STFC Website

part of UK Research & Innovation

Version 1.1.3

29th March 2023

EB13: Sparse unsymmetric: Arnoldi's method

Given a real unsymmetric \(n \times n\) matrix \(\mathbf{A} = {\{a_{ij}\}}\), this routine uses Arnoldi based methods to calculate the \(r\) eigenvalues \(\lambda _i , i = 1,..., r\), that are of largest absolute value, or are right-most, or are of largest imaginary parts. The right-most eigenvalues are those with the most positive real part. There is an option to compute the associated eigenvectors \(\mathbf{y} _i\), \(i = 1,..., r\), where \(\mathbf{Ay} _i = \lambda _i \mathbf{y} _i\). The routine may be used to compute the left-most eigenvalues of \(\mathbf{A}\) by using \(-\mathbf{A}\) in place of \(\mathbf{A}\).

The Arnoldi methods offered by EB13 are:

(1) The basic (iterative) Arnoldi method.

(2) Arnoldi’s method with Chebyshev acceleration of the starting vectors.

(3) Arnoldi’s method applied to the preconditioned matrix \(p _l (\mathbf{A})\), where \(p _l\) is a Chebyshev polynomial.

Each method is available in blocked and unblocked form.