MAGIC015: Introduction to Numerical Analysis

Course details

Semester

Autumn 2007
Monday, October 8th to Friday, December 14th

Hours

Live lecture hours
20
Recorded lecture hours
0
0

Tuesdays
11:05 - 11:55
Tuesdays
13:05 - 13:55

Description

Prerequisites

No prerequisites information is available yet.

Syllabus

This is a 20 lecture course. The aim of the course is to introduce students to a number of key ideas and methods in numerical analysis and for the students to learn to implement algorithms in Matlab.
Syllabus
Lecture 1:   Introduction and prerequisites. Description of the ideas to be covered and the assessment activities.
Lecture 2:   Stable and unstable computation, relative and absolute error, floating point computation and round off errors.
Lecture 3:   Finding roots of nonlinear equations. Bisection, secant and Newton's methods.
Lecture 4:   Approximation of functions I. Polynomial interpolation, Lagrange and Newton forms: divided differences.
Lecture 5:   Approximation of function II. Piecewise polynomial approximation. Splines and their generalisations into higher dimensions.
Lecture 6:   Approximation of functions III. Least squares and orthogonal polynomials.
Lecture 7:   Numerical integration. Newton-Cotes and Gauss formulae. Integration of periodic functions. Romburg integration.
Lecture 8:   The Fast Fourier transform.
Lecture 9:   Numerical differentiation and Richardson extrapolation.
Lecture 10:   Solving systems of linear equations I. Gauss elimination, pivoting. Cholesky factorisation.
Lecture 11:   Solving systems of linear equations II. Conditioning and error analysis.
Lecture 12:   Solving systems of linear equations II. Iterative methods: Jacobi, Gauss-Seidel, SOR.
Lecture 13:   Least squares solution and the QR algorithm.
Lecture 14:   Solving partial differential equations I. Finite difference methods for elliptic equations.
Lecture 15:   Solving partial differential equations II. The Galerkin method and finite element methods.
Lecture 16:   Solving partial differential equations III. Parabolic equations, explicit and implicit methhods. The Crank-Nicolson method.
Lecture 17:   Solving ordinary differential equations I. Taylor series methods. Runge-Kutta methods.
Lecture 18:   Solving ordinary differential equations II. Multi-step methods. Higher order differential equations.
Lecture 19:   Solving ordinary differential equations III. Boundary value problems: shooting methods, finite difference methods, collocation.
Lecture 20:   Summarising and finishing course. This lecture also allows some time if other topics take longer than expected.

There are a number of excellent books on numerical analysis and you are encouraged to consult these books for alternative and often better accounts of what you have heard in lectures. In the main I have followed Kincaid and Cheney  and Higham .
1. S. D. Conte and C. deBoor, Elementary Numerical Analysis, (3rd Ed) McGraw-Hill, 1980.
2. N. J Higham, Accuracy and Stability of Numerical Algorithms, SIAM, 1996.
3. A. Iserles, A First Course in the Numerical Analysis of Differential Equations, CUP, 1996.
4. D. R. Kincaid and E. W. Cheney, Numerical Analysis, Brooks/Cole Publishing Company, 1991.
5. E. Süli and D. Myers, An Introduction to Numerical Analysis, CUP, 2003.

Lecturer

• JL

Professor Jeremy Levesley

University
University of Leicester

Bibliography

No bibliography has been specified for this course.

Assessment

Attention needed

Assessment information will be available nearer the time.