Computing Basics 1 Sources of Error

Numerical solutions to problems differ from their analytical counterparts. Why? The reason for the difference is that most numerical computations involve error. There are two main types of error that can arise. The first, called roundoff error, arises from the fact that computers do not store real numbers instead they store a subset of real numbers: floatingpoint numbers. Thus the solution of computer arithmetic is approximated with the nearest machine-representable number. The second, called truncation error, arises from truncating infinite series, substituting for continuous functions with discrete approximations, and approximating limits of infinite processes. There are two statistics used to measure error: absolute error and relative error. Suppose a real number x is represented in the computer by the floating-point number x̃, then the absolute roundoff error is ||x−x̃|| and the relative roundoff error is ||x−x̃|| ||x|| . Similarly, suppose we approximate a function f by f̃ , then the absolute truncation error from evaluating f at some point x is ||f(x)− f̃(x)|| and the relative truncation error is ||f(x)−f̃(x)|| ||f(x)|| . Most computations will involve a combination of truncation and roundoff error. For example suppose we want to evaluate some function, f(·) at some real number x numerically. In order to do this we approximate f(·) with f̃(·) and the computer stores our real number x as the floating-point number x̃. We can bound the total error, either absolute or relative, by the sum of the roundoff error and the truncation error: total error = ||f̃(x̃)− f(x)|| ≤ ||f̃(x̃)− f(x̃)|| } {{ } truncation error + ||f(x̃)− f(x)|| } {{ } roundoff error . (1)