Module Identifier MA44520  
Academic Year 2000/2001  
Co-ordinator Professor A R Davies  
Semester Semester 1  
Other staff Professor A R Davies, Dr R J Douglas  
Pre-Requisite MA25110  
Course delivery Lecture   20 x 1hour lectures  
  Seminars / Tutorials   7 x 1hour seminars  
Assessment Exam   2 Hours (written examination)   100%  
  Resit assessment   2 Hours (written examination)   100%  

General description
The expansion of a function in terms of an infinite sequence of orthogonal functions underlies many numerical methods of approximation. The accuracy of the approximations and the efficiency of their implementation are important factors when determining the applicability of these methods in scientific computations. In this module, we make a detailed study of expansions in terms of orthogonal polynomials which yield 'spectrally accurate' approximations (ie those for which the coefficient decays faster than any inverse power of k for certain well-behaved functions).

The aim of this module is to introduce students to key issues in the numerical approximation of a given function and to demonstrate the power of certain orthogonal functions to produce rapidly convergent approximations to smooth functions.

Learning outcomes
On completion of this module students should be able to:

1. APPROXIMATION IN A HILBERT SPACE: Introduction. Definitions. Best L^{2} approximation theorem.
2. APPROXIMATION BY TRIGONOMETRIC POLYNOMIALS: Continuous Fourier expansions. Rate of decay of Fourier coefficients. Error estimates. Discrete Fourier series. Trigonometric interpolating polynomials. Differentiation of Fourier series. Fast Fourier transform.
3. APPROXIMATION BY ORTHOGONAL POLYNOMIALS: Generation of orthogonal polynomials. Properties of orthogonal polynomials. Gauss-type quadrature rules. Classical polynomials. Rate of convergence of orthogonal series expansions. Chebyshev and Legendre polynomials. Interpolation by Chebyshev polynomials. Differentiation of Chebyshev and Legendre expansions.

Reading Lists
** Recommended Text
C Canuto, M Y Hussaini, A Quarteroni and T A Zang. (1986) Spectral Methods in Fluid Dynamics. Springer Verlag
J P Boyd. (1989) Chebyshev and Fourier Spectral Methods. Springer Verlag