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General

Semester

Autumn 2009 (Monday, October 5 to Friday, December 11)

Timetable

• Tue 12:05 - 12:55
• Thu 11:05 - 11:55

Prerequisites

Undergraduate analysis and linear algebra.

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.

Reading list and references

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

 Email jl1@mcs.le.ac.uk Phone (0116) 2523897 Interests Approximation theory, numerical analysis

Students

 Hanefa Al-Qasmi (Manchester) Mohd Alazah (Reading) Sabrina Blackwell (Loughborough) Ashley Brereton (Liverpool) FAIRHURST David (Leicester) Aiman Elragig (Exeter) Saadia Farid (Manchester) Donatello Gallucci (Newcastle) PAUL GUNS (Cardiff) Raja Zafar Iqbal (Birmingham) Muhammad Anjum Javed (Newcastle) Lanchakorn Kittiratanawasin (Loughborough) Nor Alisa Mohd Damanhuri (Manchester) Kieron Rhys Moore (Loughborough) Manan'Iarivo (Manou) Rasolonjanahary (Liverpool) Stephen Rickaby (East Anglia) Badr Saad Al-Kahtani (Manchester) Mousa Shaikh-Soltan (Manchester) Chris Welshman (Manchester)

Bibliography

 Elementary Numerical Analysis Conte Accuracy and stability of numerical algorithms Higham A first course in the numerical analysis of differential equations Iserles Numerical analysis Kincaid An introduction to numerical analysis Süli and Mayers

Note:

Clicking on the link for a book will take you to the relevant Google Book Search page. You may be able to preview the book there. On the right hand side you will 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

No assessment information is available yet.

No assignments have been set for this course.

Files

Files marked L are intended to be displayed on the main screen during lectures.

Week(s) File lecture1009-notes.pdf lecture1009.pdf lecture109-notes.pdf lecture109.pdf lecture1109-notes.pdf lecture1109.pdf lecture1209.pdf lecture13-notes.pdf lecture13.pdf lecture1409-notes.pdf lecture1409.pdf lecture1509-notes.pdf lecture1509.pdf lecture1609-notes.pdf lecture1609.pdf lecture1709-notes.pdf lecture1709.pdf lecture1809-notes.pdf lecture1809.pdf lecture1909-notes.pdf L lecture1909.pdf L lecture209-notes.pdf lecture209.pdf lecture309-notes.pdf lecture309.pdf lecture409-notes.pdf lecture409.pdf lecture509-notes.pdf lecture509.pdf lecture609-notes.pdf lecture609.pdf lecture709-notes.pdf lecture709.pdf lecture809-notes.pdf lecture809.pdf lecture909-notes.pdf lecture909.pdf problems1.pdf problems2.pdf problems3.pdf problems4.pdf L quadroot.m