An Augmented Matrix-Based CJ-FEAST SVDsolver for Computing a Partial Singular Value Decomposition with the Singular Values in a Given Interval

SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 24-58, March 2024.
Abstract. 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 proposed to compute the singular triplets of a large matrix [math] with the singular values in an interval [math] contained in the singular spectrum. The new CJ-FEAST SVDsolver is a subspace iteration applied to an approximate spectral projector of the augmented matrix [math] associated with the eigenvalues in [math], and it constructs approximate left and right singular subspaces independently, onto which [math] is projected to obtain the Ritz approximations to the desired singular triplets. Compact estimates are given for the accuracy of the approximate spectral projector constructed by the Chebyshev–Jackson series expansion in terms of series degree, 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-based 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 and meanwhile demonstrate that the proposed solver is more robust and substantially more efficient than the corresponding contour integral-based versions that exploit the trapezoidal rule and the Gauss–Legendre quadrature to construct an approximate spectral projector.