Errors in Numerical Algorithms

Simulation algorithms for dynamical systems rely on the shadowing lemma to establish the proximity of a pseudo-orbit and a true orbit. The power of the shadowing lemma is tremendous in that it takes all sorts of errors in a computer simulation into account. Numerical analysis, on the other hand, is not that fortunate. There is no systematic error analysis that encompasses all possibilities. Finite element method's error equation, for example, only deals with discretization error, but ignores computational round-off error, and can not provide a good estimate of the proximity of the real solution based on mesh properties.

Here we briefly discuss the error estimate in a numerical simulation of contraction mapping. The fixed point and iterated epsilon-pseudo-orbit can be estimated to be within epsilon/(1-C) plus a arbitrarily small number, where C is the contraction constant. That is, if the contraction constant is close to 1, much smaller round-off and general computational errors are needed to guarantee the iterated simulation is within reasonable range of the true fixed point.

Most books on numerical analysis only declare the soundness of contraction mapping theorem, but leave systematic error estimate, epsilon/(1-C), out of the material. We believe this is a big mistake and rigorous error analysis is the key to a reliable numerical approximation. It's time to establish analogues of the shadowing lemma in numerical analysis.