# MAGIC015: Introduction to Numerical Analysis

## Course details

A specialist MAGIC course

### Semester

Autumn 2008
Monday, October 6th to Friday, December 12th

### Hours

Live lecture hours
20
Recorded lecture hours
0
0

Tuesdays
12:05 - 12:55
Thursdays
09:05 - 09:55

### Announcements

The assessment on this module will be via 5 sets of problems. You should submit all problems to your tutor at the end of the course. We will provide solutions for the problems for the tutor to use to mark the problems.

## Description

### 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:   Wavelets I.
Lecture 10:   Wavelets II.
Lecture 11:   Solving systems of linear equations I. Gauss elimination, pivoting. Cholesky factorisation.
Lecture 12:   Solving systems of linear equations II. Conditioning and error analysis.
Lecture 13:   Solving systems of linear equations II. Iterative methods: Jacobi, Gauss-Seidel, SOR.
Lecture 14:   Least squares solution, Schur decomposition, the QR and QZ algorithms.
Lecture 15:   Power method and singluar value decomposition.
Lecture 16:   Krylov subspace methods: Arnoldi algorithm.
Lecture 17:   Conjugate gradient method and GMres.
Lecture 18:   Functions of a matrix.
Lecture 19:   This lecture will be set aside for expansion of topics in the course previously.
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 [4] and Higham [2].
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

### Follow the link for a book to take you to the relevant Google Book Search page

You may be able to preview the book there and see links to places where you can buy the book. There is also link marked 'Find this book in a library' - this sometimes works well, but not always - you will need to enter your location, but it will be saved after you do that for the first time.

## Assessment

### Attention needed

Assessment information will be available nearer the time.