Standard Cholesky decomposition (LL^T) of a matrix and associated features.
| 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.
Public Types | |
| enum | { RowsAtCompileTime = MatrixType::RowsAtCompileTime, ColsAtCompileTime = MatrixType::ColsAtCompileTime, Options = MatrixType::Options, MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime } |
| enum | { PacketSize = internal::packet_traits<Scalar>::size, AlignmentMask = int(PacketSize)-1, UpLo = _UpLo } |
| typedef _MatrixType | MatrixType |
| typedef MatrixType::Scalar | Scalar |
| typedef NumTraits< typename MatrixType::Scalar >::Real | RealScalar |
| typedef MatrixType::Index | Index |
| typedef internal::LLT_Traits < MatrixType, UpLo > | Traits |
Public Member Functions | |
| LLT () | |
| Default Constructor. | |
| LLT (Index size) | |
| Default Constructor with memory preallocation. | |
| LLT (const MatrixType &matrix) | |
| Traits::MatrixU | matrixU () const |
| Traits::MatrixL | matrixL () const |
| template<typename Rhs > | |
| const internal::solve_retval < LLT, Rhs > | solve (const MatrixBase< Rhs > &b) const |
| template<typename Derived > | |
| void | solveInPlace (MatrixBase< Derived > &bAndX) const |
| LLT & | compute (const MatrixType &matrix) |
| Computes / recomputes the Cholesky decomposition A = LL^* = U^*U of matrix. | |
| const MatrixType & | matrixLLT () const |
| MatrixType | reconstructedMatrix () const |
| ComputationInfo | info () const |
| Reports whether previous computation was successful. | |
| Index | rows () const |
| Index | cols () const |
| template<typename VectorType > | |
| void | rankUpdate (const VectorType &vec) |
| Performs a rank one update of the current decomposition. | |
Protected Attributes | |
| MatrixType | m_matrix |
| bool | m_isInitialized |
| ComputationInfo | m_info |
| typedef MatrixType::Index Eigen::LLT::Index |
| typedef _MatrixType Eigen::LLT::MatrixType |
| typedef NumTraits<typename MatrixType::Scalar>::Real Eigen::LLT::RealScalar |
| typedef MatrixType::Scalar Eigen::LLT::Scalar |
| typedef internal::LLT_Traits<MatrixType,UpLo> Eigen::LLT::Traits |
| anonymous enum |
| Eigen::LLT::LLT | ( | ) | [inline] |
Default Constructor.
The default constructor is useful in cases in which the user intends to perform decompositions via LLT::compute(const MatrixType&).
| Eigen::LLT::LLT | ( | Index | size | ) | [inline] |
| Eigen::LLT::LLT | ( | const MatrixType & | matrix | ) | [inline] |
| LLT< MatrixType, _UpLo > & Eigen::LLT::compute | ( | const MatrixType & | a | ) |
| ComputationInfo Eigen::LLT::info | ( | ) | const [inline] |
| Traits::MatrixL Eigen::LLT::matrixL | ( | ) | const [inline] |
| const MatrixType& Eigen::LLT::matrixLLT | ( | ) | const [inline] |
| Traits::MatrixU Eigen::LLT::matrixU | ( | ) | const [inline] |
| void Eigen::LLT::rankUpdate | ( | const VectorType & | v | ) |
| MatrixType Eigen::LLT::reconstructedMatrix | ( | ) | const |
| const internal::solve_retval<LLT, Rhs> Eigen::LLT::solve | ( | const MatrixBase< Rhs > & | b | ) | const [inline] |
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:
Output:
| void Eigen::LLT::solveInPlace | ( | MatrixBase< Derived > & | bAndX | ) | const |
ComputationInfo Eigen::LLT::m_info [protected] |
bool Eigen::LLT::m_isInitialized [protected] |
MatrixType Eigen::LLT::m_matrix [protected] |
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