• 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...

Introduction

Numerical Methods Coursework

Numerical Integration

The Problem

Integration means finding the area underneath a particular region of a function. At my current knowledge of maths I am not able to integrate various functions. Therefore I am going to use knowledge of numerical methods to produce an approximation to an area which does not have an analytic solution.

In my coursework I will integrate the function:

image29.png

The graph below shows the graph of the function.

image30.jpg

The red arrow determines the region between 0.25 and 1.25, which then leads to the integral:

image37.png

I can not solve this problem using the knowledge of C1 and C2, because I am not able to integrate cos(x) yet. Due to this I suggest that this problem will be appropriate for numerical solution.

The Approximation Rules

To solve this problem, I am going to use knowledge of numerical integration studied in the “Numerical Methods” textbook. The approximate methods of definite integrals may be determined by numerical integration using:

1. The Trapezium Rule:

The Trapezium Rule divides the area underneath the curve into trapeziums. We can then use the formula image45.png (where a, b are the bases and h is the height of the trapezium) to estimate the area.

image61.jpgimage54.pngimage00.png

...read more.

Middle

image27.pngimage28.pngimage26.png

image03.pngimage02.png

image04.png

image39.jpgimage07.pngimage08.pngimage05.pngimage06.pngimage10.pngimage09.png

image13.pngimage13.pngimage11.png

image15.pngimage14.pngimage16.png

Error Analysis

Because all three rules are just estimates, there is always an error involved. In this section I will give an estimate of how much error there is in my solution of the problem.

Firstly I want to refer to my graph of my function on page 1. The section of the curve is concave that means that the Trapezium Rule gives an overestimate and the Midpoint Rule gives an underestimate. For that reason the estimates of the Trapezium Rule are decreasing and they will reach the solution from above. On the other side the Midpoint Rule estimates are increasing and they will reach the solution from below.

Error in the Trapezium Rule

In the Trapezium Rule the error is proportional to h2. We add a constant k so that:

image40.png

In general, if image41.png is the absolute error in image33.png then there is a constant (k) so that:

image42.png

Where h is the strip width corresponding to n strips.

So the Trapezium Rule with 2n strips has a strip width of image43.png, such that:

image44.png

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.

...read more.

Conclusion

This can be proved by looking on the diagram below.

image19.pngimage18.pngimage17.pngimage19.png

image20.pngimage20.pngimage20.pngimage20.png

image22.png

The distance between image50.png and image51.png is 4 times smaller than the distance between image52.png and image53.png, where X represents the actual solution. To get the next distance between image51.png and image55.png we multiply the “multiplier” 0.25 with the distance between image50.png and image51.png.

This development leads to improved solution by extrapolation, as stated in the “Formula Application” section.

Error in my solution

The error gets smaller the more strips are used as shown above. Therefore:

Rule

Error

T2, M2

image56.png

T4, M4

image57.png

T8, M8

image58.png

T4096, M4096

image59.png

---

---

S2

image57.png

S4

image60.png

S256

image62.png

Interpretation

 My solution to the integral image63.png is: 0.885909(6 decimal places)

The solution refers to:        T4096=        0.885909120684554

M4096=        0.885909049759261

S256=        0.885909075135303

The fact that the Trapezium Rule and the Midpoint Rule give the same solution till the sixth decimal place guaranties that my solution is valid. Simpson’s Rule, the weighted average, proves the solution as well.

Therefore: My solution is proved by 3 different rules.

I can say with guarantee that there is an error involved, because all three rules are just approximation rules. But I can say that my solution is not far away from the actual solution as the error for the rules with 4096 stripes is very small (actual value stated in the “Error Analysis” section).

...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. Marked by a teacher

    Estimate a consumption function for the UK economy explaining the economic theory and statistical ...

    3 star(s)

    Therefore, even the best consumption functions, which discussed above should be used with care. Abstract The project has attempted to Estimate a consumption function for the UK economy and explain economic theories by using statistical techniques. The first part of the project provides a brief introduction about the importance of consumption prediction in the economy and aims of this project.

  2. MEI numerical Methods

    The difference between the two methods is that the secant method uses two approximations which are the most recent approximations of the root. The false position method also uses two approximations; it uses the most recent approximation of the root and the most recent root which has the opposite sign of the most recent approximation of the root.

  1. C3 Coursework: Numerical Methods

    The first root I shall attempt to find is the root at point C. The initial estimate (starting point) is x=2. The tangent to the point x=2 crosses the x axis at 1.63. This gives a second estimate of 1.63.

  2. Borders Coursework

    So: 2 * 4 = 8 8 - 4 = 4 The answer matches the answer in the sequence so the equation is correct, but I will check again to be sure.

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

    15 1.529297 8.43E-05 1.529419 -0.00044 1.529358 -0.00018 6.1E-05 16 1.529297 8.43E-05 1.529358 -0.00018 1.529327 -4.6E-05 3.05E-05 17 1.529297 8.43E-05 1.529327 -4.6E-05 1.529312 1.9E-05 1.53E-05 18 1.529312 1.9E-05 1.529327 -4.6E-05 1.52932 -1.4E-05 7.63E-06 19 1.529312 1.9E-05 1.52932 -1.4E-05 1.529316 2.7E-06 3.81E-06 20 1.529316 2.7E-06 1.52932 -1.4E-05 1.529318 -5.5E-06 1.91E-06 After 20th terms and approximation of the root to is found.

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

    Thus M16 means that area under the curve is split in 16 rectangles whereas M32 means its split up in 32. Having explained the mid-point rule above, I shall now explain the trapezium rule. The two methods are quite similar to each other in the sense of approximating the definite integral.

  1. Numerical integration coursework

    make an approximation from Mâ up to Mââ, these algorithms made it much easier and quicker to come to a final answer thanks to the ability to just simply pull down the box with the algorithm in to transfer it to other boxes.

  2. C3 COURSEWORK - comparing methods of solving functions

    < -1 then the iteration will diverge away from the root in a cobweb fashion and it will become a failure case Evaluating g'(x) using calculus Since g(x) = 0.8x³+2x²–1 g'(x) = 2.4x2+4x The roots were x = -2.7876 ([-3, -2])

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