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Diffstat (limited to 'src/Eigen/src/Cholesky/LDLT.h')
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diff --git a/src/Eigen/src/Cholesky/LDLT.h b/src/Eigen/src/Cholesky/LDLT.h deleted file mode 100644 index c52b7d1..0000000 --- a/src/Eigen/src/Cholesky/LDLT.h +++ /dev/null @@ -1,604 +0,0 @@ -// This file is part of Eigen, a lightweight C++ template library -// for linear algebra. -// -// Copyright (C) 2008-2011 Gael Guennebaud <gael.guennebaud@inria.fr> -// Copyright (C) 2009 Keir Mierle <mierle@gmail.com> -// Copyright (C) 2009 Benoit Jacob <jacob.benoit.1@gmail.com> -// Copyright (C) 2011 Timothy E. Holy <tim.holy@gmail.com > -// -// 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_LDLT_H -#define EIGEN_LDLT_H - -namespace Eigen { - -namespace internal { - template<typename MatrixType, int UpLo> struct LDLT_Traits; - - // PositiveSemiDef means positive semi-definite and non-zero; same for NegativeSemiDef - enum SignMatrix { PositiveSemiDef, NegativeSemiDef, ZeroSign, Indefinite }; -} - -/** \ingroup Cholesky_Module - * - * \class LDLT - * - * \brief Robust Cholesky decomposition of a matrix with pivoting - * - * \param MatrixType the type of the matrix of which to compute the LDL^T Cholesky decomposition - * \param UpLo the triangular part that will be used for the decompositon: Lower (default) or Upper. - * The other triangular part won't be read. - * - * Perform a robust Cholesky decomposition of a positive semidefinite or negative semidefinite - * matrix \f$ A \f$ such that \f$ A = P^TLDL^*P \f$, where P is a permutation matrix, L - * is lower triangular with a unit diagonal and D is a diagonal matrix. - * - * The decomposition uses pivoting to ensure stability, so that L will have - * zeros in the bottom right rank(A) - n submatrix. Avoiding the square root - * on D also stabilizes the computation. - * - * Remember that Cholesky decompositions are not rank-revealing. Also, do not use a Cholesky - * decomposition to determine whether a system of equations has a solution. - * - * \sa MatrixBase::ldlt(), class LLT - */ -template<typename _MatrixType, int _UpLo> class LDLT -{ - public: - typedef _MatrixType MatrixType; - enum { - RowsAtCompileTime = MatrixType::RowsAtCompileTime, - ColsAtCompileTime = MatrixType::ColsAtCompileTime, - Options = MatrixType::Options & ~RowMajorBit, // these are the options for the TmpMatrixType, we need a ColMajor matrix here! - MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime, - MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime, - UpLo = _UpLo - }; - typedef typename MatrixType::Scalar Scalar; - typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar; - typedef typename MatrixType::Index Index; - typedef Matrix<Scalar, RowsAtCompileTime, 1, Options, MaxRowsAtCompileTime, 1> TmpMatrixType; - - typedef Transpositions<RowsAtCompileTime, MaxRowsAtCompileTime> TranspositionType; - typedef PermutationMatrix<RowsAtCompileTime, MaxRowsAtCompileTime> PermutationType; - - typedef internal::LDLT_Traits<MatrixType,UpLo> Traits; - - /** \brief Default Constructor. - * - * The default constructor is useful in cases in which the user intends to - * perform decompositions via LDLT::compute(const MatrixType&). - */ - LDLT() - : m_matrix(), - m_transpositions(), - m_sign(internal::ZeroSign), - m_isInitialized(false) - {} - - /** \brief Default Constructor with memory preallocation - * - * Like the default constructor but with preallocation of the internal data - * according to the specified problem \a size. - * \sa LDLT() - */ - LDLT(Index size) - : m_matrix(size, size), - m_transpositions(size), - m_temporary(size), - m_sign(internal::ZeroSign), - m_isInitialized(false) - {} - - /** \brief Constructor with decomposition - * - * This calculates the decomposition for the input \a matrix. - * \sa LDLT(Index size) - */ - LDLT(const MatrixType& matrix) - : m_matrix(matrix.rows(), matrix.cols()), - m_transpositions(matrix.rows()), - m_temporary(matrix.rows()), - m_sign(internal::ZeroSign), - m_isInitialized(false) - { - compute(matrix); - } - - /** Clear