Content

Solve a System of Linear Equations (sv)

sv (defined in namespace flens::lapack) computes the solution to a real system of linear equations

\[ A X = B, \]

where \(A\) is an \(n \times n\) matrix and \(X\) and \(B\) are \(n \times n_{rhs}\) 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 \(A X = B\).

Interface: Multiple Right-Hand Sides

A

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

piv

(output) integer valued DenseVector
The pivot indices that define the permutation matrix \(P\). Row \(i\) of the matrix was interchanged with \(piv_i\).

B

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

Return value:

\(i=0\)

Successfull 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.

Interface: Single Right-Hand Side

A

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

piv

(output) integer valued DenseVector
The pivot indices that define the permutation matrix \(P\). Row \(i\) of the matrix was interchanged with \(piv_i\).

b

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

Return value:

\(i=0\)

Successfull 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.

Notes