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Maurice Yap 6946 – Core 3 Mathematics Coursework – 4752/02 Methods for Ada!ced Mathematics

Using numerical methods to find roots of and solve polynomial equations

This report will explore and compare the advantages and disadvantages of three different numerical methods used to solve polynomial equations, where analytical methods cannot easily be used. It will explore instances where, for some reason, they fail and also examine their ease, efficiency and usefulness in solving polynomial equations.

Change of sign decimal search method

The change of sign! method can be used to find an approximation of a root to an equation to a specified accuracy, using a decimal search."olynomial equations can be illustrated graphically as the function y # f$x%, as shown below in figure &. The points where the curve intersects the x'axis are the real roots of the equation f$x% #(, because the x'axis is where y # (. If the curve crosses this line, the values for f$x% when x is slightly larger and smaller than the root will be positive and negative, either way round $given that the values chosen for x are not beyond any other roots%.) logical and systematic way to use this to solve an equation to a certain degree of accuracy is a decimal search, where having already identified integer intervals where roots occur, the interval is divided into ten, and f$x% for each of the ten new values for x is found. ) search for a change of sign $* or '% is conducted and the process is repeated in the interval where the change of sign occurs until the level of accuracy desired is achieved. )fter this, the same technique is applied to find the other roots and thereby solving the equation.

+xample of an application of the change of sign method

or example, consider solving the following equation, by first finding the greatest root to five significant figures-

6

x

5

−

9

x

4

−

4

x

3

−

20

x

+

26

=

0

It is shown in figure & that there are three roots to this equation. That which is labelled root c! will be attempted to be found.