| // This file is part of Eigen, a lightweight C++ template library |
| // for linear algebra. |
| // |
| // Copyright (C) 2008 Gael Guennebaud <g.gael@free.fr> |
| // |
| // 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 <http://www.gnu.org/licenses/>. |
| |
| #ifndef EIGEN_LLT_H |
| #define EIGEN_LLT_H |
| |
| template<typename MatrixType, int UpLo> struct LLT_Traits; |
| |
| /** \ingroup cholesky_Module |
| * |
| * \class LLT |
| * |
| * \brief Standard Cholesky decomposition (LL^T) of a matrix and associated features |
| * |
| * \param MatrixType the type of the matrix of which we are computing the LL^T Cholesky decomposition |
| * |
| * This class performs a LL^T Cholesky decomposition of a symmetric, positive definite |
| * matrix A such that A = LL^* = U^*U, where L is lower triangular. |
| * |
| * While the Cholesky decomposition is particularly useful to solve selfadjoint problems like D^*D x = b, |
| * for that purpose, we recommend the Cholesky decomposition without square root which is more stable |
| * and even faster. Nevertheless, this standard Cholesky decomposition remains useful in many other |
| * situations like generalised eigen problems with hermitian matrices. |
| * |
| * Remember that Cholesky decompositions are not rank-revealing. This LLT decomposition is only stable on positive definite matrices, |
| * use LDLT instead for the semidefinite case. Also, do not use a Cholesky decomposition to determine whether a system of equations |
| * has a solution. |
| * |
| * \sa MatrixBase::llt(), class LDLT |
| */ |
| /* HEY THIS DOX IS DISABLED BECAUSE THERE's A BUG EITHER HERE OR IN LDLT ABOUT THAT (OR BOTH) |
| * Note that during the decomposition, only the upper triangular part of A is considered. Therefore, |
| * the strict lower part does not have to store correct values. |
| */ |
| template<typename _MatrixType, int _UpLo> class LLT |
| { |
| public: |
| typedef _MatrixType MatrixType; |
| enum { |
| RowsAtCompileTime = MatrixType::RowsAtCompileTime, |
| ColsAtCompileTime = MatrixType::ColsAtCompileTime, |
| Options = MatrixType::Options, |
| MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime |
| }; |
| typedef typename MatrixType::Scalar Scalar; |
| typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar; |
| typedef typename MatrixType::Index Index; |
| |
| enum { |
| PacketSize = ei_packet_traits<Scalar>::size, |
| AlignmentMask = int(PacketSize)-1, |
| UpLo = _UpLo |
| }; |
| |
| typedef LLT_Traits<MatrixType,UpLo> Traits; |
| |
| /** |
| * \brief Default Constructor. |
| * |
| * The default constructor is useful in cases in which the user intends to |
| * perform decompositions via LLT::compute(const MatrixType&). |
| */ |
| LLT() : m_matrix(), 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 LLT() |
| */ |
| LLT(Index size) : m_matrix(size, size), |
| m_isInitialized(false) {} |
| |
| LLT(const MatrixType& matrix) |
| : m_matrix(matrix.rows(), matrix.cols()), |
| m_isInitialized(false) |
| { |
| compute(matrix); |
| } |
| |
| /** \returns a view of the upper triangular matrix U */ |
| inline typename Traits::MatrixU matrixU() const |
| { |
| ei_assert(m_isInitialized && "LLT is not initialized."); |
| return Traits::getU(m_matrix); |
| } |
| |
| /** \returns a view of the lower triangular matrix L */ |
| inline typename Traits::MatrixL matrixL() const |
| { |
| ei_assert(m_isInitialized && "LLT is not initialized."); |
| return Traits::getL(m_matrix); |
| } |
| |
