt) m_diag[k] = d; else { Index p = Lp[k]+m_nonZerosPerCol[k]++; Li[p] = k ; /* store L(k,k) = sqrt (d) in column k */ Lx[p] = internal::sqrt(d) ; } if(d == Scalar(0)) { ok = false; /* failure, D(k,k) is zero */ break; } } m_info = ok ? Success : NumericalIssue; m_factorizationIsOk = true; } namespace internal { template struct solve_retval, Rhs> : solve_retval_base, Rhs> { typedef SimplicialCholeskyBase Dec; EIGEN_MAKE_SOLVE_HELPERS(Dec,Rhs) template void evalTo(Dest& dst) const { dec().derived()._solve(rhs(),dst); } }; template struct sparse_solve_retval, Rhs> : sparse_solve_retval_base, Rhs> { typedef SimplicialCholeskyBase Dec; EIGEN_MAKE_SPARSE_SOLVE_HELPERS(Dec,Rhs) template void evalTo(Dest& dst) const { dec().derived()._solve_sparse(rhs(),dst); } }; } #endif // EIGEN_SIMPLICIAL_CHOLESKY_H