Large Scale Eigen Value Computation

The numerical solution of large scale algebraic eigen value problems is now available thanks to new methods and software. A class of methods called Krylov subspace projection methods is used. The well known Lanczos method is the first one. The Arnoldi method is a generalization of Lanczos method applied to the non-symmetric case. A variant of Arnoldi-Lonczos scheme called the Implicitly Restarted Arnoldi Method 1 is a part of public domain software package called ARPACK which is integrated in Radioss. Restarting is introduced as a way to overcome intractable storage and computational requirements in the original Arnoldi method. Implicit restarting is a variant of restarting which may be considered as a truncated form of the powerful implicitly shifted QR technique that is suitable for large scale problems. It provides a mean to approximate a few eigen values with user specified properties in space proportional to the number of eigen values required. The details of the method are not explained here.

1 Sorensen D.C., “Implicit application of polynomial filters in a k-step Arnoldi method”, SIAM J. Matrix Anal. Appl., Vol. 13, pp. 357-385, 1992.