This course is a prerequisite for Numerical Analysis II, Digital Signal Processing (Part II).

Aims

The aims of this course are to provide introductions to floating-point
arithmetic, numerical analysis and numerical software. Current
implementations of floating-point arithmetic will be described. The
basic principles of good numerical techniques will be illustrated by
examples, but it will be shown that the design of a numerical
algorithm is not necessarily straightforward, even for simple
problems. The emphasis of the course will be on principles and
practicalities rather than mathematical analysis.

Lectures

Floating-point arithmetic.
General description; the numerical analyst's view; overflow and
underflow. [0.6 lectures]

Condition and stability.
Condition of a problem; stability of an algorithm. [0.6 lectures]

Order of convergence; computational complexity. [0.3 lectures]

IEEE arithmetic.
The IEEE Floating-point standards. [1 lecture]

Simple numerical methods.
Differentiation; finite differences; splines. Linear and
non-linear equations.
Gaussian elimination; Choleski factorisation; linear least
squares;
Newton-Raphson iteration. Integration. Quadrature rules;
summation of series. [3 lectures]

Numerical software.
Portability; languages; the Brown model; implementation
issues for IEEE arithmetic, automatic quadrature, BLAS. [0.7 lectures]

Objectives

At the end of the course students should

appreciate both the historical significance of numerical
computation and its continued relevance to the solution of
mathematical problems

understand the advantages and limitations of IEEE arithmetic

be able to apply a small number of numerical techniques with
an understanding of their underlying principles

understand the special considerations needed when implementing
floating-point algorithms in re-usable software

Recommended reading

Conte, S.D. & Boor, C. de (1980). Elementary numerical analysis. McGraw-Hill.
Shampine, L.F., Allen, R.C. Jr & Pruess, S. (1997). Fundamentals of numerical computing. Wiley.