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)

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