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Testing Root Methods

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Introduction

Pure 2 Coursework: Testing Root Methods A. Hayton Pure Mathematics 2 Coursework: Testing root methods Aim: My aim is to find the advantages and disadvantages of three different root-finding methods: The Change of sign method, The Newton-Raphson method and Fixed Point Iteration. Method 1: Decimal Search (Change of Sign Method) The decimal search method is used to find the first root of y=x5-x+0.2 - the table of our working is shown below. As you can see, when we find the sign changes we can then move on and find the next decimal place. Error Bounds of this root As we can see from the table, the error bounds of this root are -1.044762 and -1.044761 (the root lies in between these two points because y is negative for one and positive for the other). The root is found in 49 iterations. Failure of the Change of Sign Method The root closest to x=1 cannot be found because the decimal search method will stop when it reaches around x=0.3 (finding the first root in between 0 and 1) ...read more.

Middle

Method 3: Rearranging the formula In order to solve graphs by this method, the equation needs to be rearranged so that it crosses y=x. Adding an x to both sides of the equation enables us to do this. For example, y=x5+1.4x2-2x+0.5 becomes x=(x5+1.4x2+0.5)/2. This means that when the new graph crosses y=x the old graph will cross y=0 at the same point. Here is a visual representation of the new graph, along with y=x. If we use the Fixed Point Iteration function of Autograph, we can see the way it works out the graph points: x Change in x 0 N/a 0.25 0.25 0.294238 0.0442383 0.311706 0.0174678 0.319484 0.0077777 0.323113 0.00362939 0.324842 0.00172927 0.325674 0.000831971 0.326076 0.000402127 0.326271 0.000194798 0.326366 9.44653E-05 0.326412 4.58339E-05 0.326434 2.22439E-05 0.326445 1.07966E-05 0.32645 5.24073E-06 0.326452 2.54394E-06 0.326454 1.23489E-06 0.326454 5.99452E-07 The root we have found is x=0.326454. If this value of x were put into the old equation we would find that this is a root of that too. If we find g'(x) for our starting value (x0), it turns out to be 0. ...read more.

Conclusion

It also does Newton-Raphson and Fixed Point iterations in a matter of seconds. It works to a large number of decimal places if this is set in the options. Newton-Raphson Iterations do not need the original equation to be changed in any way and all the roots can be found easily by specifying a starting value close to the root. This is by far the most preferred method out of the three. Fixed-Point Iterations need the equation to be changed before a root can be found. Also, two graphs need to be drawn which can take time in Autograph. Both graphs then need to be selected instead of just one, but from there the computer does all the work. If we want to find all the roots this can take a long time as the graph needs to be redrawn as many as 3 times in some cases! Using the spreadsheet helps to speed up the Decimal Search method, but it cannot be done in Autograph. This means as well as taking longer than the other two it is also more difficult to complete, as all the boxes need to be filled in manually. ?? ?? ?? ?? ...read more.

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