Abstract

D.L. Harrar II and M.R. Osborne
School of Mathematical Sciences
Australian National University
May 14, 2001

Discretizations of differential equation eigenvalue problems have a sensitivity to perterbations which is asymptotically least as h 0 when the differential equation is in first order form. Both second and fourth order accurate discretizations of first order systems are straight forward to derive and lead to generalised eigenvalue problems of the form

((A(h) - *B(h)) v = 0

where both A and B are narrow banded, block bidiagonal - hence unsymmetric - and typically B is singular. Solutions of the differential equation associated with small eigenvalues are best determined by the discretization so Krylov subspace methods (for example) can be applied to find large solutions of

((1/)*I - A^-1)v = 0

This already involves rational methods in principle. Thus, for the narrow banded systems relevant here, rapidly convergent methods based on inverse iteration applied to the original formulation as a nonstandard generalised eigenvalue problem prove attractive. We have worked with a method combining aspects of Newton's method and inverse iteration and having a convergence rate of 3.56. In our implementation this basic algorithm is combined with a limiting form of Wielandt deflation to find a sequence of eigenvalues. It has proved extremely satisfactory in a range of applications. The formulation has the advantage that it is easy to insert the eigenvalue calculation into an outer loop to satisfy a constraint on an auxiliary parameter. Also, the deflation procedure has been rigourously tested and has proved very satisfactory.

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