Coursework for Numerical Methods

Numerical Methods Coursework

© MEI September 2005

Coursework for Numerical Methods

Introduction

The 2004 specification contains two modules in Numerical Mathematics. In Curriculum 2000 there were three modules, and previously there was only one. The two modules are called Numerical Methods and Numerical Computation. In Numerical Methods there is a coursework requirement. For this module a single piece of coursework is required; it will contribute 20% to the assessment.

The coursework should be clearly related to the content of the module. However, since there is no coursework beyond Numerical Methods, it is permissible to go outside the content of that module in coursework. Previously any task that covered any part of the syllabus of the Numerical Analysis module could be submitted as coursework for that module and therefore was banned from the Numerical Methods module. This is no longer case and so tasks suggested in previous coursework banks for that module may also be submitted.

This revised edition of the Coursework Bank for Numerical Methods reflects the changes in the coursework requirements for this module.

Centres may develop their own coursework tasks but are strongly advised to seek advice from OCR before embarking on tasks that are significantly different from those suggested in this bank.

Teachers are invited to photocopy the Student Guidelines incorporated into this bank, but please note that no other part of the bank should be made available to students.

MEI would be pleased to update this bank in accordance with the needs of teachers. Any comments or suggestions should be sent to Richard Lissaman,

Teachers’ Notes

What is required?

“Candidates are expected to investigate a problem which is suitable for numerical solution, using one of the methods in the specification.”

Rationale

The module aims to develop skills in the areas of problem identification, use of numerical methods and control of error in practice.

The coursework assignment should enable students to demonstrate a facility with technology and an awareness of the difficulties that can arise when computers are used to do mathematics.

The coursework counts for 20% of the assessment and should take of the order of 6 - 8 hours.

Description

Students are expected to investigate a problem which is suitable for numerical solution, using one of the methods in the specification. Problems which have an analytical solution are acceptable only if the analytical solution is too time consuming, or too advanced to be feasible.

Students should use a computer to develop a solution which is both effective and accurate. In particular they must show how the desired accuracy has been achieved, either by means of sufficient iterations of the numeric process to ensure that accuracy has been achieved, or by means of a theoretical analysis of errors.

It is expected that coursework will usually be done using a spreadsheet for the following reasons.

• Spreadsheets avoid the need for students to program their own control structures and to design

their own output layout.

• Using a spreadsheet is a natural way for students to investigate new techniques when a

module is being learned.

A piece of coursework which uses a spreadsheet should include print-outs not just of the values obtained but also the formulae used to obtain them. It is not necessary, however, to print out everything; often just a few lines of formulae will be sufficient to indicate that a correct method has been used. Similarly, if a process takes a long time to converge it may not be necessary to print out all the values; some early values and some later values will suffice.

As with any computer routine, careful annotation is important if it is to be intelligible. Spreadsheet formulae can be difficult to interpret unless annotation makes it clear what each cell contains.

The report by which the task is assessed should be well written and easily readable to anyone who has not seen the task being completed.

The report should be word-processed and contain the following:

(i)Any graphs, using a software package (e.g Autograph)

(ii)Easily readable script with all formulae written at least in Equation Editor if not

Mathtype or equivalent

(iii)Spreadsheet work with some formulae given.

The Oral Communication element of the assessment is the same as for other modules.

As with most other coursework tasks, the criteria for the assessment are laid out within a number of domains on the coursework assessment sheet. This sheet is found on page 199 of the specification booklet. A version of this assessment sheet is included in this bank.

The task

Problem Specification

The problem needs to be able to fulfil the criteria.

If the problem is the solution of an equation then there must be the opportunity for error analysis. If the problem is numerical integration then it needs to be of a function that cannot be integrated analytically. It is not necessary to demonstrate that the integral cannot be done by analytical means known to the student.

Strategy

It is necessary to explain why they have chosen your particular strategy - e.g. why did you choose Simpson’s Rule to do a numerical integration?

Note that doing Midpoint, Trapezium and Simpson without any particular reason is not using an “appropriate procedure” but covering all options!! So the first mark would be gained but not the second.

Formula application

This needs to be substantial. An application of Simpson, for instance, that goes as far as S16 would gain only 1 mark.

Taking the work through to, say, S64, or to a stage where successive values are the same to 6 significant figures would gain the second mark.

To extrapolate to the “best estimate” would gain the third mark.

Use of technology

The use of a spreadsheet is not specifically mentioned but should be taken to be obligatory!!

The formulae used to obtain the spreadsheet values should be explained. All formulae for every line that covers a number of pages should be avoided!

Error analysis

Demonstration of the order of the method is required here.

Note that any values used need to be taken from within the work being done. To compare a numerical integration with a known value (obtained analytically perhaps) or by some other means (e.g. using “Derive” on a calculator) is obviously not acceptable.

Interpretation

Students often forget to say exactly what the answer is!! What are they looking for, so what have they found!

Some discussion needs to take place as to the validity of the method and hence the possible limitations of solution

Teachers are advised to look at the exemplar coursework published by OCR. It is expected that MEI will be able to offer additional exemplar material in due course. Teachers are also advised to consult the Principal Moderator’s reports published by OCR after each examination session.

