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In this investigation I aim to investigate three methods of finding the roots to equations and then compare them. I aim to be able to use fully and understand all three techniques.

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Introduction

Mathematics Coursework Aim In this investigation I aim to investigate three methods of finding the roots to equations and then compare them. I aim to be able to use fully and understand all three techniques. The three methods that I am going to examine are: ?Decimal Search/Change of Sign Method ?The Newton-Raphson Method ?The Re-arrangement Method. Initial Exploration of Methods When I started this project I did not have a deep understanding of these methods I was using. In order to develop my understanding I experimented with the methods and also with the equations that I was going to use so that I knew what they would look like, making it easier for me to create equations to my own specifications. Decimal Search/Change of Sign Method In the piece of course work we are asked to find (or not to find) the roots of equations that we choose. But what are the roots. To put it simply they are the values that can be given to x so that f(x) is equal to zero. This can be displayed graphically by plotting the values of x against values of f(x). This will produce a line that will be continuous (will not be using any discontinuous equations in this project). ...read more.

Middle

If you look at my TABLES section you will see that I have easily found the root at roughly 0.14. But if we look at the two charts in green we can see that this method does not find any negative roots even when the resolution is increased. It is only when, with information, not from the search method but from omnigraph, that we can find the solutions between -1.1 -1. The solutions can be seen closer in a graphical from here. I have not gone further with the decimal search method to find them, because I have already proved that I can find roots and all I needed to do here was to prove that first it seemed as is they didn't exist and then that they did. The Newton-Raphson method The Newton-Raphson method is much more sophisticated than the Decimal Search method. It involves a very clever method to find out where the roots of an equation are. It works on the principle that a curve will be pointing (it's tangent) at the root of an equation. To find the roots it takes the derivative of the equation and then calculates where the tangent to the line at a given point will cross the x-axis. ...read more.

Conclusion

I have demonstrated four failures with the Newton-Raphson Method, but there I still one more even in this equation! The Newton-Raphson method can in fact find a root that doesn't even exist. If you look in the tables section you will see one root that is found where I haven't tested form accuracy. This root does not exist. This is because the root almost exists, it is where it looks like a double root on the large scale picture But as you can see from this graph it is about a 1trillionth away from the x-axis, but does not touch and then due to software and/or hardware limitations the spreadsheet program thinks that there is a double root at this point and creates a non-existent root, with is a far worse fault than missing out on a root that does exist. Software/Hardware limitations are a large problem in situations like this. Most programs will only use a certain number of bits to determine a point and that number may further be limited if it being used on a low quality computer. If you look on the graph above you can see the resolution. The values all seem to be distinct trillionths away from each other and this can be problem in scenarios like this. The Re-arrangement Method Original y=(x^6-2*x^4+4*x^2-8x+3.141592654)/20 Success y=(x^6-2*x^4+4*x^2+3.141592654)/8 Failure y=((x{^{6}+4x{^{2}-8x+3.141592654)/2){^{0.25} Comparison of Methods ...read more.

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