## Numerical Solutions to Equations

It is nice to be able to solve equations exactly – it is aesthetic and the question is clearly answered. In practice however, exact answers are often not possible and we need to know how to solve equations numerically to a sufficient degree of accuracy – in practice, to so many significant figures or decimal places. One method of finding numerical solutions to equations is shown here. The idea is to rearrange an equation to make a particular occurrence of the variable to be solved for – usually – the subject, and solve iteratively starting from a particular initial value.

Example: a)Show that has a zero between and b)Show that a possible solution is given by c)Use the iterative formula Starting from the initial value find and to four decimal places.

d)Solve the equation and give the solution to three decimal places.

a)  There is a sign change for between and so somewhere in between these two values for there is a value of for which b)     c)   1. We continue until two successive iterations agree to 3 decimal places.   to 3 decimal places. 