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

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 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 \( A = L L^T \).

B

(input/output) real valued GeMatrix
On entry, the \( n \times n_{\text{rhs}} \) right hand side matrix \( B \).+ On exit, if the return value equals zero, the \( n \times n_{\text{rhs}} \) solution matrix \( X \).

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

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 \( A = L L^T \).

B

(input/output) complex valued GeMatrix
On entry, the \( n \times n_{\text{rhs}} \) right hand side matrix \( B \).+ On exit, if the return value equals zero, the \( n \times n_{\text{rhs}} \) solution matrix \( X \).

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 Right-Hand Side (Real and Complex Variant)

A

(input/output) real valued SyMatrix or complex valued HeMatrix
On entry, the symmetric or 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 \( A = U^H U \) or \( A = L L^H \).

b

(input/output) real or complex valued DenseVector
On entry, the right hand side vector \( b \).+ On exit, if the return value equals zero, the solution vector \( x \).

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.