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Eigen
3.3.4
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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 |
_UpLo | the triangular part that will be used for the decompositon: Lower (default) or Upper. The other triangular part won't be read. |
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.
Example:
Output:
The matrix A is 4 -1 2 -1 6 0 2 0 5 The Cholesky factor L is 2 0 0 -0.5 2.4 0 1 0.209 1.99 To check this, let us compute L * L.transpose() 4 -1 2 -1 6 0 2 0 5 This should equal the matrix A
This class supports the inplace decomposition mechanism.
Public Types | |
typedef Eigen::Index | Index |
Public Member Functions | |
const LLT & | adjoint () const |
template<typename InputType > | |
LLT< MatrixType, _UpLo > & | compute (const EigenBase< InputType > &a) |
ComputationInfo | info () const |
Reports whether previous computation was successful. More... | |
LLT () | |
Default Constructor. More... | |
LLT (Index size) | |
Default Constructor with memory preallocation. More... | |
template<typename InputType > | |
LLT (EigenBase< InputType > &matrix) | |
Constructs a LDLT factorization from a given matrix. More... | |
Traits::MatrixL | matrixL () const |
const MatrixType & | matrixLLT () const |
Traits::MatrixU | matrixU () const |
template<typename VectorType > | |
LLT< _MatrixType, _UpLo > | rankUpdate (const VectorType &v, const RealScalar &sigma) |
RealScalar | rcond () const |
MatrixType | reconstructedMatrix () const |
template<typename Rhs > | |
const Solve< LLT, Rhs > | solve (const MatrixBase< Rhs > &b) const |
typedef Eigen::Index Eigen::LLT< _MatrixType, _UpLo >::Index |
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Default Constructor.
The default constructor is useful in cases in which the user intends to perform decompositions via LLT::compute(const MatrixType&).
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Default Constructor with memory preallocation.
Like the default constructor but with preallocation of the internal data according to the specified problem size.
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Constructs a LDLT factorization from a given matrix.
This overloaded constructor is provided for inplace decomposition when MatrixType
is a Eigen::Ref.
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*this
, that is, a const reference to the decomposition itself as the underlying matrix is self-adjoint.This method is provided for compatibility with other matrix decompositions, thus enabling generic code such as:
LLT<MatrixType,_UpLo>& Eigen::LLT< _MatrixType, _UpLo >::compute | ( | const EigenBase< InputType > & | a | ) |
Computes / recomputes the Cholesky decomposition A = LL^* = U^*U of matrix
Example:
Output:
Here is the matrix A: 2 -1 -1 3 Here is the right hand side b: 1 2 3 1 Computing LLT decomposition... The solution is: 1.2 1.4 1.4 0.8 The matrix A is now: 2 -1 -1 4 Computing LLT decomposition... The solution is now: 1 1.29 1 0.571
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Reports whether previous computation was successful.
Success
if computation was succesful, NumericalIssue
if the matrix.appears to be negative.
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TODO: document the storage layout
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LLT<_MatrixType,_UpLo> Eigen::LLT< _MatrixType, _UpLo >::rankUpdate | ( | const VectorType & | v, |
const RealScalar & | sigma | ||
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Performs a rank one update (or dowdate) of the current decomposition. If A = LL^* before the rank one update, then after it we have LL^* = A + sigma * v v^* where v must be a vector of same dimension.
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*this
is the Cholesky decomposition. MatrixType Eigen::LLT< MatrixType, _UpLo >::reconstructedMatrix | ( | ) | const |
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Since this LLT class assumes anyway that the matrix A is invertible, the solution theoretically exists and is unique regardless of b.
Example:
Output:
2.02 2.97