Globalizing the convergence of Newtons Method

One of the main problems with Newton s method is the choice of a starting value In the case of highly nonlinear problems or due to a poor approximation B x) of the method might converge only for starting values in a very small 

From the physical point of view —F xq) is a force which has to be added to F(x) in order to keep the system in a non-equilibrium position. The goal of the homotopy is then to successively reduce this force to zero by incrementally changing s. 

The embedding chosen in (3.8.1) is called a global homotopy. It is a special case of a more general class of embeddings, the so-called convex homotopy 

In the introduction to this chapter we saw already a third type of homotopy, which is based on a physical parameter in the physical problem. There, in Example 3.1.2 this parameter s was related to the gravitational constant pgr in such a way that for s = 0 the truck was given in its known assembly configuration while for s = 1 the truck is in its unknown equilibrium position, s reflects the amount of loading STYiiggr considered in the system. A continuation method gradually increases this loading. 

in general a homotopy function for a continuation method is defined as 

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