# Triangular Factorization of Positive Definite Matrices (trf)

trf computes the Cholesky factorization of a real symmetric positive definite matrix $$A$$.

## Real Variant

The Cholesky decomposition is used to factor $$A$$ as

• $$A = U^T U$$, if $$A$$ is stored in the upper triangular part,

• $$A = L L^T$$, if $$A$$ is stored in the lower triangular part,

where $$U$$ is an upper triangular matrix and $$L$$ is a lower triangular matrix.

 A (input/output) real valued SyMatrix On entry, the symmetric matrix $$A$$ stored either in the upper or lower triangular part of $$A$$. The other part is not referenced. On exit, if the return value is zero, the factor $$U$$ or $$L$$ from the Cholesky factorization $$A = U^T U$$ or $$A = L L^T$$.

Return value:

 i=0 Successful exit. i>0 The leading minor of order $$i$$ is not positive definite, and the factorization could not be completed.

## Complex Variant

The Cholesky decomposition is used to factor $$A$$ as

• $$A = U^H U$$, if $$A$$ is stored in the upper triangular part,

• $$A = L L^H$$, if $$A$$ is stored in the lower triangular part,

where $$U$$ is an upper triangular matrix and $$L$$ is a lower triangular matrix. The factored form of $$A$$ is then used to solve the system of equations $$A X = B$$.

 A (input/output) complex valued HeMatrix On entry, the hermitian matrix $$A$$ stored either in the upper or lower triangular part of $$A$$. The other part is not referenced. On exit, if the return value is zero, the factor $$U$$ or $$L$$ from the Cholesky factorization $$A = U^T U$$ or $$A = L L^T$$.

Return value:

 i=0 Successful exit. i>0 The leading minor of order $$i$$ is not positive definite, and the factorization could not be completed.