Solving a General System of Linear Equations (sv)

sv computes the solution to a real system of linear equations \( AX=B \), where \( A \) is an \( n \times n \) matrix and \( X \) and \( B \) are \( n \times n_{\text{rhs}} \) matrices.

The \( LU \) decomposition with partial pivoting and row interchanges is used to factor \( A \) as

\[ A = P L U \]

where \( P \) is a permutation matrix, \( L \) is unit lower triangular, and \( U \) is upper triangular. The factored form of \( A \) is then used to solve the system of equations \( A X = B \).

Multiple Right-Hand Sides

A

(input/output) real or complex valued GeMatrix
On entry, the \( m \times n \) matrix to be factored.
On exit, the factors \( L \) and \( U \) from the factorization \( A = PLU \). The unit diagonal elements of \( L \) are not stored.

piv

(output) integer valued DenseVector
The pivot indices. For \( 1 \leq i \leq \min\{m, n\} \), row \( i \) of the matrix was interchanged with row \( piv_i \).

B

(input/output) real or complex valued GeMatrix

On entry, the \( n \times n_{\text{rhs}} \) matrix of 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

\( U_{i,i} \) is exactly zero. The factorization has been completed, but the factor \( U \) is exactly singular, so the solution could not be computed.

Single Right-Hand Side

A

(input/output) real or complex valued GeMatrix
On entry, the \( m \times n \) matrix to be factored.
On exit, the factors \( L \) and \( U \) from the factorization \( A = PLU \). The unit diagonal elements of \( L \) are not stored.

piv

(output) integer valued DenseVector
The pivot indices. For \( 1 \leq i \leq \min\{m, n\} \), row \( i \) of the matrix was interchanged with row \( piv_i \).

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

\( U_{i,i} \) is exactly zero. The factorization has been completed, but the factor \( U \) is exactly singular, so the solution could not be computed.