It is shown how piecewise differentiable functions F : R n → R m that are defined by evaluation programs can be approximated locally by a piecewise linear model based on a pair of sample pointsx andx. We show that the discrepancy between function and model at any point x is of the bilinear order O( x −x x −x ). As an application of the piecewise linearization procedure we devise a generalized Newton's method based on successive piecewise linearization and prove for it sufficient conditions for convergence and convergence rates equaling those of semismooth Newton. We conclude with the derivation of formulas for the numerically stable implementation of the aforedeveloped piecewise linearization methods.

In this article we analyze a generalized trapezoidal rule for initial value problems with piecewise smooth right hand side F : R n → R n . When applied to such a problem the classical trapezoidal rule suffers from a loss of accuracy if the solution trajectory intersects a nondifferentiability of F . The advantage of the proposed generalized trapezoidal rule is threefold: Firstly we can achieve a higher convergence order than with the classical method. Moreover, the method is energy preserving for piecewise linear Hamiltonian systems. Finally, in analogy to the classical case we derive a third order interpolation polynomial for the numerical trajectory. In the smooth case the generalized rule reduces to the classical one. Hence, it is a proper extension of the classical theory. An error estimator is given and numerical results are presented.

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