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RQ factorization¶

Just like an LQ factorization, but the orthogonalization process starts from the bottom row and produces a much sparser triangular factor when the matrix is wider than it is tall.

Factorization¶

C++ API¶

void RQ(Matrix<F> &A, Matrix<F> &t, Matrix<Base<F>> &d)¶

void RQ(AbstractDistMatrix<F> &A, AbstractDistMatrix<F> &t, AbstractDistMatrix<Base<F>> &d)¶: Overwrite the matrix \(A\) with \(R\) and the Householder reflectors representing \(\hat Q\). The scalings for the Householder reflectors are stored in the vector t and the unitary diagonal matrix which forces \(R\) to be positive is defined by the vector d.

void rq::ExplicitTriang(Matrix<F> &A)¶

void rq::ExplicitTriang(AbstractDistMatrix<F> &A)¶: Overwrite A with the triangular factor, R.

C API¶

ElError ElRQ_s(ElMatrix_s A, ElMatrix_s t, ElMatrix_s d)¶

ElError ElRQ_d(ElMatrix_d A, ElMatrix_d t, ElMatrix_d d)¶

ElError ElRQ_c(ElMatrix_c A, ElMatrix_c t, ElMatrix_s d)¶

ElError ElRQ_z(ElMatrix_z A, ElMatrix_z t, ElMatrix_d d)¶

ElError ElRQDist_s(ElDistMatrix_s A, ElDistMatrix_s t, ElDistMatrix_s d)¶

ElError ElRQDist_d(ElDistMatrix_d A, ElDistMatrix_d t, ElDistMatrix_d d)¶

ElError ElRQDist_c(ElDistMatrix_c A, ElDistMatrix_c t, ElDistMatrix_s d)¶

ElError ElRQDist_z(ElDistMatrix_z A, ElDistMatrix_z t, ElDistMatrix_d d)¶: Overwrite the matrix \(A\) with \(R\) and the Householder reflectors representing \(\hat Q\). The scalings for the Householder reflectors are stored in the vector t and the unitary diagonal matrix which forces \(R\) to be positive is defined by the vector d.

ElError ElRQExplicitTriang_s(ElMatrix_s A)¶

ElError ElRQExplicitTriang_d(ElMatrix_d A)¶

ElError ElRQExplicitTriang_c(ElMatrix_c A)¶

ElError ElRQExplicitTriang_z(ElMatrix_z A)¶: Overwrite A with the triangular factor, R.

Apply the factored matrix¶

C++ API¶

void rq::ApplyQ(LeftOrRight side, Orientation orientation, const Matrix<F> &A, const Matrix<F> &t, const Matrix<Base<F>> &d, Matrix<F> &B)¶

void rq::ApplyQ(LeftOrRight side, Orientation orientation, const AbstractDistMatrix<F> &A, const AbstractDistMatrix<F> &t, const AbstractDistMatrix<Base<F>> &d, AbstractDistMatrix<F> &B)¶: Applies the implicitly-defined \(Q\) (or its adjoint) stored within A, t, and d from either the left or the right to \(B\).

C API¶

ElError ElApplyQAfterRQ_s(ElLeftOrRight side, ElOrientation orientation, ElConstMatrix_s A, ElConstMatrix_s t, ElConstMatrix_s d, ElMatrix_s B)¶

ElError ElApplyQAfterRQ_d(ElLeftOrRight side, ElOrientation orientation, ElConstMatrix_d A, ElConstMatrix_d t, ElConstMatrix_d d, ElMatrix_d B)¶

ElError ElApplyQAfterRQ_c(ElLeftOrRight side, ElOrientation orientation, ElConstMatrix_c A, ElConstMatrix_c t, ElConstMatrix_s d, ElMatrix_c B)¶

ElError ElApplyQAfterRQ_z(ElLeftOrRight side, ElOrientation orientation, ElConstMatrix_z A, ElConstMatrix_z t, ElConstMatrix_d d, ElMatrix_z B)¶

ElError ElApplyQAfterRQDist_s(ElLeftOrRight side, ElOrientation orientation, ElConstDistMatrix_s A, ElConstDistMatrix_s t, ElConstDistMatrix_s d, ElDistMatrix_s B)¶

ElError ElApplyQAfterRQDist_d(ElLeftOrRight side, ElOrientation orientation, ElConstDistMatrix_d A, ElConstDistMatrix_d t, ElConstDistMatrix_d d, ElDistMatrix_d B)¶

ElError ElApplyQAfterRQDist_c(ElLeftOrRight side, ElOrientation orientation, ElConstDistMatrix_c A, ElConstDistMatrix_c t, ElConstDistMatrix_s d, ElDistMatrix_c B)¶

ElError ElApplyQAfterRQDist_z(ElLeftOrRight side, ElOrientation orientation, ElConstDistMatrix_z A, ElConstDistMatrix_z t, ElConstDistMatrix_d d, ElDistMatrix_z B)¶

Solve linear systems with the factored matrix¶

C++ API¶

void rq::SolveAfter(Orientation orientation, const Matrix<F> &A, const Matrix<F> &t, const Matrix<Base<F>> &d, const Matrix<F> &B, Matrix<F> &X)¶

void rq::SolveAfter(Orientation orientation, const AbstractDistMatrix<F> &A, const AbstractDistMatrix<F> &t, const AbstractDistMatrix<Base<F>> &d, const AbstractDistMatrix<F> &B, AbstractDistMatrix<F> &X)¶: Solves a set of linear systems using an existing packed RQ factorization given by \(A\) and the vectors \(t\) and \(d\). \(B\) is the matrix of input vectors and \(X\) is the matrix of solutions.

C API¶

ElError ElSolveAfterRQ_s(ElOrientation orientation, ElConstMatrix_s A, ElConstMatrix_s t, ElConstMatrix_s d, ElConstMatrix_s B, ElMatrix_s X)¶

ElError ElSolveAfterRQ_d(ElOrientation orientation, ElConstMatrix_d A, ElConstMatrix_d t, ElConstMatrix_d d, ElConstMatrix_d B, ElMatrix_d X)¶

ElError ElSolveAfterRQ_c(ElOrientation orientation, ElConstMatrix_c A, ElConstMatrix_c t, ElConstMatrix_s d, ElConstMatrix_c B, ElMatrix_c X)¶

ElError ElSolveAfterRQ_z(ElOrientation orientation, ElConstMatrix_z A, ElConstMatrix_z t, ElConstMatrix_d d, ElConstMatrix_z B, ElMatrix_z X)¶

ElError ElSolveAfterRQDist_s(ElOrientation orientation, ElConstDistMatrix_s A, ElConstDistMatrix_s t, ElConstDistMatrix_s d, ElConstDistMatrix_s B, ElDistMatrix_s X)¶

ElError ElSolveAfterRQDist_d(ElOrientation orientation, ElConstDistMatrix_d A, ElConstDistMatrix_d t, ElConstDistMatrix_d d, ElConstDistMatrix_d B, ElDistMatrix_d X)¶

ElError ElSolveAfterRQDist_c(ElOrientation orientation, ElConstDistMatrix_c A, ElConstDistMatrix_c t, ElConstDistMatrix_s d, ElConstDistMatrix_c B, ElDistMatrix_c X)¶

ElError ElSolveAfterRQDist_z(ElOrientation orientation, ElConstDistMatrix_z A, ElConstDistMatrix_z t, ElConstDistMatrix_d d, ElConstDistMatrix_z B, ElDistMatrix_z X)¶