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Solution of equations by Numerical Methods.

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Introduction

Ashley Hilsdon Div 3

Pure 2 Coursework

Solution of equations by Numerical Methods

Method 1: The Change of Sign method

The simplest method for solving an f(x) function is to use a change of sign method; these include the methods of bisection, decimal search and linear interpolations.

Unfortunately, as well as being the simplest methods, they are also relatively cumbersome. The bisection was employed below for the function of f(x)=x5+3x2+x+2. This was also displayed graphically in the graph below.

image00.png

The actual calculations for this method are summarised below in this table. I placed these with relevant diagrams

The error bounds for this result are conveniently provided by the use of this method, and are -1.485<x<-1.475. The maximum error is therefore ± 0.005.

However, this method is clearly not particularly easy to work with. In addition, as with many methods there are some functions for x, which do not work. An example is the function f(x)=x3-2x2-x. This is shown below.

image01.png

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Middle

Using an estimate for (a.) as 0, the Newton-Raphson iterative formula was then used.

This next diagram gives some indication as to how the Newton- Raphson method actually works. What it basically does is to ‘slide’ the tangent across and is indeed sometimes known as ‘tangent sliding’.

image03.png

For this method, the root can be calculated much more quickly then using the Change of Sign method. In accordance with the question the root was calculated to five significant figures, this means that the real value of x exists between 0.16655 and 0.16665, which give an error value of ± 0.00005.

This method, despite its speed in calculation a root, does not work for every function; this is illustrated below.

Method 3: Fixed Point Estimation- Rearranging the function

The third and final method for numerically solving a function of x is by rearranging an existing function of x and then using fixed-point iteration. This is illustrated for the function f(x)=x3+x-3 below.

Diagram of f(x)=x3+x-3

As is displayed graphically, the function with regards to (g)

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Conclusion

Rearranging the function is another useful technique, especially when a ‘cobweb’ effect is produced, as this gives natural error bounds for the root. However, it relies on an important rearrangement of the function, in conjunction with the plotting of at least two graphs. Once this has been done, the iterative formula is used, though within my calculations did not prove as fast as the Newton-Raphson method.

As I have already mentioned, the rearrangement method, relies more heavily on suitable graphical generating software or hardware; this is therefore a clear disadvantage that it faces. However, the change of sign method requires an interval for the root and the Newton-Raphson method requires an approximation to the result. Though, as this investigation has proved clear, regardless of the method, it has been necessary to have some idea of the function before attempting calculations. This is particularly important in the case of turning points and roots close to each other.

...read more.

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