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

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. It is subspace iteration in nature applied to an approximate spectral projector associated with the cross-product matrix A T A and constructs approximate left and right singular subspaces corresponding to the desired singular values, onto which A is projected to obtain approximations to the desired singular triplets. Approximate spectral projectors are constructed using the Chebyshev-Jackson series expansion other than contour integration and quadrature rules, and they are proven to be always symmetric positive semi-definite with the eigenvalues in [0, 1]. Compact estimates are established for pointwise approximation errors of a specific step function that corresponds to the exact spectral projector, the accuracy of the approximate spectral projector, the number of desired singular triplets, the distance between the desired right singular subspace and the subspace generated each iteration, and the convergence of the FEAST SVDsolver. Practical selection strategies are proposed for the series degree and the subspace dimension. Numerical experiments illustrate that the FEAST SVDsolver is robust and efficient.

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

“…, m + n for later use, whose labeling order is postponed to Section 5. The authors in [13] have proposed an S C -based Chebshev-Jackson series FEAST (CJ-FEAST) SVDsolver, an adaptation of the FEAST eigensolver [16] to the concerning SVD problem. The FEAST eigensolver was introduced by Polizzi [16] in 2009 and has been developed in [4,6,14,17,23], and it performs on subspaces of a fixed dimension p, and uses subspace iteration [5,19,21] on an approximate spectral projector associated with the eigenvalues in a given region to generate a sequence of subspaces, onto which the Rayleigh-Ritz projection of the original matrix or matrix pair is realized.…”

“…The S C -based CJ-FEAST SVDsolver then constructs the corresponding approximate left singular space by premultiplying the right one with A, realize the Rayleigh-Ritz projection of A onto the left and right subspaces constructed, and compute the Rayleigh-Ritz approximations to the desired singular triplets. We have numerically observed in [13] that the S C -based CJ-FEAST SVDsolver is often a few to tens times more efficient than the contour integral-based IFEAST [4] adapted to the SVD problem when the interval [a, b] is inside the singular spectrum and it is competitive with the latter when the desired singular values are extreme ones. We have theoretically argued and numerically confirmed in [13] that the CJ-FEAST SVDsolver is more robust than contour integral based SVDsolvers.…”

“…We have numerically observed in [13] that the S C -based CJ-FEAST SVDsolver is often a few to tens times more efficient than the contour integral-based IFEAST [4] adapted to the SVD problem when the interval [a, b] is inside the singular spectrum and it is competitive with the latter when the desired singular values are extreme ones. We have theoretically argued and numerically confirmed in [13] that the CJ-FEAST SVDsolver is more robust than contour integral based SVDsolvers.…”

“…To overcome the above robustness deficiency of the S C -based CJ-FEAST SVDsolver, we will exploit S A to propose a new effective CJ-FEAST SVDsolver in this paper. In order to distinguish the two solvers, we abbreviate the S C -based CJ-FEAST SVDsolver in [13] and the S A -based CJ-FEAST SVDsolver to be proposed in this paper as the CJ-FEAST SVDsolverC and the CJ-FEAST SVDsolverA, respectively. Unlike the CJ-FEAST SVDsolverC, we will construct an approximation P to the spectral projector P SA of S A associated with the eigenvalues σ ∈ [a, b] by the Chebyshev-Jackson series expansion.…”

“…We will prove that they share some similar properties. For instance, the approximate spectral projector constructed is unconditionally symmetric positive semi-definite (SPSD), its eigenvalues always lies in the interval [0, 1], and the strategies on degree choices of Chebyshev-Jackson polynomial series developed in [13] can be directly adapted to the CJ-FEAST SVD-solverA. We can estimate n sv by this approximate spectral projector and Monte-Carlo methods [1,2], as done in [13].…”