Wednesday, May 3, Lori Student Center, Room 228, 9:00am

Matrix-Free Numerical Continuation and Bifurcation

Kurt Georg
Department of Mathematics
Colorado State University

A numerical continuation method traces solution branches of a nonlinear system H(lambda,x) = 0 where H: R times R^N -> R which is typically obtained by discretizing a parameter dependent operator equation. The method is called matrix-free if the Jacobian of H is not calculated explicitly, but its action on a vector is given via a difference approximation of a directional derivative. In connection with modern (transpose-free) iterative linear solvers, this is a suitable approach for large systems. We give an introduction into the technique of matrix-free numerical continuation and analyze those recent approaches to numerical bifurcation which permit a matrix-free approach. A sequence of MATLAB codes, which can be viewed as a blueprint for the numerical implementation of this approach, is currently under construction and will be made available on the internet. Parts of this talk are based on a 200 page survey by Allgower/Georg in the Handbook of Numerical Analysis.
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