This chapter contains two investigations which use graphics: A, on spirographs and zigzags, and B, on the problem of determining the shape of a wire which gives the fastest time of descent for a bead sliding down the wire.

A Spirographs and zigzags

Aims of the project

The idea here is to draw a zigzag line determined by a simple rule. The interest lies in determining how many times the line must zig and zag before it closes up, and in working out how large the resulting picture is, so that it can be drawn fitting neatly on the screen. We shall also investigate the connection between the zigzags and certain spirograph curves (epicycloids). The basic idea for this project comes from the book [1].

Mathematical ideas used

Vectors in the plane, linear equations and 2 × 2 matrices, regular polygons, gcds, trigonometric formulae and ellipses all come into this project.

MATLAB techniques used

This is a project about drawing sequences of lines. You need to amend a given program to make it do successively more tasks automatically. One of these involves calculating gcds, and another involves setting the screen size to fit the zigzag.

Construction of the zigzag

This project involves drawing zigzag lines according to a simple rule, and examining the mathematics behind these constructions. There is an intimate connection between the zigzags and certain epicycloids, which are obtained by rolling one circle on another and drawing the path traced out by a point rigidly attached to the rolling circle.

In this chapter, we shall introduce further MATLAB structures in the context of some properties of whole numbers.

A loop to calculate Fibonacci numbers

Type in succession

» f = [11]

» f(3) = f(1) + f(2)

» f

» f(4) = f(2) + f(3);

» f

The last command produces the vector [1 1 2 3]. Thus f (1) refers to the first entry of the vector f, f (2) to the second, etc.

We can do this over and over in a loop:

» f = [11]

for k = 1:15

f(k+2) = f(k+1) + f(k)

end

» f

Note that MATLAB suppresses the » prompt until the loop is completed by end. What the for … end loop does is to take in succession the values 1,2, …, 15 for the variable k and to augment the vector f by a new entry f (k + 2) each time round. Thus k = 1 makes f emerge as [1 1 2]; k = 2 makes it emerge as [1 1 2 3], and so on. The semi-colon after the equation for f (k;+ 2) suppresses output during the 15 times the loop is executed.

Typing

plot(f, '*')

gives a plot of the values, placing an asterisk at each point (i,f(i)). Just plot(f) joins these points up with straight segments, producing a steeply sloping curve. The entries of the vector f are the Fibonacci numbers, 1,1,2,3,5,8,13,21,34, … The rule for forming them is, as above, f(1) = f(2) = 1; f (k + 2) = f(k + 1) + f(k) for k ≥ 1.

The above commands can be put into an M-file. As a variant on the above we could use a ‘while loop’, as follows:

f=[1 1];

k=1;

while f(k) > 1000

f(k+2)=f(k+1)+f(k);

k=k+1;

end

f

plot(f)

This M-file is stored under the name fibno.m and is executed by typing fibno.

First steps with MATLAB

If you haven't already done so, you should start MATLAB by doubleclicking the relevant icon with your mouse or by asking a friend sitting next to you to show you how. Asking a friend is often the quickest way to obtain help and, in what follows, we will encourage you to take this route when all else fails. If ‘clicking’ and ‘icons’ mean nothing to you, you may need some extra help in getting started with Windows. It might also be that your system doesn't use Microsoft Windows and that simply typing matlab will do the trick. For example, if you are using a Unix system of some kind this may be the case. If all goes well you will see a MATLAB prompt

>>

inviting you to initiate a calculation. In what follows, any line beginning with >> indicates typed input to MATLAB. You are expected to type what follows but not the >> prompt itself. MATLAB supplies that automatically.

Arithmetic with MATLAB

MATLAB understands the basic arithmetic operations: add is +, subtract is −, multiply is * and divide is /. Powers are indicated with ^, thus typing

results in

If you typed the above line and nothing happened, perhaps you omitted to press the <Enter> key at the end of the line. The laws of precedence are built in but, if in doubt, you should put in the brackets. For example

The sort of elementary functions familiar on hand calculators are also available.

Linear systems of algebraic equations are one of the most important subjects in mathematics, since most other subjects, methods and problems involve or reduce to this subject.

Aims of the project

As numerical methods involve truncations and finite precisions, we investigate their effect on solution accuracy. Most of us know some theory about linear systems but may not be aware of the good or bad choice of solution methods on computers, what determines the accuracy of the numerical solution and whether an obtained solution can be improved. Large scale problems arising from practice often involve sparse matrices and special techniques can be developed. This project addresses all such issues.

Mathematical ideas used

You have learnt that to solve Ax = b you just type x = A\b. To investigate the sensitivity of the solution x with regard to the matrix A (or its condition number) and the right-hand side vector b, we use a controllable number of digits in our calculations. The use of iterative refinements is illustrated. Finally we discuss how a sparse matrix may be condensed towards band forms by use of permutation matrices.

