# A Chebyshev--Jackson series based block SS--RR algorithm for computing partial eigenpairs of real symmetric matrices

## TL;DR

This paper introduces a Chebyshev--Jackson series-based block SS--RR algorithm for efficiently computing partial eigenpairs of large symmetric matrices within a specific interval, avoiding expensive linear system solutions.

## Contribution

It develops a novel CJ series expansion approach for moments in the block SS--RR method, improving efficiency over traditional contour integral techniques.

## Key findings

- The CJ series converges pointwise to moments with quantifiable rates.
- The new algorithm outperforms the trapezoidal rule-based method in numerical tests.
- Theoretical convergence results extend to higher-order moments.

## Abstract

This paper considers eigenpair computations of large symmetric matrices with the desired eigenvalues lying in a given interval using the contour integral-based block SS--RR method, a Rayleigh--Ritz projection onto a certain subspace generated by moment matrices. Instead of using a numerical quadrature to approximately compute the moments by solving a number of large shifted complex linear systems at each iteration, we make use of the Chebyshev--Jackson (CJ) series expansion to approximate the moments, which only involves matrix-vector products and avoids expensive solutions of the linear systems. We prove that the CJ series expansions pointwise converge to the moments as the series degree increases, but at different convergence rates depending on point positions and moment orders. These extend the available convergence results on the zeroth moment of CJ series expansions to higher order ones. Based on the results established, we develop a CJ--SS--RR algorithm. Numerical experiments illustrate that the new algorithm is more efficient than the contour integral-based block SS--RR algorithm with the trapezoidal rule.