Version 1.1.3
29th March 2023 User documentation
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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 rightmost, or are of largest imaginary parts. The rightmost 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 leftmost 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.