| /** \returns the solution x of \f$ A x = b \f$ using the current decomposition of A. |
| * |
| * Since this LLT class assumes anyway that the matrix A is invertible, the solution |
| * theoretically exists and is unique regardless of b. |
| * |
| * Example: \include LLT_solve.cpp |
| * Output: \verbinclude LLT_solve.out |
| * |
| * \sa solveInPlace(), MatrixBase::llt() |
| */ |
| template<typename Rhs> |
| inline const ei_solve_retval<LLT, Rhs> |
| solve(const MatrixBase<Rhs>& b) const |
| { |
| ei_assert(m_isInitialized && "LLT is not initialized."); |
| ei_assert(m_matrix.rows()==b.rows() |
| && "LLT::solve(): invalid number of rows of the right hand side matrix b"); |
| return ei_solve_retval<LLT, Rhs>(*this, b.derived()); |
| } |
| |
| template<typename Derived> |
| bool solveInPlace(MatrixBase<Derived> &bAndX) const; |
| |
| LLT& compute(const MatrixType& matrix); |
| |
| /** \returns the LLT decomposition matrix |
| * |
| * TODO: document the storage layout |
| */ |
| inline const MatrixType& matrixLLT() const |
| { |
| ei_assert(m_isInitialized && "LLT 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(); } |
| |
| protected: |
| /** \internal |
| * Used to compute and store L |
| * The strict upper part is not used and even not initialized. |
| */ |
| MatrixType m_matrix; |
| bool m_isInitialized; |
| }; |
| |
| // forward declaration (defined at the end of this file) |
| template<int UpLo> struct ei_llt_inplace; |
| |
| template<> struct ei_llt_inplace<Lower> |
| { |
| template<typename MatrixType> |
| static bool unblocked(MatrixType& mat) |
| { |
| typedef typename MatrixType::Scalar Scalar; |
| typedef typename MatrixType::RealScalar RealScalar; |
| typedef typename MatrixType::Index Index; |
| ei_assert(mat.rows()==mat.cols()); |
| const Index size = mat.rows(); |
| for(Index k = 0; k < size; ++k) |
| { |
| Index rs = size-k-1; // remaining size |
| |
| 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); |
| |
| RealScalar x = ei_real(mat.coeff(k,k)); |
| if (k>0) x -= mat.row(k).head(k).squaredNorm(); |
| if (x<=RealScalar(0)) |
| return false; |
| mat.coeffRef(k,k) = x = ei_sqrt(x); |
| if (k>0 && rs>0) A21.noalias() -= A20 * A10.adjoint(); |
| if (rs>0) A21 *= RealScalar(1)/x; |
| } |
| return true; |
| } |
| |
| template<typename MatrixType> |
| static bool blocked(MatrixType& m) |
| { |
| typedef typename MatrixType::Index Index; |
| ei_assert(m.rows()==m.cols()); |
| Index size = m.rows(); |
| if(size<32) |
| return unblocked(m); |
| |
| Index blockSize = size/8; |
| blockSize = (blockSize/16)*16; |
| blockSize = std::min(std::max(blockSize,Index(8)), Index(128)); |
| |
| for (Index k=0; k<size; k+=blockSize) |
| { |
| Index bs = std::min(blockSize, size-k); |
| Index rs = size - k - bs; |
| |
| Block<MatrixType,Dynamic,Dynamic> A11(m,k, k, bs,bs); |
| Block<MatrixType,Dynamic,Dynamic> A21(m,k+bs,k, rs,bs); |
| Block<MatrixType,Dynamic,Dynamic> A22(m,k+bs,k+bs,rs,rs); |
| |
| if(!unblocked(A11)) return false; |
| if(rs>0) A11.adjoint().template triangularView<Upper>().template solveInPlace<OnTheRight>(A21); |
| if(rs>0) A22.template selfadjointView<Lower>().rankUpdate(A21,-1); // bottleneck |
| } |
| return true; |
| } |
| }; |
| |
| template<> struct ei_llt_inplace<Upper> |
| { |
| template<typename MatrixType> |
| static EIGEN_STRONG_INLINE bool unblocked(MatrixType& mat) |
| { |
| Transpose<MatrixType> matt(mat); |
| return ei_llt_inplace<Lower>::unblocked(matt); |
| } |
| template<typename MatrixType> |