MATLAB techniques used

Seven M-files lin_solv.m, chop.m, lu2.m, lu3.m, lu4.m, solv6.m and spar_ex.m are used to assist this project. The last two M-files are listed in the chapter. Here chop.m, used to fix digits, is used by lu2.m, lu3.m and lu4.m. You will get useful experience of MATLAB's easy and simple commands for both dense and sparse matrices.

The name MATLAB stands for matrix laboratory. MATLAB was originally written to provide easy access to matrix software developed by the LINPACK and EISPACK projects, which together represent the state of the art in software for matrix computation. MATLAB converts user commands (including M-files) into C or C++ codes before calling other program modules (subroutines). Today MATLAB has been developed to cover many application areas. You may search the internet with the key word MATLAB to see some evidence. To learn more about the product developers, check the web page

• http://www.mathworks.com/

If you hope to find out more of, or wish to use directly, the LAPACK and EISPACK programs, check the following

• http://www.netlib.org/lapack/

For users who would like a gentle and easy introduction to MATLAB, try the following

• http://www.ius.cs.cmu.edu/help/Math/vascJielp-matlab.html

• http://www.unm.edu/cirt/info/software/apps/matlab.html

For novice users who prefer short guides to MATLAB with concrete and simple examples, try

• http://www.liv.ac.uk/CSD/acukJitml/486.dir/486.html

• http://www-math.cc.utexas.edu/math/Matlab/Matlab.html

• http://www.indiana.edu/∼statmath/math/matlab/index.html

Advanced users of MATLAB may check the following pages for more comprehensive guides to MATLAB

• http://www.rrz.uni-hamburg.de/RRZ/software/Matlab/

• http://www.engin.umich.edu/group/ctm/

• ftp://ftp.math.ufl.edu/pub/matlab/

Even more sophisticated examples may be available on the internet; e.g. try the following for more M-files

• ftp://ftp.cc.tut.fi/pub/math/piche/numanal/

• http://www.ma.man.ac.uk/∼higham/testmat.html

• http://www.tc.cornell.edu/∼anne/projects/MM.html

• http://users.comlab.ox.ac.uk/nick.trefethen/multimatlab.html

• ftp://ftp.mathworks.com/pub/mathworks/toolbox/matlab/sparfun/

The following sites contain technical information on MATLAB and FAQ (frequently asked questions) pages

• http://www.mathworks.com/

• http://www-europe.mathworks.com/

• http://www.math.ufl.edu/help/matlab-faq.html

Aims of the project

Magic squares have been known for centuries. This project explores their properties from the perspective of matrix algebra, that is, using addition and multiplication of matrices. The project is not concerned with the number-theoretic problem of finding magic squares containing consecutive integers. The project is self-contained, but it may be of interest to know that several of the mathematical results come from the article [16]. This article also contains other results on the same subject.

Mathematical ideas used

Matrix multiplication, row reduced echelon form and solution of linear equations are used. Also, for example, 3 × 3 matrices are regarded as lying in nine-dimensional space R9, and subspaces of R9 are considered. (There is no requirement to know the definition of an abstract vector space: all spaces are contained in some Rn.) The ideas of linear independence and basis are used. It is necessary to know that, in a subspace X of dimension r in Rn, a set of r vectors in X which is linearly independent automatically spans X and so forms a basis. It is necessary to know the definitions of eigenvalues and eigenvectors, and to use these in a simple argument involving powers of matrices.

MATLAB techniques used

The project is about matrices, so you will need the techniques described in Chapter 2. At one point there is an M-file with several ‘for’ loops, and ‘if’ statements, so you will need to understand these ideas. See Chapter 3. Note that the project is somewhat ‘open-ended’: students who work quickly might like to go on to the final section, on 5 × 5 magic squares, which could be regarded as optional.

Aims of the project

You are invited to study the solutions of a list of ordinary differential equations using whatever methods you have at your disposal.

Mathematical ideas used

Some of the equations are capable of analytic solution using such mathematical tools as: separation of variables, integrating factors or series solutions. Others are examples of homogeneous, or constant coefficient differential equations. What you can bring to bear will very much depend on your mathematical background at this point.

MATLAB techniques used

All the numerical techniques required have been introduced in Chapter 7. For example, grain (or phase) plot analysis and the numerical solution of coupled first order equations. You may find helpful the M-files associated with that work (fodesol.m, species.m, vderpol.m,…). You can use these directly or copy and modify them as required.

Strategy

Your aim, for each equation in the list of exercises, is to provide the following information as appropriate:

1. (a) For first order equations, a grain plot with typical solutions superimposed. For second order, or coupled first order equations, sketch of a typical phase plot.

2. (b) An analytic general solution if yon can find one.

3. (c) The particular solution for the specified initial conditions.

4. (d) Any other comments you wish to make on the nature of the solutions, their stability etc.

In each case, start by classifying the type of the differential equation. Is it linear? What order is it? Has it got constant coefficients? Is it homogeneous? If it is of a type which you recognise, then try to solve it ‘analytically’, that is, by paper and pencil.