Orthogonal polynomials of the R-linear generalized minimal residual method

Abstract

The speed of convergence of the R-linear GMRES method 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.

On a Weyl–von Neumann type theorem for antilinear self-adjoint operators

“…However, diagonalizable antilinear operators are a lot more scarce. The probability of a randomly picked antilinear operator on an n-dimensional Hilbert space to be diagonalizable is 2 −n(n−1)/2 , see [19] for details.…”