# I am going to solve equations by using three different numerical methods in this coursework and compare the methods which are Bisection, Newton-Raphson, and Rearranging method.

Introduction-Solving equations by numerical methods

Numerical Methods are used for solving equations which solutions are not possible to be solved by algebraic or analytical methods. Now I am going to solve equations by using three different numerical methods in this coursework and compare the methods which are Bisection, Newton-Raphson, and Rearranging method. The comparison is about their ease of use and speed of convergence. In this coursework, I will use some software which are Microsoft Excel (used for calculations) and Autograph 3.0 (used for drawing graphs) to help me complete the coursework.

Bisection method

Bisection Method is looking for a sign change of a continuous function. Actually, we couldn’t use some normal methods which are algebraic or analytical methods to solve some particular cases. I have chosen an equation y=2x³-3x²-8x+7 which is a non-trivial equation. By using the autograph, we can see the graph (Below) cross the x axis and there are three roots. Now take the root in the interval [-2,-1] and start by taking the mid-point of the interval,-1.5.

f(-1.5)=5.5,so f(-1.5)>0.Since f(-2)<0,the root is in[-2,-1.5].

Now take the mid-point of this second interval,-1.75.

f(-1.75)=1.09375,so f(-1.75)>0.Since f(-2)<0,the root is in[-2,-1.75].

And if we continue this method, than we can find out the root that I have shown on the graph.

I am going to combine the graph and the figure I have worked out in Excel. As the figures show blow:

In this spreadsheet, a and b are the two values of intervals. (a+b)/2 is the mid-point.

By using the Excel, we can easily find the x value in the many terms satisfy my required degree of accuracy which is answer to 4 decimal places.

Root is -1.8007 to 4d.p

Error bounds is -1.8007±0.00005

Root bounds is -1.80075<x<-1.80065

Check     X           Y

-1.80075     -0.000688(negative)

-1.80065      0.001538(positive)

Therefore, my answer is correct and it lies between this interval.

Below shows the formulae for using in the Excel:

Bisection failure

I am going to show the failure of bisection method. I have chosen a non-trivial equation y=10(x–1.32)(x–1.98)(x–1.55)+0.1.And the graph shows:

We can see these three roots are very close together. They are all in the interval [1, 2].By using the Excel:

This is a bisection failure. As the graph shows, we can't solve the other roots. We can only find one root, so the failure exists.

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