Next: Tridiagonal Decomposition of Real Symmetric Matrices, Previous: Singular Value Decomposition, Up: Linear Algebra [Index]
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.
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.
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
.
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.
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.
Next: Tridiagonal Decomposition of Real Symmetric Matrices, Previous: Singular Value Decomposition, Up: Linear Algebra [Index]