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

TL;DR

This paper introduces a robust FEAST SVDsolver using Chebyshev--Jackson series for efficiently computing partial singular triplets of large matrices, especially effective for extreme and interior singular values.

Contribution

It develops a new spectral projector approximation method with Chebyshev--Jackson polynomials, providing convergence analysis, error bounds, and practical strategies, improving robustness and efficiency over existing contour integral-based FEAST methods.

Findings

The new FEAST SVDsolver is more efficient for extreme and interior singular values.

It offers robust convergence and reliable estimates for the number of singular triplets.

Numerical experiments show superior performance compared to traditional methods.

Abstract

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.