• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month
Page
1. 1
1
2. 2
2
3. 3
3
4. 4
4
5. 5
5
6. 6
6

# Numerical Methods Coursework

Extracts from this document...

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

.

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.

Middle

= k²   =

This means that halving h, or equivalently doubling n will reduce the error by a factor of = 0.25. Since 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 = 0.25. Since 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.

Conclusion

The problem specified. can also be written as   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;

)² -

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 , 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.

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

# Related AS and A Level Core & Pure Mathematics essays

1. ## MEI numerical Methods

This is then divided by the function of the second approximation minus the function of the first approximation to give the third approximation. I can use this method because my x(0) is equal to ?/2, and x(1) is equal to 0; hence if we were to substitute these values into

2. ## Solving Equations. Three numerical methods are discussed in this investigation. There are advantages and ...

( it appears as ############ which means there is no space to show the value of g(xn).). We can then consider to the gradient and one conditions shows why method is fail. The reason why it is not work is because there is one restriction - the gradient of, cannot be greater than 1 or less.

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

x r l = ? x r r = ? r = Area of semi circle = 1/2 x pi x r2 ?� radius Area 180 3.183099 31.83099 Formulae used in excel for this calculation is given in Appendix 2. By the use of conditional formatting of the entire spreadsheet I have found that the

2. ## Numerical Methods coursework

so that: Where h is the strip width corresponding to n strips. So the Trapezium Rule with 2n strips has a strip width of , such that: That shows that halving h or doubling n will reduce the error by a factor of 0.25. Therefore the "error multiplier" is 0.25.

1. ## Numerical integration can be described as set of algorithms for calculating the numerical value ...

f(??? ?f(??) > the area of the underneath the curve in figure 1.0 is thus h f(??) + h f(??????h f(??) + h f(??? ???h f(??) > this can be generalised to Mn= h [f(??) + f(?????(??) + f(???????????f(?n)] Note that Mn is dependent on the number of rectangles.

2. ## 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.

1. ## Numerical integration coursework

Trapezium rule The trapezium rule is very similar to the mid-point rule in that it also divides the area up into strips; however the difference is that instead of using rectangles trapeziums are used, as shown.

2. ## C3 COURSEWORK - comparing methods of solving functions

22 0.876 -0.02565 23 0.877 -0.01809 The root lies between 0.879 and 0.880 24 0.878 -0.01051 25 0.879 -0.00293 26 0.8791 -0.00217 27 0.8792 -0.00141 28 0.8793 -0.00065 29 0.87931 -0.00057 30 0.87932 -0.0005 31 0.87933 -0.00042 The root lies between 0.8793 and 0.8794 32 0.87934 -0.00034 33 0.87935 -0.00027

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