*Lecturer: Dr M.R. O'Donohoe*
(`mro2@cam.ac.uk`)

*No. of lectures:* 8

*This course is a prerequisite for Numerical Analysis II (Part II and
Diploma).*

**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]**Errors in numerical methods.**Machine epsilon; error analysis; solving quadratics; convergence; error testing; rounding error; norms. [1.8 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 books**

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.