| static EIGEN_STRONG_INLINE bool blocked(MatrixType& mat) |
| { |
| Transpose<MatrixType> matt(mat); |
| return ei_llt_inplace<Lower>::blocked(matt); |
| } |
| }; |
| |
| template<typename MatrixType> struct LLT_Traits<MatrixType,Lower> |
| { |
| typedef TriangularView<MatrixType, Lower> MatrixL; |
| typedef TriangularView<typename MatrixType::AdjointReturnType, Upper> MatrixU; |
| inline static MatrixL getL(const MatrixType& m) { return m; } |
| inline static MatrixU getU(const MatrixType& m) { return m.adjoint(); } |
| static bool inplace_decomposition(MatrixType& m) |
| { return ei_llt_inplace<Lower>::blocked(m); } |
| }; |
| |
| template<typename MatrixType> struct LLT_Traits<MatrixType,Upper> |
| { |
| typedef TriangularView<typename MatrixType::AdjointReturnType, Lower> MatrixL; |
| typedef TriangularView<MatrixType, Upper> MatrixU; |
| inline static MatrixL getL(const MatrixType& m) { return m.adjoint(); } |
| inline static MatrixU getU(const MatrixType& m) { return m; } |
| static bool inplace_decomposition(MatrixType& m) |
| { return ei_llt_inplace<Upper>::blocked(m); } |
| }; |
| |
| /** Computes / recomputes the Cholesky decomposition A = LL^* = U^*U of \a matrix |
| * |
| * |
| * \returns a reference to *this |
| */ |
| template<typename MatrixType, int _UpLo> |
| LLT<MatrixType,_UpLo>& LLT<MatrixType,_UpLo>::compute(const MatrixType& a) |
| { |
| assert(a.rows()==a.cols()); |
| const Index size = a.rows(); |
| m_matrix.resize(size, size); |
| m_matrix = a; |
| |
| m_isInitialized = Traits::inplace_decomposition(m_matrix); |
| return *this; |
| } |
| |
| template<typename _MatrixType, int UpLo, typename Rhs> |
| struct ei_solve_retval<LLT<_MatrixType, UpLo>, Rhs> |
| : ei_solve_retval_base<LLT<_MatrixType, UpLo>, Rhs> |
| { |
| typedef LLT<_MatrixType,UpLo> LLTType; |
| EIGEN_MAKE_SOLVE_HELPERS(LLTType,Rhs) |
| |
| template<typename Dest> void evalTo(Dest& dst) const |
| { |
| dst = rhs(); |
| dec().solveInPlace(dst); |
| } |
| }; |
| |
| /** 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 LLT::solve(), MatrixBase::llt() |
| */ |
| template<typename MatrixType, int _UpLo> |
| template<typename Derived> |
| bool LLT<MatrixType,_UpLo>::solveInPlace(MatrixBase<Derived> &bAndX) const |
| { |
| ei_assert(m_isInitialized && "LLT is not initialized."); |
| ei_assert(m_matrix.rows()==bAndX.rows()); |
| matrixL().solveInPlace(bAndX); |
| matrixU().solveInPlace(bAndX); |
| return true; |
| } |
| |
| /** \returns the matrix represented by the decomposition, |
| * i.e., it returns the product: L L^*. |
| * This function is provided for debug purpose. */ |
| template<typename MatrixType, int _UpLo> |
| MatrixType LLT<MatrixType,_UpLo>::reconstructedMatrix() const |
| { |
| ei_assert(m_isInitialized && "LLT is not initialized."); |
| return matrixL() * matrixL().adjoint().toDenseMatrix(); |
| } |
| |
| /** \cholesky_module |
| * \returns the LLT decomposition of \c *this |
| */ |
| template<typename Derived> |
| inline const LLT<typename MatrixBase<Derived>::PlainObject> |
| MatrixBase<Derived>::llt() const |
| { |
| return LLT<PlainObject>(derived()); |
| } |
| |
| /** \cholesky_module |
| * \returns the LLT decomposition of \c *this |
| */ |
| template<typename MatrixType, unsigned int UpLo> |
| inline const LLT<typename SelfAdjointView<MatrixType, UpLo>::PlainObject, UpLo> |
| SelfAdjointView<MatrixType, UpLo>::llt() const |
| { |
| return LLT<PlainObject,UpLo>(m_matrix); |
| } |
| |
| #endif // EIGEN_LLT_H |