Summary of the course: Mathematical Methods (and Graphics)

Comp.145  October 1996  Dr. A.J. Mestel

There will be about 25 lectures and 15 tutorials on the mathematical methods part of this course. During the first part of this term there will be two lectures a week and one class, then one lecture and one class. In January we will return to two lectures and one class for the first part of term. Dr Gillies will teach the Graphics part of the course next term.

Absence: Dr Walton has very kindly agreed to give the first two lectures. Aims of the course: To develop the mathematical skills of A-level to the point where

they can be used to analyse the behaviour of computational algorithms and to enable their construction. In particular, to understand the relationship between functions of continuous and discrete variables. To understand three-dimensional geometric modelling.

Contents:

(1) Vector algebra. Scalar and vector products. Planes, lines and spheres in vector form.

(2) Matrix algebra. Multiplication, 3 × 3 matrices viewed as transformations of 3-vectors. Invertible and non-invertible transformations.

(3) Inverses and determinants of matrices. Efficiency of row operative methods.

(4) Gaussian elimination. Solutions of simultaneous linear equations of full or deficient rank. Effect of finite precision arithmetic.

(5) Eigenvalues and eigenvectors. Powers of matrices. Behaviour of repetitive processes.

(6) Complex numbers. Modulus and argument. De Moivre’s theorem and complex roots.

(7) Recurrence relations, general and particular solutions. Relation to finite difference methods for differential equations.

(8) Limits: convergence of sequences and series. Ratio and comparison tests. Power series, radius of convergence. Taylor series.

(9) Functions of many variables: Partial differentiation. Simple use of chain rule. Maxima, minima and saddle points. Taylor series.

(10) Iterative methods for solution of nonlinear equations. Newton’s method. Analysis of errors. More than one variable.

Books: No single book is recommended for the course as a whole. For the first part of the course, any book with ‘Vector’ ‘Matrix’ or ‘Linear Algebra’ in the title should be adequate. In general, do not be afraid of book titles with ‘engineer’ or ‘scientist’ in them. These will usually be of more use than titles which sound more computing orientated, as these latter tend to over-emphasise discrete mathematics. Examples of useful books: Elementary linear algebra by H. Anton (J. Wiley) Mathematical methods for scientists and engineers by G. Stephenson (Longmans)

Printed notes: From time to time I shall hand out printed notes. These will often be highly condensed and will not cover all aspects of the course; it will be neither necessary nor sufficient to know everything upon them. The course is defined by the lectures given, so it is very important to take comprehensive and accurate notes yourself. If you miss lectures, arrange to copy the missed notes. Example sheets for the tutorial classes will be handed out; usually solutions also, eventually. There will be two pieces of assessed work per term. Probably, one will take the form of a mini-examination in the tutorial, while the other will require handing in after a week. Probable office hour: 10.30 - 11.30 on Friday.