Numerical Methods with Python

5 Iteration

While solving linear systems is probably the most common task in applied mathematics, another often-used concept for solving problems algorithmically is iteration. We use iteration primarily for two functions:

• to advance in time;

• to improve upon a previous approximation.

To introduce iteration as a process, we begin by showing a powerful but simple method to find the roots of a function.

5.1 Finding roots: the bisection method

Suppose we have a continuous function $\mathit{f}\mathrm{(}\mathit{x}\mathrm{)}$, as shown below, and we wish to find the roots (or zeros) of the function, indicated by black dots. That is, we wish to solve

$\begin{array}{c}\mathit{f}\mathrm{(}\mathit{x}\mathrm{)}=0.\end{array}$

If we can find ...