Numerical Analysis 2

Martin Lotz – School of Mathematics – The University of Manchester

About

Lecture Tuesday 1-2pm Simon Building 3.40
Thursday 3-4pm Zochonis Theater D
Tutorials Thursday 4-5pm Zochonis Theater D

This course introduces numerical methods for approximating functions and data, evaluating integrals and solving ordinary differential equations. It continues the introduction to numerical analysis begun in MATH20602 – Numerical Analysis 1. It provides theoretical analysis of the problems along with algorithms for their solution. Insight into the algorithms will be given through MATLAB, Julia and Python illustrations, but the course does not require any programming. Among all the Numerical Analysis texts, the one that is closest to the course is:

Some more general information on the course can be found on the course description page. This course is part of the Numerical Analysis Pathway.

Lecture Notes

Lecture notes will be published regularly, usually right before or after the lecture. The weekly blog entries with the lecture content can be accessed through the Lectures menu or on Blackboard. A complete and revised set of lecture notes in one document will be made available towards the end of the lectures.

Prerequisites

The course requires MATH20602 (Numerical Analysis 1) and basic knowledge of multivariate calculus, analysis and linear algebra as provided first and second year modules (MATH10121 or MATH10131, MATH10202 or MATH10212). A summary of important results from analysis can be found here. Students are expected to know these results and should be prepared to use them during the course wherever necessary.

Content

  • Approximation and Curve Fitting. Best approximation in the 1-norm. Weierstrass theorem, equioscillation theorem, Chebyshev polynomials. Best approximation in the 2-norm. Orthogonal polynomials. Rational approximation; Padé approximants. [6 hours]
  • Numerical Integration Interpolatory rules. The Romberg scheme: extrapolation using the Euler-Maclaurin summation formula. Gaussian quadrature. Adaptive quadrature. [6 hours]
  • Initial Value Problems for ODEs Introduction and existence theorem. Numerical methods: one step methods and multistep methods. Eulers method. Local truncation error, convergence, local error. Taylor series method. Runge-Kutta methods. Trapezium rule. Functional iteration and predictor-corrector PE(CE)m implementations. Absolute stability. Linear multistep methods. Higher order systems. [10 hours]

Intended Learning Outcomes

  • characterise the best approximation of a function using different norms;
  • compute Padé approximations and evaluate their quality;
  • derive quadrature rules and their error bounds;
  • apply the Trapezium rule, Gauss quadrature and adaptive quadrature to compute integrals;
  • describe the Romberg scheme in the context of extrapolation;
  • analyse and apply one-step, multi-step, and the Euler method for solving ordinary differential equations (ODE);
  • solve ODE numerically using Runge-Kutta, Trapezium and higher-order methods;
  • quantify the error and convergence of numerical solvers for ODE;
  • recognize some of the difficulties that can occur in the numerical solution of problems arising in science and engineering.

Problem Sheets

The problem sheets contain a set of problems to be worked on at home, and some that will be worked on during the example classes, starting in Week 2. Solutions will be made available. The problems will be both theoretical and computational. The problem sheets are available on the Problems menu item.

Computing

Numerical Analysis is about computation, and it is therefore helpful to be acquainted with a programming language or a computing system. In this course we will use MATLAB, Julia or Python. More information can be found in the Computing menu.

Exams

Unless stated otherwise, all the covered material will be relevant for the exam. The exam will not include programming tasks. Previous exams can be found in the Assessment menu. A coursework test will be announced in due course.

Recommended Literature

  1. Endre Süli and David F. Mayers. An Introduction to Numerical Analysis. Cambridge University Press, Cambridge, UK, 2003. ISBN 0-521-00794-1. x+433 pp.
  2. Richard L. Burden and J. Douglas Faires. Numerical Analysis. Brooks/Cole, Pacific Grove, CA, USA, seventh edition, 2001. ISBN 0-534-38216-9. xiii+841 pp.
  3. James L. Buchanan and Peter R. Turner. Numerical Methods and Analysis. McGraw-Hill, New York, 1992. ISBN 0-07-008717-2, 0-07-112922-7 (international paperback edition). xv+751 pp.
  4. David Kincaid and Ward Cheney. Numerical Analysis: Mathematics of Scientific Computing. Brooks/Cole, Pacific Grove, CA, USA, third edition, 2002. ISBN 0-534-38905-8. xiv+788 pp.
  5. David Nelson, editor. The Penguin Dictionary of Mathematics. Penguin, London, fourth edition, 2008. ISBN 978-0-141-03023-4. 480 pp.