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.