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Investigate the three different numerical methods used to solve equations.

Extracts from this document...

Introduction

Solving Equations by Numerical Methods Introduction In this coursework I am going to investigate the three different numerical methods used to solve equations. These include the change of sign method, Newton Raphson method and the Rearranging method. To carry out the investigation I will explain how each method works, using an example of a working equation in each case. I will also show when each of the methods will not work with other equations. I will then compare all three of the methods. Change of sign method Decimal search and Interval bisection are both ways of finding an interval where there is a change of sign. The change of sign can be found on a graph when the line crosses the x-axis. Wherever there is a change of sign there will be a root. These two methods find the interval where the root lies. Decimal Search To find the roots using decimal search, the y-values must be found, using the values of X from 0.1, 0.2, 0.3, all the way up to 1. When a change of sign occurs, this means there is a root lying here. Once the root interval to 1 decimal place has been found, it must then be found to 2 and 3 decimal places and so on to the required number of places. ...read more.

Middle

n Xn Xn+1 1 -2 -8.6 2 -8.6 877.8978571 3 877.8978571 -3417800597 4 -3417800597 1.99623E+29 5 1.99623E+29 -3.97741E+88 6 -3.97741E+88 #NUM! 7 #NUM! #NUM! After entering my formula into the spreadsheet, the table confirms what the graph displayed. The curve of the equation is too steep on both sides of the root. When a tangent is drawn at -2 it crosses the x-axis way past the root at -8.6 and so it is diverging away from the root. This confirms that the Newton Raphson method will not find the roots in this case. Rearranging Equations If you use any f(x) equation and rearrange it into the form x = g(x) the point on the graph f(x) where y = 0 (The root) is the same x-value as the point where y = g(x) crosses the line y = x. To find the roots, I need to find the point where the line y = g(x) crosses the line y = x. I will put in an estimate value of x (X0) and it will lead to a better estimate (X1). I have chosen the equation f(x) = x3-3x+1 This can then be rearranged into the form: 3x = x3+1 x = (x3+1) ...read more.

Conclusion

From the table it can be concluded that the Newton Raphson method is the quickest of the three as it shows convergence very quickly each time. The decimal search is the slowest by far of the three methods. It can take around 25 steps to home in on the root. At first the Rearrangement and Newton Raphson methods appear to be the most difficult to use, since the formulas must be calculated before hand. Once the formula has been calculated and inputted for the first estimate, after this it is very simple. When using a spreadsheet it can then just be dragged down until it produces convergence. Even including the calculations needed for the Newton Raphson and rearrangement method, they still prove to be far quicker than either of the change of sign methods. The main advantage of the change of sign method over the other 2 methods is that it is easy to find all of the roots to almost any equation. Overall I would say that the Newton Raphson method is the most efficient of the three. Although there is some preparation before using the method, it still ends up being the quickest. Pure Mathematics 2 Coursework Michael Hatt ...read more.

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