Solving a Positive Definite System of Linear Equations (sv)
sv computes the solution to a real (or complex) system of linear equations \( A X = B \), where \( A \) is a \( n \times n \) symmetric (or hermitian) positive definite matrix and \( X \) and \( B \) are \( n \times n_{\text{rhs}} \) matrices.
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. The factored form of \( A \) is then used to solve the system of equations \( A X = B \).
A 
(input/output) real valued SyMatrix On exit, if the return value is zero, the factor \( U \) or \( L \) from the Cholesky factorization \( A = U^T U \) or \( A = L L^T \). 
B 
(input/output) real valued GeMatrix 
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, and the solution has not been computed. 
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 exit, if the return value is zero, the factor \( U \) or \( L \) from the Cholesky factorization \( A = U^T U \) or \( A = L L^T \). 
B 
(input/output) complex valued GeMatrix 
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, and the solution has not been computed. 
Single RightHand Side (Real and Complex Variant)
A 
(input/output) real valued SyMatrix or complex valued HeMatrix On exit, if the return value is zero, the factor \( U \) or \( L \) from the Cholesky factorization \( A = U^H U \) or \( A = L L^H \). 
b 
(input/output) real or complex valued DenseVector 
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, and the solution has not been computed. 