Newton’s Method

Abstract

Many of the equations arising in practical problems are of a type difficult or impossible to solve by the standard algebraic methods. For example, the equations:

$$ 2\sin {\kern 1pt} x - x = 0{\kern 1pt} {\kern 1pt} {e^x} - 2x - 1 = 0,{\kern 1pt} {\kern 1pt} {x^6} - 3x + 1 = 0 $$

have roots which we may estimate, by graphing the functions and finding where the graphs cut the x-axis, but which we cannot find exactly. In these cases a numerical procedure known as Newton’s method allows us to use a value x ₀ which is an approximate root of the equation:

$$ f(x) = 0 $$

in order to obtain a better approximation x ₁.