Pseudoinverse

General

Computes the pseudoinverse of a general matrix through computing its SVD, modifying the singular values with the function

\[\begin{split}f(\sigma) = \left\{\begin{array}{cc} 1/\sigma, & \sigma \ge \epsilon \, n \, \| A \|_2 \\ 0, & \mbox{otherwise} \end{array}\right.,\end{split}\]

where \(\epsilon\) is the relative machine precision, \(n\) is the height of \(A\), and \(\| A \|_2\) is the maximum singular value. If a nonzero value for tolerance was specified, it is used instead of \(\epsilon n \| A \|_2\).

C++ API

void Pseudoinverse(Matrix<F> &A, Base<F> tolerance = 0)

void Pseudoinverse(AbstractDistMatrix<F> &A, Base<F> tolerance = 0)

C API

ElError ElPseudoinverse_s(ElMatrix_s A)

ElError ElPseudoinverse_d(ElMatrix_d A)

ElError ElPseudoinverse_c(ElMatrix_c A)

ElError ElPseudoinverse_z(ElMatrix_z A)

ElError ElPseudoinverseDist_s(ElDistMatrix_s A)

ElError ElPseudoinverseDist_d(ElDistMatrix_d A)

ElError ElPseudoinverseDist_c(ElDistMatrix_c A)

ElError ElPseudoinverseDist_z(ElDistMatrix_z A)

Hermitian

Computes the pseudoinverse of a Hermitian matrix through a customized version of HermitianFunction() which used the eigenvalue mapping function

\[\begin{split}f(\omega) = \left\{\begin{array}{cc} 1/\omega, & |\omega| \ge \epsilon \, n \, \| A \|_2 \\ 0, & \mbox{otherwise} \end{array}\right.,\end{split}\]

where \(\epsilon\) is the relative machine precision, \(n\) is the height of \(A\), and \(\| A \|_2\) can be computed as the maximum absolute value of the eigenvalues of \(A\). If a nonzero value for tolerance is specified, it is used instead of \(\epsilon n \| A \|_2\).

C++ API

void HermitianPseudoinverse(UpperOrLower uplo, Matrix<F> &A, Base<F> tolerance = 0)

void HermitianPseudoinverse(UpperOrLower uplo, AbstractDistMatrix<F> &A, Base<F> tolerance = 0)

C API

ElError ElHermitianPseudoinverse_s(ElUpperOrLower uplo, ElMatrix_s A)

ElError ElHermitianPseudoinverse_d(ElUpperOrLower uplo, ElMatrix_d A)

ElError ElHermitianPseudoinverse_c(ElUpperOrLower uplo, ElMatrix_c A)

ElError ElHermitianPseudoinverse_z(ElUpperOrLower uplo, ElMatrix_z A)

ElError ElHermitianPseudoinverseDist_s(ElUpperOrLower uplo, ElDistMatrix_s A)

ElError ElHermitianPseudoinverseDist_d(ElUpperOrLower uplo, ElDistMatrix_d A)

ElError ElHermitianPseudoinverseDist_c(ElUpperOrLower uplo, ElDistMatrix_c A)

ElError ElHermitianPseudoinverseDist_z(ElUpperOrLower uplo, ElDistMatrix_z A)