A FEAST SVDsolver for the computation of singular value decompositions of large matrices based on the Chebyshev--Jackson series expansion

Zhongxiao Jia, Kailiang Zhang

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 $A^TA$ 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.

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