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 thathas a zerobetweenand

b)Show that a possible solution is given by

c)Use the iterative formulaStarting from the initial valuefindandto four decimal places.

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

a)

There is a sign change forbetweenandso somewhere in between these two values forthere is a value offor which

b)

c)

500. We continue until two successive iterations agree to 3 decimal places.

to 3 decimal places.