Standard Cholesky decomposition (LL^T) of a matrix and associated features. More...
Public Member Functions | |
LLT & | compute (const MatrixType &matrix) |
ComputationInfo | info () const |
Reports whether previous computation was successful. | |
LLT () | |
Default Constructor. | |
LLT (Index size) | |
Default Constructor with memory preallocation. | |
Traits::MatrixL | matrixL () const |
const MatrixType & | matrixLLT () const |
Traits::MatrixU | matrixU () const |
MatrixType | reconstructedMatrix () const |
template<typename Rhs > | |
const internal::solve_retval < LLT, Rhs > | solve (const MatrixBase< Rhs > &b) const |
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.
LLT | ( | ) | [inline] |
Default Constructor.
The default constructor is useful in cases in which the user intends to perform decompositions via LLT::compute(const MatrixType&).
LLT | ( | Index | size | ) | [inline] |
Default Constructor with memory preallocation.
Like the default constructor but with preallocation of the internal data according to the specified problem size.
LLT< MatrixType, _UpLo > & compute | ( | const MatrixType & | a | ) |
Computes / recomputes the Cholesky decomposition A = LL^* = U^*U of matrix
ComputationInfo info | ( | ) | const [inline] |
Reports whether previous computation was successful.
Success
if computation was succesful, NumericalIssue
if the matrix.appears to be negative. Traits::MatrixL matrixL | ( | void | ) | const [inline] |
const MatrixType& matrixLLT | ( | ) | const [inline] |
TODO: document the storage layout
Traits::MatrixU matrixU | ( | ) | const [inline] |
MatrixType reconstructedMatrix | ( | ) | const |
const internal::solve_retval<LLT, Rhs> solve | ( | const MatrixBase< Rhs > & | b | ) | const [inline] |
Since this LLT class assumes anyway that the matrix A is invertible, the solution theoretically exists and is unique regardless of b.
Example:
typedef Matrix<float,Dynamic,2> DataMatrix; // let's generate some samples on the 3D plane of equation z = 2x+3y (with some noise) DataMatrix samples = DataMatrix::Random(12,2); VectorXf elevations = 2*samples.col(0) + 3*samples.col(1) + VectorXf::Random(12)*0.1; // and let's solve samples * [x y]^T = elevations in least square sense: Matrix<float,2,1> xy = (samples.adjoint() * samples).llt().solve((samples.adjoint()*elevations)); cout << xy << endl;
Output:
2.02 2.97