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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`) - Function:
*int***gsl_linalg_complex_cholesky_decomp***(gsl_matrix_complex **`A`) These functions factorize the symmetric, positive-definite square matrix

`A`into the Cholesky decomposition*A = L L^T*(or*A = L L^H*for the complex case). On input, the values from the diagonal and lower-triangular part of the matrix`A`are used (the upper triangular part is ignored). On output the diagonal and lower triangular part of the input matrix`A`contain the matrix*L*, while the upper triangular part of the input matrix is overwritten with*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`

.When testing whether a matrix is positive-definite, disable the error handler first to avoid triggering an error.

- Function:
*int***gsl_linalg_cholesky_solve***(const gsl_matrix **`cholesky`, const gsl_vector *`b`, gsl_vector *`x`) - Function:
*int***gsl_linalg_complex_cholesky_solve***(const gsl_matrix_complex **`cholesky`, const gsl_vector_complex *`b`, gsl_vector_complex *`x`) These functions solve the system

*A x = b*using the Cholesky decomposition of*A*held in the matrix`cholesky`which must have been previously computed by`gsl_linalg_cholesky_decomp`

or`gsl_linalg_complex_cholesky_decomp`

.

- Function:
*int***gsl_linalg_cholesky_svx***(const gsl_matrix **`cholesky`, gsl_vector *`x`) - Function:
*int***gsl_linalg_complex_cholesky_svx***(const gsl_matrix_complex **`cholesky`, gsl_vector_complex *`x`) These functions solve the system

*A x = b*in-place using the Cholesky decomposition of*A*held in the matrix`cholesky`which must have been previously computed by`gsl_linalg_cholesky_decomp`

or`gsl_linalg_complex_cholesky_decomp`

. On input`x`should contain the right-hand side*b*, which is replaced by the solution on output.

- Function:
*int***gsl_linalg_cholesky_invert***(gsl_matrix **`cholesky`) - Function:
*int***gsl_linalg_complex_cholesky_invert***(gsl_matrix_complex **`cholesky`) These functions compute the inverse of a matrix from its Cholesky decomposition

`cholesky`, which must have been previously computed by`gsl_linalg_cholesky_decomp`

or`gsl_linalg_complex_cholesky_decomp`

. On output, the inverse is stored in-place in`cholesky`.