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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