Small errors imply large evaluation instabilities

Abstract Numerical analysts and scientists working in applications often observe that once they improve their techniques to get a better accuracy, some instability of the evaluation creeps through back door. This paper shows for large class numerical methods such Trade-off Principle between error stability is unavoidable. It an instance no free lunch theorem . Here, mathematical map takes input data output data. independent from routine calculates output. Therefore, different computational stability. The setting confined recovery functions data, but it includes solving differential equations by writing as under constraints imposed operators boundary values. trade-off principle bounds product two terms below. first related errors, second turns out be instability. Under certain conditions satisfied splines kernel-based interpolation, both can minimized. Then lower bound attained, term inverse term. As byproduct, shown Kansa’s Unsymmetric Collocation Method sacrifices accuracy improved stability, when compared symmetric collocation.