• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

Numerical Methods Coursework

Extracts from this document...


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.






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.


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.


= 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.


 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.

This student written piece of work is one of many that can be found in our AS and A Level Core & Pure Mathematics section.

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related AS and A Level Core & Pure Mathematics essays

  1. Best shape for gutter and further alegbra - using Excel to solve some mathematical ...

    The benefit of the replication of formulae is also a benefit to using a spreadsheet in question 2. The first two rows were calculated first using the formula given in the table on the previous page. By filling down with this formula, the other rows contain the same calculation for

  2. Arctic Research (Maths Coursework)

    order for the researchers to arrive at observation site C, their plane must leave the base camp at a bearing of 90o and must expect 0.16 hrs (0.16 hrs � 60 = 9.68 minutes) for the journey in which they will travel with a R.V.

  1. Experimentally calculating the wavelength of an He-Ne laser by means of diffraction gratings

    Materials: * 1 diffraction grating of 600 lines/mm * 1 diffraction grating of 300 lines/mm * 1 diffraction grating of 100 lines/mm * 1 grating holder * 1 He-Ne laser * 1 meter-stick Method: 1. Place the He-Ne laser on a level surface, facing a screen.

  2. MEI numerical Methods

    To use fix point iteration we do rearrange f(x) = 0, to G(x) = x. We then do G(x1), and so on; this is called a recurrence relation. By doing this, most of the time the sequence gets closer to the final answer thus it converges.

  1. Although everyone who gambles at all probably tries to make a quick mental marginal ...

    To make the games equal in odds, the entry price to Case 2 must be raised substantially: when a is the reciprocal of the entry price(the odds). Here a comes to equal which implies that the entry price would have to be in the range from $131073 and $262144 .

  2. Numerical Methods coursework

    * There is a connection between the Trapezium Rule and the Midpoint Rule which can be used to shorten calculations: * Quick calculation of using and * All three rules simplify the working out on the spreadsheet due to less difficult formulae Formula Application I used the programme "Microsoft Excel 2003" to produce the spreadsheet.

  1. C3 Coursework: Numerical Methods

    The able shows that there is change of sign from 0.752 to 0.753. This means that the root of the equation must lie between 0.752 and 0.753. Our estimate of the root is with a maximum error of � 0.0005.

  2. Numerical Method (Maths Investigation)

    (See Diag NRM-01) However, Graphmatica only allows me to draw one tangent which is not enough to show how Newton-Raphson Method work, which I may need at least two tangents, one at Xn and another one at Xn+1. Microsoft Excel XP has contributed 35% of my Coursework.

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work