any existing decomposition - * \sa rankUpdate(w,sigma) - */ - void setZero() - { - m_isInitialized = false; - } - - /** \returns a view of the upper triangular matrix U */ - inline typename Traits::MatrixU matrixU() const - { - eigen_assert(m_isInitialized && "LDLT is not initialized."); - return Traits::getU(m_matrix); - } - - /** \returns a view of the lower triangular matrix L */ - inline typename Traits::MatrixL matrixL() const - { - eigen_assert(m_isInitialized && "LDLT is not initialized."); - return Traits::getL(m_matrix); - } - - /** \returns the permutation matrix P as a transposition sequence. - */ - inline const TranspositionType& transpositionsP() const - { - eigen_assert(m_isInitialized && "LDLT is not initialized."); - return m_transpositions; - } - - /** \returns the coefficients of the diagonal matrix D */ - inline Diagonal<const MatrixType> vectorD() const - { - eigen_assert(m_isInitialized && "LDLT is not initialized."); - return m_matrix.diagonal(); - } - - /** \returns true if the matrix is positive (semidefinite) */ - inline bool isPositive() const - { - eigen_assert(m_isInitialized && "LDLT is not initialized."); - return m_sign == internal::PositiveSemiDef || m_sign == internal::ZeroSign; - } - - #ifdef EIGEN2_SUPPORT - inline bool isPositiveDefinite() const - { - return isPositive(); - } - #endif - - /** \returns true if the matrix is negative (semidefinite) */ - inline bool isNegative(void) const - { - eigen_assert(m_isInitialized && "LDLT is not initialized."); - return m_sign == internal::NegativeSemiDef || m_sign == internal::ZeroSign; - } - - /** \returns a solution x of \f$ A x = b \f$ using the current decomposition of A. - * - * This function also supports in-place solves using the syntax <tt>x = decompositionObject.solve(x)</tt> . - * - * \note_about_checking_solutions - * - * More precisely, this method solves \f$ A x = b \f$ using the decomposition \f$ A = P^T L D L^* P \f$ - * by solving the systems \f$ P^T y_1 = b \f$, \f$ L y_2 = y_1 \f$, \f$ D y_3 = y_2 \f$, - * \f$ L^* y_4 = y_3 \f$ and \f$ P x = y_4 \f$ in succession. If the matrix \f$ A \f$ is singular, then - * \f$ D \f$ will also be singular (all the other matrices are invertible). In that case, the - * least-square solution of \f$ D y_3 = y_2 \f$ is computed. This does not mean that this function - * computes the least-square solution of \f$ A x = b \f$ is \f$ A \f$ is singular. - * - * \sa MatrixBase::ldlt() - */ - template<typename Rhs> - inline const internal::solve_retval<LDLT, Rhs> - solve(const MatrixBase<Rhs>& b) const - { - eigen_assert(m_isInitialized && "LDLT is not initialized."); - eigen_assert(m_matrix.rows()==b.rows() - && "LDLT::solve(): invalid number of rows of the right hand side matrix b"); - return internal::solve_retval<LDLT, Rhs>(*this, b.derived()); - } - - #ifdef EIGEN2_SUPPORT - template<typename OtherDerived, typename ResultType> - bool solve(const MatrixBase<OtherDerived>& b, ResultType *result) const - { - *result = this->solve(b); - return true; - } - #endif - - template<typename Derived> - bool solveInPlace(MatrixBase<Derived> &bAndX) const; - - LDLT& compute(const MatrixType& matrix); - - template <typename Derived> - LDLT& rankUpdate(const MatrixBase<Derived>& w, const RealScalar& alpha=1); - - /** \returns the internal LDLT decomposition matrix - * - * TODO: document the storage layout - */ - inline const MatrixType& matrixLDLT() const - { - eigen_assert(m_isInitialized && "LDLT is not initialized."); - return m_matrix; - } - - MatrixType reconstructedMatrix() const; - - inline Index rows() const { return m_matrix.rows(); } - inline Index cols() const { return m_matrix.