On the Role of Computable Error Estimates in the Analysis of Numerical Approximation Algorithms

Abstract

Truncation error estimation for methods of numerical approximation, i.e., estimation of their errors of approximation, is an important constituent of numerical analysis. The error estimates obtained can be dichotomized according to whether an error estimate is computable from information that can be used for computing the approximations. For example, the standard error estimate for the bisection method for finding a zero of a function f(x) in an interval [a, b], under the assumption that f is continuous on [a, b] with f(a)f(b) < 0, is computable. The method proceeds as follows. Let [a ₀, b ₀] = [a, b]. At Step k (k ≥ 0), compute f(x _k ), where x _k = (a _k + b _k )/2. If f(x _k ) = 0, an exact zero is found. Otherwise, let [a _k+1, b _k+1] = [a _k , x _k ] if f(a ₀)f(x _k ) < 0, or [a _k+1, b _k+1] = [x _k , b _k ] if f(b _k ) < 0, and proceed to Step k + 1. Now let x _k be the kth approximation to an zero of f in [a, b]. Then there exists $\bar{x} \in [a,b]$ such that $f(\bar{x}) = 0$ and

$$\left| {\bar x - x_k } \right| \leqslant \frac{{b - a}} {{2^k }}.$$

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