MATRIX-FREE NUMERICAL CONTINUATION AND BIFURCATION*
Kurt Georg · Numerical Functional Analysis and Optimization · 2001
A numerical continuation method traces solution branches of a nonlinear system 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 manuscript are based on the 200 page article [2] Allgower, E. L. and Georg, K. 1997. “Numerical path following”. In Handbook of Numerical Analysis Edited by: Ciarlet, P. G. and Lions, J. L. Vol. 5, 3–207. North-Holland. [Crossref] , [Google Scholar].