Approximate Numerical Methods for Root Finding

Consider a function $f$, then $r$ is called a root of the function if

$$f(r) = 0.$$ (1.5)

In most cases the root must be obtained by numerical methods using a recipe or algorithm. This means that we do not have an exact value for the root and only an approximate value. However, if the nuerical method involves iteration then the root can be approximated to whatever accuracy we desire. In order to understand numerical approximations and accuracy we need to understand the meaning different errors due to (i) rounding and truncation, (ii) error estimation, (iii) order of convergence and (iv) stability. These ideas are introduced below but first a reminder of the different forms of numbers.