Numerical Computation of Jacobians

Newton s method or its simplified versions require the knowledge of the Jacobian F (x) at some points. Unfortunately, today s multibody formalisms and programs don t provide this extra information, though at least by applying tools for automatic differentiation this would be possible in principle [Gri89]. Normally, the Jacobian is approximated by finite differences with being the unit 

More generally, the derivative in the direction of s can be obtained by using 

The increment rj has to be chosen such that the influence of the approximation error e rj) = A]F x) — can be neglected. e r ) consists of truncation errors 

In Fig. 3.3 the overall error for the example sin (l) is given. In the left part of the figure the roundoff error dominates and in the right part the truncation error. The slopes in double logarithmic representation are -1 and -f 1 for the roundoff and approximation errors which can be expected from (3.5.1). 

When neglecting and higher order terms this bound is minimized if rj is selected according the rule of thumb 

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