Non-linear algebraic equations

Abstract

One of the fundamental problems of mathematics is that of solving equations of the form

$$ f(x) = 0 $$

(3.1)

where f is a real valued function of a real variable. Any number α satisfying (3.1) is called a root of the equation or a zero of f. If

$$ f(x) = mx + d $$

for some given constants m and d, then f is said to be linear; otherwise it is said to be non-linear. Most equations arising in practice are non-linear and are rarely of a form which allows the roots to be determined exactly. Consequently, numerical techniques must be used to find them. These techniques can be divided into two categories; two-point and one-point methods. At each stage of a two-point method, a pair of numbers is calculated defining an interval within which a root is known to lie. The length of this interval generally decreases as the method proceeds, enabling the root to be calculated to any prescribed accuracy. Examples are the bisection and false position methods. One-point methods, on the other hand, generate a single sequence of numbers which converge to one of the roots. Examples are the fixed point, Newton and secant methods.