Numerical Methods Coursework

Student Guidelines

Introduction

There is one piece of coursework required for this module. You should take a problem which is suitable for numerical solution using one of the methods in the syllabus for this module. The main aim of this module is to introduce you to real-life problems that do not have nice, neat solutions and which need numerical methods. A crucial part of the process of finding a solution to a problem is to determine the precision of that solution. A solution to an equation is of little value unless we know the bounds of error for it.

The problems you choose to tackle should not be “standard” problems that are used in text books but also need not be over-complicated.

You should be careful to ensure:

• that the problem you choose to solve does not have an analytical solution,

• any equations that you solve involves work beyond the requirements for the coursework in the module Methods for Advanced Mathematics, C3.

If at all possible, you should use a spreadsheet. This will enable you to design your own output layout as well as simplifying all the computation.

When you write your report you should bear in mind that it should be informative. In other words it is not simply to demonstrate that you have done the coursework, but should inform the reader what you have done and what conclusions you have reached.

However, pages of numbers will not prove very informative. Your print-outs should be as concise as possible and with explanations and formulae used within the spreadsheet work. Annotations by hand on your computer print-outs are acceptable and may be the most informative way to describe what you have done.

It may not be necessary to print out everything. A balance needs to be found between including all that is done and producing pages of numbers that would confuse the report.

Make sure you have a copy of the assessment sheet when you start the assignment so that you know the criteria by which it will be marked.

Phase 1. Problem Specification

You need to identify a problem that is suitable for the assignment and fits into the criteria given above. You need to explain any context from which the problem has arisen and why it is appropriate.

Phase 2. Strategy

The way in which you intend to solve the problem needs to be explained carefully. Note, however, that mere bookwork is not required.

Phase 3. Formula Application

This you will do on a spreadsheet. Your report should make clear what you are doing.

Phase 4. Use of technology.

A brief explanation of what technology and programs you are using should be given.

Phase 5. Error Analysis

A number that represents “the answer” is of little value unless you give the bounds of possible error. This will depend on the method, the formulae used, the decisions you made regarding the number of decimal places being worked and the speed of convergence.

Phase 6. Interpretation

Make sure you express your solution clearly and accurately! It may be helpful to refer the solution back to the original problem if it came out of a particular context. The limitations of the solution need also to be discussed.

Phase 7. Oral Communication

You will receive marks for Oral communication. You should be prepared to discuss the work with your teacher. For example:

(i)Why did you do the task?

(ii)What are its uses and limitations?

(iii)Is it the best solution?

(iv)Could the task have been extended in any way?

and finally … don't forget to number the pages of your work!

Numerical Methods

Suggested tasks

The suggestions for tasks in this bank are indicative only, in order to show the type and depth of treatment expected in the investigation.

It is not expected that candidates will necessarily restrict themselves to tasks from this bank.

Solution of equations

Care must be taken not to merely repeat the coursework assignment for C3. No credit can be given for this. The task must be of a much more substantial nature.

(a)Find all the roots of the equation for 0.05 < x < 1 and for given

(small)k in range 1 < k < 1.

(b)Obtain the first 10 positive roots of the equation x – tanx = 0, using fixed point iteration.

Explain carefully how starting values are chosen and how convergence may be

achieved.

(c)Investigate the solution of for varied values of the

parameter k.

(d)In the Newton-Raphson method, it is sometimes convenient to approximate f ’(xr) using the forward or central difference method. Investigate the effect this has on the speed of convergence.

Errors

(a)Spreadsheets store numbers to a fixed precision. Investigate the accuracy with which operations such as the following can be carried out using a spreadsheet.

(i)Using a quadratic formula to solve the equation x2 – bx + 1 = 0 when b

is very large.

(ii)Subtracting nearly equal quantities; e.g. the forward difference and central difference formulae for numerical differentiation.

(b)Examine the accuracy of a formula from Mechanics. For example, in the formula

, examine the effect on s from errors in u, g, t.

(c)The cosine rule and the sine rule are used in surveying. Examine how calculated lengths

and angles are affected by errors of measurement.

Numerical Integration

Students should not use numerical methods simply to evaluate a definite integral which they know how to integrate analytically.

(a)Evaluate to as many decimal places as you can using an appropriate integral.

(b)Evaluate for a valuea between 1 and 2.

Draw a graph of I against a.

(c)Find from a Normal Distribution table correct to 5 decimal places.

(d)Is

Finite Differences

(a)Use Newton’s forward difference formula to approximate a function which cannot be

integrated analytically . ().

Use Newton’s approximation to estimate the integral for a range of different limits.

Determine the error in the estimate using another integration rule.

Numerical Methods (NM) Coursework: Assessment Sheet

Task: Candidates are expected to investigate a problem which is suitable for numerical solution, using one of the methods in the specification.

Domain

Mark

Description

Comment

Mark

Total

Authentication should be given on the appropriate sheet, for teachers and candidates

Coursework must be available for moderation by OCR.

NMcoursework/MEIsept2005