MATH30003 Approximation Theory and Numerical Integration SEMESTER: First CONTACT: Dr Ruth Thomas (M/O16) CREDIT RATING: 10
 Aims: To introduce some methods for approximating a function by polynomials, splines and rational functions. To introduce the concept of best approximation. To introduce Gaussian quadrature rules. Intended Learning Outcomes: On successful completion of this module students will: Have acquired active knowledge and understanding of basic approximation theory (including the formulation of some best approximation problems) and of an advanced technique in numerical integration. Be able to approximate a function by a polynomial, piecewise polynomial or Padé approximant. Pre-requisites: 153, 157, 211 (ex-UMIST), MT1202 Sequences and Series (ex-VUM) This course is NOT available to students who have taken MT2181 in previous years. Dependent Courses: Course Description: The first half of the course is concerned with approximation theory. The aim is to approximate a complicated function by a much simpler function (such as a polynomial), which is easier to evaluate, differentiate and integrate. In numerical integration, the course builds on the ideas introduced in Module 157. A family of integration rules, known as Gaussian quadrature rules, is introduced. Teaching Mode: 2 Lectures per week 1 Tutorial per week Private Study: 5 hours per week Recommended Texts: E, Suli and D Mayers.  An Introduction to Numerical Analysis.  CUP, 2003. Assessment Methods: Coursework: 20% Coursework Mode: Test in Week 7: Students will be asked to write out their solutions to a small number of problems chosen from a set notified in advance. Examination: 80% Examination is of 2 hours duration at the end of the First Semester.
 No of lectures: Syllabus 2 Introduction and revision of function norms; Weierstrass' Theorem for polynomial approximation. 2 Best minimax (L \infty) polynomial approximation: the equioscillation property and de la Vallée Poussin's Theorem. 3 Chebyshev polynomials: definition and properties. Best L \infty approximation to x n+1 by a polynomial of degree n or less. Economisation of power series. Lanczos \tau method. 4 Best Euclidean (L2) polynomial approximation: construction and properties of orthogonal polynomials (the Gram-Schmidt process). 2 Introduction to Hermite interpolation. 3 Approximation by piecewise polynomials (splines): definition and construction. 2 Rational approximation: Padé approximants. 4 Gaussian quadrature.

Last revised August, 2006