C3 Coursework Notes



 Maurice Yap 6946 – Core 3 Mathematics Coursework – 4752/02 Methods for Ada!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-
















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.


OCR MEI GCE Unit 4753/02 Methods for Advanced Mathematics




OCR MEI GCE Unit 4753/02 Methods for Advanced Mathematics



The solution of equations are commonly found analytically, using a predetermined formula such as thequadratic formula or by algebraic manipulation, with the roots of a cubic, quartic or quintic polynomialequation being denoted by letters of the Greek alphabet. However with far more complex equations,analytical methods prove to be inadequate and this is where numerical methods become of value.Numerical methods are methods of finding numerical approximations of solutions. Numericalmethods should be used where analytical methods are unavailable.

I will use three distinct examples of numerical methodsi.

Systematic search for change of sign, using decimal searchii.

Fixed point iteration using the Newton-Raphson methodiii.

Fixed point iteration after rearranging the equation f (x) = 0 into the form x = g (x)I will use these three different methods to find the roots of complex equations, in order to show howeach deem successful. I will also show instances where each method fails comparing it numerically andgraphically. After using these methods, I will then compare the method by applying them on the sameequation to show which is faster and more reliable.

Numerical Methods| Change of sign (Decimal Search)

The decimal search method works by giving values for x for which values of f (x) change. When thevalues of 

f(x) =



7x+5 change from a positive to a negative or from a positive to a negative, thisindicates that the interval contains a root. Using the decimal search method I am going to find theroots of the equation x


7x+5 = 0 where f(x)=0

As you can see from the table, the highlighted cells indicate that the equation x


7x+5 = 0 has rootsbetween the intervals [-2,-1],[0,1] and [1,2]x


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