Contents Summary

Title Page
1. Introduction
2. Key Idea
3. Root finding in one dimension
4. Linear equations
5. Numerical integration
6. First order ordinary differential equations
7. Higher order ordinary differential equations
8. Partial differential equations

1. Introduction

These lecture notes are written for the Numerical Methods course as part of the Natural Sciences Tripos, Part IB. The notes are intended to compliment the material presented in the lectures rather than replace them.

1.1 Objective

An understanding of how a method works aids in choosing a method. It can also provide an indication of what can and will go wrong, and of the accuracy which may be obtained.

Unfortunately the course is now examinable and therefore the material must be presented in a manner consistent with this.

1.2 Books


More specialised:

For many people, Numerical Recipes is the bible for simple numerical techniques. It contains not only detailed discussion of the algorithms and their use, but also sample source code for each. Numerical Recipes is available for three tastes: Fortran, C and Pascal, with the source code examples being taylored for each.

1.3 Programming

While a number of programming examples are given during the course, the course and examination do not require any knowledge of programming. Numerical results are given to illustrate a point and the code used to compute them presented in these notes purely for completeness.

1.4 Tools

Unfortunately this course is too short to be able to provide an introduction to the various tools available to assist with the solution of a wide range of mathematical problems. These tools are widely available on nearly all computer platforms and fall into two general classes:

1.4.1 Software libraries

These are intended to be linked into your own computer program and provide routines for solving particular classes of problems.

The first two are commercial packages providing object libraries, while the final of these libraries mirrors the content of the Numerical Recipes book and is available as source code.

1.4.2 Maths systems

These provide a shrink-wrapped solution to a broad class of mathematical problems. Typically they have easy-to-use interfaces and provide graphical as well as text or numeric output. Key features include algebraic analytical solution. There is fierce competition between the various products available and, as a result, development continues at a rapid rate.

1.5 Course Credit

Prior to the 1995-1996 academic year, this course was not examinable. Since then, however, there have been two examination questions each year. Some indication of the type of exam questions may be gained from earlier tripos papers and from the later examples sheets. Note that there has, unfortunately, been a tendency to

concentrate on the more analysis side of the course in the examination questions.

Some of the topics covered in these notes are not examinable. This situation is indicated by an asterisk at the end of the section heading.

1.6 Versions

These lecture notes are written in Microsoft Word 7.0 for Windows 95. The same Word document is used as the source for both printed and HTML versions. Conversion from Word to HTML is achieved through a combination of custom macros to adjust the formatting and Microsoft Internet Assistant for Word.

1.6.1 Word version

The Word version of the notes is available for those who may wish to alter or print it out. The Word 7.0 file format is interchangeable with Word 6.0.

1.6.2 Notation in HTML formatted notes

The source Word document contains graphics, display equations and inline equations and symbols. All graphics and complex display equations (where the Microsoft Equation Editor has been used) are converted to GIF files for the HTML version. However, many of the simpler equations and most of the inline equations and symbols do not use the Equation Editor as this is very inefficient. As a consequence, they appear as characters rather than GIF files in the HTML document. This has major advantages in terms of document size, but can cause problems with older World Wide Web browsers.

Due to limitations in HTML and many older World Wide Web browsers, Greek and Symbols used within the text and single line equations may not be displayed correctly. Similarly, some browsers do not handle superscript and subscript. To avoid confusion when using older browsers, all Greek and Symbols are formatted in Green. Thus if you find a green Roman character, read it as the Greek equivalent. Table 1of the correspondences is given below. Variables and normal symbols are treated in a similar way but are coloured dark Blue to distinguish them from the Greek. The context and colour should distinguish them from HTML hypertext links. Similarly, subscripts are shown in dark Cyan and superscripts in dark Magenta. Greek subscripts and superscripts are the same Green as the normal characters, the context providing the key to whether it is a subscript or superscript. For a similar reason, the use of some mathematical symbols (such as less than or equal to) has been avoided and their Basic computer equivalent used in stead.

Fortunately many newer browsers (Microsoft Internet Explorer 3.0 and Netscape 3.0 on the PC, but on many Unix platforms the Greek and Symbol characters are unavailable) do not have the same character set limitations. The colour is still displayed, but the characters appear as intended.

Greek/Symbol character Name
a alpha
b beta
d delta
D Delta
e epsilon
j phi
F Phi
l lambda
m mu
p pi
q theta
s sigma
y psi
Y Psi
<= less than or equal to
>= greater than or equal to
<> not equal to
=~approximately equal to
vector vectors are represented as bold

Table 1 : Correspondence between colour and characters.

1.6.3 Copyright

These notes may be duplicated freely for the purposes of education or research. Any such reproductions, in whole or in part, should contain details of the author and this copyright notice.

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Stuart Dalziel, last page update: 17 February 1998