Generalized Chebyshev Acceleration
Nurgül Gökgöz
TL;DR
The paper addresses speeding up simple stationary solvers for linear systems, notably the Jacobi method, by polynomial acceleration using generalized Chebyshev tools tied to the root system $A_2$. It extends classical Chebyshev acceleration by constructing $p_m$ from the generalized polynomials $f_m$, and introduces an auxiliary matrix $\tilde{M}$ to handle complex spectra, yielding a three-term semi-iterative scheme. The main result provides a concrete update formula that guarantees convergence acceleration and demonstrates dramatic reductions in the dominant error factor on example systems. The work broadens Chebyshev acceleration to higher-symmetry polynomial families, offering a pathway to faster convergence for certain non-Hermitian linear systems.
Abstract
We use generalized Chebyshev polynomials, associated with the root system $A_2$, to provide a new semi-iterative method for accelerating simple iterative methods for solving linear systems. We apply this semi-iterative method to the Jacobi method, and give an example. There are certain restrictions but the resulting acceleration is rather high.