Cholesky Decomposition
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,
A = L 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.
- Function: int gsl_linalg_cholesky_decomp (gsl_matrix * A)
-
This function factorizes the positive-definite square matrix A
into the Cholesky decomposition A = L L^T. On output the diagonal
and lower triangular part of the input matrix A contain the matrix
L. The upper triangular part of the input matrix contains
L^T, the diagonal terms being identical for both L and
L^T. If the matrix is not positive-definite then the
decomposition will fail, returning the error code
GSL_EDOM
.
- Function: int gsl_linalg_cholesky_solve (const gsl_matrix * cholesky, const gsl_vector * b, gsl_vector * x)
-
This function solves the system A x = b using the Cholesky
decomposition of A into the matrix cholesky given by
gsl_linalg_cholesky_decomp
.
- Function: int gsl_linalg_cholesky_svx (const gsl_matrix * cholesky, gsl_vector * x)
-
This function solves the system A x = b in-place using the
Cholesky decomposition of A into the matrix cholesky given
by
gsl_linalg_cholesky_decomp
. On input x should contain
the right-hand side b, which is replaced by the solution on
output.