Path: |
rdoc/cholesky_complex.rdoc |

Last Update: |
Sun Nov 14 15:15:45 -0800 2010 |

A symmetric, positive definite square matrix `A` has a Cholesky
decomposition into a product of a lower triangular matrix `L` and
its transpose `L^T`. This is sometimes referred to as taking the
square-root of a matrix. The Cholesky decomposition can only be carried out
when all the eigenvalues of the matrix are positive. This decomposition can
be used to convert the linear system `A x = b` into a pair of
triangular systems `L y = b, L^T x = y`, which can be solved by
forward and back-substitution.

- GSL::Linalg::Complex::Cholesky::decomp(A)
- GSL::Linalg::Complex::cholesky_decomp(A)
Factorize the positive-definite square matrix

`A`into the Cholesky decomposition`A = L L^H`. On input only the diagonal and lower-triangular part of the matrix`A`are needed. The diagonal and lower triangular part of the returned matrix contain the matrix`L`. The upper triangular part of the returned matrix contains L^T, and the diagonal terms being identical for both L and L^T. If the input matrix is not positive-definite then the decomposition will fail, returning the error code`GSL::EDOM`.

- GSL::Linalg::Complex::Cholesky::solve(chol, b, x)
- GSL::Linalg::Complex::cholesky_solve(chol, b, x)
Solve the system

`A x = b`using the Cholesky decomposition of`A`into the matrix`chol`given by`GSL::Linalg::Complex::Cholesky::decomp`.

- GSL::Linalg::Complex::Cholesky::svx(chol, x)
- GSL::Linalg::Complex::cholesky_svx(chol, x)
Solve the system

`A x = b`in-place using the Cholesky decomposition of`A`into the matrix`chol`given by`GSL::Linalg::Complex::Cholesky::decomp`. On input`x`should contain the right-hand side`b`, which is replaced by the solution on output.