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 && "LDLT is not initialized."); - return Success; - } - - protected: - - /** \internal - * Used to compute and store the Cholesky decomposition A = L D L^* = U^* D U. - * The strict upper part is used during the decomposition, the strict lower - * part correspond to the coefficients of L (its diagonal is equal to 1 and - * is not stored), and the diagonal entries correspond to D. - */ - MatrixType m_matrix; - TranspositionType m_transpositions; - TmpMatrixType m_temporary; - internal::SignMatrix m_sign; - bool m_isInitialized; -}; - -namespace internal { - -template<int UpLo> struct ldlt_inplace; - -template<> struct ldlt_inplace<Lower> -{ - template<typename MatrixType, typename TranspositionType, typename Workspace> - static bool unblocked(MatrixType& mat, TranspositionType& transpositions, Workspace& temp, SignMatrix& sign) - { - using std::abs; - typedef typename MatrixType::Scalar Scalar; - typedef typename MatrixType::RealScalar RealScalar; - typedef typename MatrixType::Index Index; - eigen_assert(mat.rows()==mat.cols()); - const Index size = mat.rows(); - - if (size <= 1) - { - transpositions.setIdentity(); - if (numext::real(mat.coeff(0,0)) > 0) sign = PositiveSemiDef; - else if (numext::real(mat.coeff(0,0)) < 0) sign = NegativeSemiDef; - else sign = ZeroSign; - return true; - } - - for (Index k = 0; k < size; ++k) - { - // Find largest diagonal element - Index index_of_biggest_in_corner; - mat.diagonal().tail(size-k).cwiseAbs().maxCoeff(&index_of_biggest_in_corner); - index_of_biggest_in_corner += k; - - transpositions.coeffRef(k) = index_of_biggest_in_corner; - if(k != index_of_biggest_in_corner) - { - // apply the transposition while taking care to consider only - // the lower triangular part - Index s = size-index_of_biggest_in_corner-1; // trailing size after the biggest element - mat.row(k).head(k).swap(mat.row(index_of_biggest_in_corner).head(k)); - mat.col(k).tail(s).swap(mat.col(index_of_biggest_in_corner).tail(s)); - std::swap(mat.coeffRef(k,k),mat.coeffRef(index_of_biggest_in_corner,index_of_biggest_in_corner)); - for(int i=k+1;i<index_of_biggest_in_corner;++i) - { - Scalar tmp = mat.coeffRef(i,k); - mat.coeffRef(i,k) = numext::conj(mat.coeffRef(index_of_biggest_in_corner,i)); - mat.coeffRef(index_of_biggest_in_corner,i) = numext::conj(tmp); - } - if(NumTraits<Scalar>::IsComplex) - mat.coeffRef(index_of_biggest_in_corner,k) = numext::conj(mat.coeff(index_of_biggest_in_corner,k)); - } - - // partition the matrix: - // A00 | - | - - // lu = A10 | A11 | - - // A20 | A21 | A22 - Index rs = size - k - 1; - Block<MatrixType,Dynamic,1> A21(mat,k+1,k,rs,1); - Block<MatrixType,1,Dynamic> A10(mat,k,0,1,k); - Block<MatrixType,Dynamic,Dynamic> A20(mat,k+1,0,rs,k); - - if(k>0) - { - temp.head(k) = mat.diagonal().real().head(k).asDiagonal() * A10.adjoint(); - mat.coeffRef(k,k) -= (A10 * temp.head(k)).value(); - if(rs>0) - A21.noalias() -= A20 * temp.head(k); - } - - // In some previous versions of Eigen (e.g., 3.2.1), the scaling was omitted if the pivot - // was smaller than the cutoff value. However, soince LDLT is not rank-revealing - // we should only make sure we do not introduce INF or NaN values. - // LAPACK also uses 0 as the cutoff value. - RealScalar realAkk = numext::real(mat.coeffRef(k,k)); - if((rs>0) && (abs(realAkk) > RealScalar(0))) - A21 /= realAkk; - - if (sign == PositiveSemiDef) { - if (realAkk < 0) sign = Indefinite; - } else if (sign == NegativeSemiDef) { - if (realAkk > 0) sign = Indefinite; - } else if (sign == ZeroSign) { - if (realAkk > 0) sign = PositiveSemiDef; - else if (realAkk < 0) sign = NegativeSemiDef; - } - } - - return true; - } - - // Reference for the algorithm: Davis and Hager, "Multiple Rank - // Modifications of a Sparse Cholesky Factorization" (Algorithm 1) - // Trivial rearrangements of their computations (Timothy E. Holy) - // allow their algorithm to work for rank-1 updates even if the - // original matrix is not of full rank. - // Here only rank-1 updates are implemented, to reduce the - // requirement for intermediate storage and improve accuracy - template<typename MatrixType, typename WDerived> - static bool updateInPlace(MatrixType& mat, MatrixBase<WDerived>& w, const typename MatrixType::RealScalar& sigma=1) - { - using numext::isfinite; - typedef typename MatrixType::Scalar Scalar; - typedef typename MatrixType::RealScalar RealScalar; - typedef typename MatrixType::Index Index; - - const Index size = mat.rows(); - eigen_assert(mat.cols() == size && w.size()==size); - - RealScalar alpha = 1; - - // Apply the update - for (Index j = 0; j < size; j++) - { - // Check for termination due to an original decomposition of low-rank - if (!