File: cholesky_complex.rdoc [gsl-1.15.3 Documentation]

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### Cholesky decomposition (>= GSL-1.10)

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.

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