Generalized RQ factorization

Implementation

The generalized RQ factorization of a pair of matrices \((A,B)\) is analogous to an RQ factorization of \(A B^{-1}\) but does not require that \(B\) is square or invertible: unitary matrices \(Q\) and \(Z\), and (right) upper-triangular matrices \(R\) and \(T\), are computed such that

\[A = R Q\]

and

\[B = Z T Q.\]

Thus, is \(B\) was square and invertible, then the RQ factorization of \(A B^{-1}\) would be given by \((R T^{-1}) Z^H\).

C++ API

void GRQ(Matrix<F> &A, Matrix<F> &tA, Matrix<Base<F>> &dA, Matrix<F> &B, Matrix<F> &tB, Matrix<Base<F>> &dB)

void GRQ(AbstractDistMatrix<F> &A, AbstractDistMatrix<F> &tA, AbstractDistMatrix<Base<F>> &dA, AbstractDistMatrix<F> &B, AbstractDistMatrix<F> &tB, AbstractDistMatrix<Base<F>> &dB)

Overwrite A with both R and the (scaled) Householder vectors which, along with the scalings tA and sign changes dA, define Q. Likewise, B is overwritten with both T and the (scaled) Householder vectors which define Z.

void grq::ExplicitTriang(Matrix<F> &A, Matrix<F> &B)

void grq::ExplicitTriang(AbstractDistMatrix<F> &A, AbstractDistMatrix<F> &B)

Overwrite A with R and B with T.

C API

ElError ElGRQ_s(ElMatrix_s A, ElMatrix_s tA, ElMatrix_s dA, ElMatrix_s B, ElMatrix_s tB, ElMatrix_s dB)

ElError ElGRQ_d(ElMatrix_d A, ElMatrix_d tA, ElMatrix_d dA, ElMatrix_d B, ElMatrix_d tB, ElMatrix_d dB)

ElError ElGRQ_c(ElMatrix_c A, ElMatrix_c tA, ElMatrix_s dA, ElMatrix_c B, ElMatrix_c tB, ElMatrix_s dB)

ElError ElGRQ_z(ElMatrix_z A, ElMatrix_z tA, ElMatrix_d dA, ElMatrix_z B, ElMatrix_z tB, ElMatrix_d dB)

ElError ElGRQDist_s(ElDistMatrix_s A, ElDistMatrix_s tA, ElDistMatrix_s dA, ElDistMatrix_s B, ElDistMatrix_s tB, ElDistMatrix_s dB)

ElError ElGRQDist_d(ElDistMatrix_d A, ElDistMatrix_d tA, ElDistMatrix_d dA, ElDistMatrix_d B, ElDistMatrix_d tB, ElDistMatrix_d dB)

ElError ElGRQDist_c(ElDistMatrix_c A, ElDistMatrix_c tA, ElDistMatrix_s dA, ElDistMatrix_c B, ElDistMatrix_c tB, ElDistMatrix_s dB)

ElError ElGRQDist_z(ElDistMatrix_z A, ElDistMatrix_z tA, ElDistMatrix_d dA, ElDistMatrix_z B, ElDistMatrix_z tB, ElDistMatrix_d dB)

Overwrite A with both R and the (scaled) Householder vectors which, along with the scalings tA and sign changes dA, define Q. Likewise, B is overwritten with both T and the (scaled) Householder vectors which define Z.

ElError ElGRQExplicitTriang_s(ElMatrix_s A, ElMatrix_s B)

ElError ElGRQExplicitTriang_d(ElMatrix_d A, ElMatrix_d B)

ElError ElGRQExplicitTriang_c(ElMatrix_c A, ElMatrix_c B)

ElError ElGRQExplicitTriang_z(ElMatrix_z A, ElMatrix_z B)

ElError ElGRQExplicitTriangDist_s(ElDistMatrix_s A, ElDistMatrix_s B)

ElError ElGRQExplicitTriangDist_d(ElDistMatrix_d A, ElDistMatrix_d B)

ElError ElGRQExplicitTriangDist_c(ElDistMatrix_c A, ElDistMatrix_c B)

ElError ElGRQExplicitTriangDist_z(ElDistMatrix_z A, ElDistMatrix_z B)

Overwrite A with R and B with T.