(isfinite)(alpha)) - break; - - // Update the diagonal terms - RealScalar dj = numext::real(mat.coeff(j,j)); - Scalar wj = w.coeff(j); - RealScalar swj2 = sigma*numext::abs2(wj); - RealScalar gamma = dj*alpha + swj2; - - mat.coeffRef(j,j) += swj2/alpha; - alpha += swj2/dj; - - - // Update the terms of L - Index rs = size-j-1; - w.tail(rs) -= wj * mat.col(j).tail(rs); - if(gamma != 0) - mat.col(j).tail(rs) += (sigma*numext::conj(wj)/gamma)*w.tail(rs); - } - return true; - } - - template<typename MatrixType, typename TranspositionType, typename Workspace, typename WType> - static bool update(MatrixType& mat, const TranspositionType& transpositions, Workspace& tmp, const WType& w, const typename MatrixType::RealScalar& sigma=1) - { - // Apply the permutation to the input w - tmp = transpositions * w; - - return ldlt_inplace<Lower>::updateInPlace(mat,tmp,sigma); - } -}; - -template<> struct ldlt_inplace<Upper> -{ - template<typename MatrixType, typename TranspositionType, typename Workspace> - static EIGEN_STRONG_INLINE bool unblocked(MatrixType& mat, TranspositionType& transpositions, Workspace& temp, SignMatrix& sign) - { - Transpose<MatrixType> matt(mat); - return ldlt_inplace<Lower>::unblocked(matt, transpositions, temp, sign); - } - - template<typename MatrixType, typename TranspositionType, typename Workspace, typename WType> - static EIGEN_STRONG_INLINE bool update(MatrixType& mat, TranspositionType& transpositions, Workspace& tmp, WType& w, const typename MatrixType::RealScalar& sigma=1) - { - Transpose<MatrixType> matt(mat); - return ldlt_inplace<Lower>::update(matt, transpositions, tmp, w.conjugate(), sigma); - } -}; - -template<typename MatrixType> struct LDLT_Traits<MatrixType,Lower> -{ - typedef const TriangularView<const MatrixType, UnitLower> MatrixL; - typedef const TriangularView<const typename MatrixType::AdjointReturnType, UnitUpper> MatrixU; - static inline MatrixL getL(const MatrixType& m) { return m; } - static inline MatrixU getU(const MatrixType& m) { return m.adjoint(); } -}; - -template<typename MatrixType> struct LDLT_Traits<MatrixType,Upper> -{ - typedef const TriangularView<const typename MatrixType::AdjointReturnType, UnitLower> MatrixL; - typedef const TriangularView<const MatrixType, UnitUpper> MatrixU; - static inline MatrixL getL(const MatrixType& m) { return m.adjoint(); } - static inline MatrixU getU(const MatrixType& m) { return m; } -}; - -} // end namespace internal - -/** Compute / recompute the LDLT decomposition A = L D L^* = U^* D U of \a matrix - */ -template<typename MatrixType, int _UpLo> -LDLT<MatrixType,_UpLo>& LDLT<MatrixType,_UpLo>::compute(const MatrixType& a) -{ - eigen_assert(a.rows()==a.cols()); - const Index size = a.rows(); - - m_matrix = a; - - m_transpositions.resize(size); - m_isInitialized = false; - m_temporary.resize(size); - - internal::ldlt_inplace<UpLo>::unblocked(m_matrix, m_transpositions, m_temporary, m_sign); - - m_isInitialized = true; - return *this; -} - -/** Update the LDLT decomposition: given A = L D L^T, efficiently compute the decomposition of A + sigma w w^T. - * \param w a vector to be incorporated into the decomposition. - * \param sigma a scalar, +1 for updates and -1 for "downdates," which correspond to removing previously-added column vectors. Optional; default value is +1. - * \sa setZero() - */ -template<typename MatrixType, int _UpLo> -template<typename Derived> -LDLT<MatrixType,_UpLo>& LDLT<MatrixType,_UpLo>::rankUpdate(const MatrixBase<Derived>& w, const typename NumTraits<typename MatrixType::Scalar>::Real& sigma) -{ - const Index size = w.rows(); - if (m_isInitialized) - { - eigen_assert(m_matrix.rows()==size); - } - else - { - m_matrix.resize(size,size); - m_matrix.setZero(); - m_transpositions.resize(size); - for (Index i = 0; i < size; i++) - m_transpositions.coeffRef(i) = i; - m_temporary.resize(size); - m_sign = sigma>=0 ? internal::PositiveSemiDef : internal::NegativeSemiDef; - m_isInitialized = true; - } - - internal::ldlt_inplace<UpLo>::update(m_matrix, m_transpositions, m_temporary, w, sigma); - - return *this; -} - -namespace internal { -template<typename _MatrixType, int _UpLo, typename Rhs> -struct solve_retval<LDLT<_MatrixType,_UpLo>, Rhs> - : solve_retval_base<LDLT<_MatrixType,_UpLo>, Rhs> -{ - typedef LDLT<_MatrixType,_UpLo> LDLTType; - EIGEN_MAKE_SOLVE_HELPERS(LDLTType,Rhs) - - template<typename Dest> void evalTo(Dest& dst) const - { - eigen_assert(rhs().rows() == dec().matrixLDLT().rows()); - // dst = P b - dst = dec().transpositionsP() * rhs(); - - // dst = L^-1 (P b) - dec().matrixL().solveInPlace(dst); - - // dst = D^-1 (L^-1 P b) - // more precisely, use pseudo-inverse of D (see bug 241) - using std::abs; - using std::max; - typedef typename LDLTType::MatrixType MatrixType; - typedef typename LDLTType::Scalar Scalar; - typedef typename LDLTType::RealScalar RealScalar; - const typename Diagonal<const MatrixType>::RealReturnType vectorD(dec().vectorD()); - // In some previous versions, tolerance was set to the max of 1/highest and the maximal diagonal entry * epsilon - // as motivated by LAPACK's xGELSS: - // RealScalar tolerance = (max)(vectorD.array().abs().maxCoeff() *NumTraits<RealScalar>::epsilon(),RealScalar(1) / NumTraits<RealScalar>::highest()); - // However, LDLT is not rank revealing, and so adjusting the tolerance wrt to the highest - // diagonal element is not well justified and to numerical issues in some cases. - // Moreover, Lapack's xSYTRS routines use 0 for the tolerance. - RealScalar tolerance = RealScalar(1) / NumTraits<RealScalar>::highest(); - - for (Index i = 0; i < vectorD.size(); ++i) { - if(abs(vectorD(i)) > tolerance) - dst.row(i) /= vectorD(i); - else - dst.row(i).setZero(); - } - - // dst = L^-T (D^-1 L^-1 P b) - dec().matrixU().solveInPlace(dst); - - // dst = P^-1 (L^-T D^-1 L^-1 P b) = A^-1 b - dst = dec().transpositionsP().transpose() * dst; - } -}; -} - -/** \internal use x = ldlt_object.solve(x); - * - * This is the \em in-place version of solve(). - * - * \param bAndX represents both the right-hand side matrix b and result x. - * - * \returns true always! If you need to check for existence of solutions, use another decomposition like LU, QR, or SVD. - * - * This version avoids a copy when the right hand side matrix b is not - * needed anymore. - * - * \sa LDLT::solve(), MatrixBase::ldlt() - */ -template<typename MatrixType,int _UpLo> -template<typename Derived> -bool LDLT<MatrixType,_UpLo>::solveInPlace(MatrixBase<Derived> &bAndX) const -{ - eigen_assert(m_isInitialized && "LDLT is not initialized."); - eigen_assert(m_matrix.rows() == bAndX.rows()); - - bAndX = this->solve(bAndX); - - return true; -} - -/** \returns the matrix represented by the decomposition, - * i.e., it returns the product: P^T L D L^* P. - * This function is provided for debug purpose. */ -template<typename MatrixType, int _UpLo> -MatrixType LDLT<MatrixType,_UpLo>::reconstructedMatrix() const -{ - eigen_assert(m_isInitialized && "LDLT is not initialized."); - const Index size = m_matrix.rows(); - MatrixType res(size,size); - - // P - res.setIdentity(); - res = transpositionsP() * res; - // L^* P - res = matrixU() * res; - // D(L^*P) - res = vectorD().real().asDiagonal() * res; - // L(DL^*P) - res = matrixL() * res; - // P^T (LDL^*P) - res = transpositionsP().transpose() * res; - - return res; -} - -/** \cholesky_module - * \returns the Cholesky decomposition with full pivoting without square root of \c *this - */ -template<typename MatrixType, unsigned int UpLo> -inline const LDLT<typename SelfAdjointView<MatrixType, UpLo>::PlainObject, UpLo> -SelfAdjointView<MatrixType, UpLo>::ldlt() const -{ - return LDLT<PlainObject,UpLo>(m_matrix); -} - -/** \cholesky_module - * \returns the Cholesky decomposition with full pivoting without square root of \c *this - */ -template<typename Derived> -inline const LDLT<typename MatrixBase<Derived>::PlainObject> -MatrixBase<Derived>::ldlt() const -{ - return LDLT<PlainObject>(derived()); -} - -} // end namespace Eigen - -#endif // EIGEN_LDLT_H |