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Numerical Methods Coursework

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Introduction

Paul Koleoso        30083490

For this coursework, I am going to use knowledge of numerical methods to produce an approximation to an area which does not have an analytic solution.

Problem

image05.png.image06.png

image15.png

image00.png

image22.png

I would be finding the approximation of the integral on the graph between the shaded portion of the graph above which is from 0 to 1 using numerical integration.

This problem was chosen because it cannot be integrated by any analytical method, therefore approximation method would be used. Due to this I suggest that this problem will be appropriate for numerical solution.

Strategy

To solve this problem, I am going to use knowledge of numerical integration studied in the Numerical Methods textbook.

Numerical integration is a method used to approximate an area under the graph. According to syllabus on NM module, the approximate methods of definite integrals may be determined by numerical integration using;

  1. Mid – point rule
  2. Trapezium rule
  3. Simpson’s rule

Since there are lots mathematical functions which can not be integrated in real life, an alternative approach to these problems are to sub-divide the area under the graph into strips or shapes such as rectangles which approximately covers the area.

...read more.

Middle

= kimage08.png²   = image09.pngimage10.pngimage11.png

This means that halving h, or equivalently doubling n will reduce the error by a factor of image12.png= 0.25. image13.pngSince the absolute error is proportional to h². It is a second order method.

Also trapezium rule is similar to the mid- point rule. Error is also proportional to h². Therefore this also means halving h, or equivalently doubling n will reduce the error by a factor of image12.png= 0.25. image13.pngSince the absolute error is proportional to h². It is a second order method.

Viewing error in terms of differences and ratio differences

When the values for the number of strips double, the ratio difference between successive estimates is the same.

...read more.

Conclusion

 The problem specifiedimage05.png.image06.png can also be written as  image23.png ³.                            Microsoft Excel might find it complicated to solve square roots. So I suggest error might have occurred, therefore affecting the validity of my result.

For example;

image24.png)² - image25.pngimage26.pngimage25.png

If I had more time I could have increased the accuracy of my result by finding the real answer and not using extrapolated values to find “M16”, “T32” ,“M32”and “T64”, as this could have improved the validity of my result and therefore making my answer to the solution more accurate by producing the solution to a higher degree of significant figures. Also to improve the validity of my result Simpson’s rule could have been used. This is because in Simpson’s rule error is proportional to image27.png, which means it is a fourth order method  and also when you halve the width of the strip or you equivalently doubled the number of strips, this would reduced the absolute error with a scale factor of 0.0625.                    

...read more.

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