Orthogonal polynomials of the R-linear generalized minimal residual method

TL;DR

This paper analyzes the convergence of the R-linear GMRES method using orthogonal polynomials, establishing bounds and a recurrence relation, and explores the probabilistic aspects of condiagonalizability in complex symmetric matrices.

Contribution

It introduces a polynomial approximation framework for R-linear GMRES convergence and links it to orthogonal polynomials and complex symmetric Jacobi matrices.

Findings

Convergence bounds are similar to classical GMRES estimates for condiagonalizable matrices.

A three-term recurrence for orthogonal polynomials is derived, connecting to complex symmetric Jacobi matrices.

The probability of a matrix being condiagonalizable is estimated using random matrix theory.

Abstract

The speed of convergence of the R-linear GMRES is bounded in terms of a polynomial approximation problem on a finite subset of the spectrum. This result resembles the classical GMRES convergence estimate except that the matrix involved is assumed to be condiagonalizable. The bounds obtained are applicable to the CSYM method, in which case they are sharp. Then a three term recurrence for generating a family of orthogonal polynomials is shown to exist, yielding a natural link with complex symmetric Jacobi matrices. This shows that a mathematical framework analogous to the one appearing with the Hermitian Lanczos method exists in the complex symmetric case. The probability of being condiagonalizable is estimated with random matrices.