Volume 40, pp. 373-380, 2013.

A note on the relation between the Newton homotopy method and the damped Newton method

Xuping Zhang and Bo Yu

Abstract

The homotopy continuation method and the damped Newton method are two known methods for circumventing the drawback of local convergence of the standard Newton method. Although some relations between these two methods have already been obtained, these relations are mainly for the differential equations which determine the paths followed by the two methods, rather than the sequences generated by the algorithms. In this paper, these sequences are investigated and some further relations are explored in terms of the marching directions and the step sizes during the iteration processes. Numerical solution of a semilinear elliptic equation is included to illustrate the relations discovered.

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Key words

Homotopy continuation, damped Newton method, domain of convergence, nonlinear algebraic equations, semilinear elliptic equations, finite element method.

AMS subject classifications

65H10, 65H20.

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