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Computing the Inverse of a Matrix (tri)

tri (defined in namespace flens::lapack) computes the inverse of a matrix \(A\). For the computation of \(A^{-1}\) two variants are provided:

For convenience further sub-variants are provided that implicitly create temporary workspaces.

Inversion of a General Matrix

For \(A\) containing the \(LU\) factorization this method inverts \(U\) and then computes \(A^{-1}\) by solving the system \(A^{-1} L = U^{-1}\) for \(A^{-1}\).

Real Variant

template <typename MA, typename VPIV, typename VWORK>
    typename RestrictTo<IsRealGeMatrix<MA>::value
                     && IsIntegerDenseVector<VPIV>::value
                     && IsRealDenseVector<VWORK>::value,
             typename RemoveRef<MA>::Type::IndexType>::Type
    tri(MA          &&A,
        const VPIV  &piv,
        VWORK       &&work);

A

(input/output) real valued GeMatrix
On entry, the \(n \times n\) matrix \(A\) contains the factors \(L\) and \(U\) from the factorization \(A = P L U\) as computed by trf.
On exit, if the return value equals , the inverse of the original matrix \(A\).

piv

(input) integer valued DenseVector
The pivot indices from trf. For \(1 \leq i \leq n\) row \(i\) of the original matrix \(A\) was interchanged with \(piv_i\).

work

(input) real valued DenseVector
For optimal performance the length of the workspace should be at least \(n \cdot n_b\), where \(n_b\) is the optimal blocksize returned by ilaenv.

Note: If \(work\) has length zero the optimal length gets computed and vector \(work\) gets resized. Opposed to the worksize queries in LAPACK the function does not stop after computing the worksize. Instead it will continue the computation with the resized workspace.

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.

Complex Variant

template <typename MA, typename VPIV, typename VWORK>
    typename RestrictTo<IsComplexGeMatrix<MA>::value
                     && IsIntegerDenseVector<VPIV>::value
                     && IsComplexDenseVector<VWORK>::value,
             typename RemoveRef<MA>::Type::IndexType>::Type
    tri(MA          &&A,
        const VPIV  &piv,
        VWORK       &&work);

A

(input/output) complex valued GeMatrix
On entry, the \(n \times n\) matrix \(A\) contains the factors \(L\) and \(U\) from the factorization \(A = P L U\) as computed by trf.
On exit, if the return value equals , the inverse of the original matrix \(A\).

piv

(input) integer valued DenseVector
The pivot indices from trf. For \(1 \leq i \leq n\) row \(i\) of the original matrix \(A\) was interchanged with \(piv_i\).

work

(input) complex valued DenseVector
For optimal performance the length of the workspace should be at least \(n \cdot n_b\), where \(n_b\) is the optimal blocksize returned by ilaenv.

Note: If \(work\) has length zero the optimal length gets computed and vector \(work\) gets resized. Opposed to the worksize queries in LAPACK the function does not stop after computing the worksize. Instead it will continue the computation with the resized workspace.

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.

Variant with Implicit Workspace Creation

This variant implicitly creates a temporary workspace and calls the real or complex variant.

template <typename MA, typename VPIV>
    typename RestrictTo<IsGeMatrix<MA>::value
                     && IsIntegerDenseVector<VPIV>::value,
             typename RemoveRef<MA>::Type::IndexType>::Type
    tri(MA          &&A,
        const VPIV  &piv);

A

(input/output) real or complex valued GeMatrix
On entry, the \(n \times n\) matrix \(A\) contains the factors \(L\) and \(U\) from the factorization \(A = P L U\) as computed by trf.
On exit, if the return value equals , the inverse of the original matrix \(A\).

piv

(input) integer valued DenseVector
The pivot indices from trf. For \(1 \leq i \leq n\) row \(i\) of the original matrix \(A\) was interchanged with \(piv_i\).

Inversion of a Triangular Matrix

Real Variant

template <typename MA>
    typename RestrictTo<IsRealTrMatrix<MA>::value,
             typename RemoveRef<MA>::Type::IndexType>::Type
    tri(MA &&A);

A

(input/output) real valued TrMatrix
On entry, the triangular \(n \times n\) matrix \(A\).
On exit, if the return value equals , the inverse of matrix \(A\).

Complex Variant

template <typename MA>
    typename RestrictTo<IsComplexTrMatrix<MA>::value,
             typename RemoveRef<MA>::Type::IndexType>::Type
    tri(MA &&A);

A

(input/output) complex valued TrMatrix
On entry, the triangular \(n \times n\) matrix \(A\).
On exit, if the return value equals , the inverse of matrix \(A\).

Notes

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