Computazionale-OLD/1. Error Analysis/1.2. Errori aritmetici in virgola mobile e errori di arrotondamento

Error Analysis

In computational applications there are two main sources of error that we need to worry about:

1. roundoff (or rounding) errors
2. approximation errors

Rounding errors originate from the necessity of having to represent numbers in a computer using a finite number of digits. Approximation errors, instead, refer to those errors that are intrinsic of the way we numerically solve a specific problem. In particular, these would be present even in the ideal case where we were able to carry out the computation without rounding errors.

A simple example of approximation error can be seen in the case where we estimate the value of the function

e^x = \sum_{n=0}^{\infty} {x^n\over n!} = 1 + x + {x^2\over 2} + {x^3\over 3!} + \ldots \ \approx
\sum_{n=0}^{n_{\rm max}} {x^n\over n!},,

by evaluating the series only up to a finite order n_{\rm max}<\infty. In this case, the approximation comes with a truncation error of {\rm O}(x^{n_{\rm max}+1}).

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Note

Let f(x) and g(x) be two functions defined on some unbounded subset D of the positive real numbers, and g(x) be non-zero in a neighborhood of a\in D (often a=0). We write f(x)={\rm O}(g(x)) \quad \text{as} \quad x\to a

if there exist positive numbers \delta and M such that for all x with 0 < |x-a| < \delta,

|f(x)| \leq M |g(x)|,.

Equivalently, we may say that f(x)={\rm O}(g(x)) if

\underset{x\to a}{\lim \sup}{|f(x)|\over |g(x)|}<\infty,.

If, instead, it holds that

\underset{x\to a}{\lim}{f(x)\over g(x)}=1,,

we write f(x) \sim g(x) as x\to a.

In this chapter, we present a general discussion on roundoff errors, not tied to a specific numerical problem we want to solve. In later chapters, we discuss several different methods to solve specific problems and study their approximation errors in detail.

Exercises

1. Consider the function f(x)=e^x in the interval x\in [0,1]. Write a program that calculates the corresponding approximating series:

   g_N(x)=\sum_{n=0}^N {x^n\over n!},.

   (a) Verify that the absolute error \Delta=|f(x)-g_N(x)| scales approximately with x^{N+1}/(N+1)! for N=1,2,3,4.

   (b) The error \Delta, in the given interval in x, differs from x^{N+1}/(N+1)! . Why is that and for which values of x?