arXiv subjects

Kailiang Zhang

Publications and source records attributed to Kailiang Zhang.

3 recordsLinked to original sources

arXiv2023

A CJ-FEAST GSVDsolver for computing a partial GSVD of a large matrix pair with the generalized singular values in a given interval

We propose a CJ-FEAST GSVDsolver to compute a partial generalized singular value decomposition (GSVD) of a large matrix pair $(A,B)$ with the generalized singular values in a given interval. The solver is a highly nontrivial extension of the FEAST eigensolver for the (generalized) eigenvalue problem and CJ-FEAST SVDsolver for the SVD problem. For a partial GSVD problem, given three left and right searching subspaces, we propose a general projection method that works on $(A,B)$ {\em directly}, and computes approximations to the desired GSVD components. For the concerning GSVD problem, we exploit the Chebyshev--Jackson (CJ) series to construct an approximate spectral projector of the generalized eigenvalue problem of the matrix pair $(A^TA,B^TB)$ associated with the generalized singular values of interest, and use subspace iteration on it to generate a right subspace. Premultiplying it with $A$ and $B$ constructs two left subspaces. Applying the general projection method to the subspaces constructed leads to the CJ-FEAST GSVDsolver. We derive accuracy estimates for the approximate spectral projector and its eigenvalues, and establish a number of convergence results on the underlying subspaces and the approximate GSVD components obtained by the CJ-FEAST GSVDsolver. We propose general-purpose choice strategies for the series degree and subspace dimension. Numerical experiments illustrate the efficiency of the CJ-FEAST GSVDsolver.

math.NA↗

arXiv2023

An augmented matrix-based CJ-FEAST SVDsolver for computing a partial singular value decomposition with the singular values in a given interval

The cross-product matrix-based CJ-FEAST SVDsolver proposed previously by the authors is shown to compute the left singular vector possibly much less accurately than the right singular vector and may be numerically backward unstable when a desired singular value is small. In this paper, an alternative augmented matrix-based CJ-FEAST SVDsolver is considered to compute the singular triplets of a large matrix $A$ with the singular values in an interval $[a,b]$ contained in the singular spectrum. The new CJ-FEAST SVDsolver is a subspace iteration applied to an approximate spectral projector of the augmented matrix $[0, A^T; A, 0]$ associated with the eigenvalues in $[a,b]$, and constructs approximate left and right singular subspaces with the desired singular values independently, onto which $A$ is projected to obtain the Ritz approximations to the desired singular triplets. Compact estimates are given for the accuracy of the approximate spectral projector, and a number of convergence results are established. The new solver is proved to be always numerically backward stable. A convergence comparison of the cross-product and augmented matrix-based CJ-FEAST SVDsolvers is made, and a general-purpose choice strategy between the two solvers is proposed for the robustness and overall efficiency. Numerical experiments confirm all the results.

math.NA↗

arXiv2022

A FEAST SVDsolver based on Chebyshev--Jackson series for computing partial singular triplets of large matrices

The FEAST eigensolver is extended to the computation of the singular triplets of a large matrix $A$ with the singular values in a given interval. The resulting FEAST SVDsolver is subspace iteration applied to an approximate spectral projector of $A^TA$ corresponding to the desired singular values in a given interval, and constructs approximate left and right singular subspaces corresponding to the desired singular values, onto which $A$ is projected to obtain Ritz approximations. Differently from a commonly used contour integral-based FEAST solver, we propose a robust alternative that constructs approximate spectral projectors by using the Chebyshev--Jackson polynomial series, which are symmetric positive semi-definite with the eigenvalues in $[0,1]$. We prove the pointwise convergence of this series and give compact estimates for pointwise errors of it and the step function that corresponds to the exact spectral projector. We present error bounds for the approximate spectral projector and reliable estimates for the number of desired singular triplets, establish numerous convergence results on the resulting FEAST SVDsolver, and propose practical selection strategies for determining the series degree and for reliably determining the subspace dimension. The solver and results on it are directly applicable or adaptable to the real symmetric and complex Hermitian eigenvalue problem. Numerical experiments illustrate that our FEAST SVDsolver is at least competitive with and is much more efficient than the contour integral-based FEAST SVDsolver when the desired singular values are extreme and interior ones, respectively, and it is also more robust than the latter.

math.NA↗