# Solution of equations by numerical methods.

P2 coursework                David Effah

PURE MAHEMATICS 2 COURSEWORK

SOLUTION OF EQUATIONS BY NUMRICAL METHODS

1. Change of sign Method
• Decimal Search
• Chosen Equation: Y = 2X3 – 5X -2

There are three possible roots for this equation as its graph crosses the x-axis three times. By looking at the graph it is possible to estimate the where about of the roots. Furthermore by testing for the existence of change of sign between any two chosen points I could be able to decide if there is a root or not. Looking at the graph of any equation is also vital as there might be some discontinuities.

I will be testing if there is a root between points [-1,0].

1. For [-1,0] – Substituting this two points in the equation Y = 2X3 – 5X-2     gives:

As it can be seen there is a change of sign, which implies that there is a root between the two points. To find this point to a limited degree of accuracy I will be using decimal search, whose working out is shown below.

Location of root: -0.4324 < x < -0.4323

The estimated root is –0.432 to three decimal places.

Error Bound: -0.432 + 0.0005

A case of failure for the Decimal Search method

To demonstrate this I have chosen the equation Y= [(X2 +1) / (5X4 - 2)] -1. It has got two vertical asymptotes. As a result the decimal search method converges to a false root.

We can see that the equation has got a root and an asymptote between 0 and 1. Let us see if the decimal search method can give us any root between [0,1]. Firstly let’s check for a change of sign between 0 and 1.

There is no change of sign that implies there is no root between the two points, which is not true. This is where drawing the graph of any equation before doing any calculations comes into use. I will carry on using the decimal search method to find a root.

Looking at the decimal search was useful. As it is possible to witness there are two alleged roots- one between 0.7 and 0.8, and the other between 0.9 and 1. In this case I will try to find the one between 0.7 and 0.8.

The decimal search method suggests that the location of the root is:

0.7952< x < 0.7953.

To three decimal places the estimated root is 0.795.

Error Bound: 0.795 + 0.0005

This is impossible as there is a vertical asymptote at about this point.

Reason: As there is a change of sign at either side of the asymptote, the method converges to the point where vertical asymptote crosses the x-axis.

Demonstration:

In order for Y= [(X2 +1) / (5X4 - 2)] -1 to be undefined the denominator has to be zero.

5X4 – 2 = 0

5X4 = 2

X4 = 2/5

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