Matematik och Datavetenskap, Chalmers Tekniska Högskola och Göteborgs Universitet

TMA740B Analys i en variabel Kf Kb, del B, 1999, vecka 3

Mathematics

Goal: Quadrature. The exponential function. Introduction to the project.

Theory: Read Chapter 28.1-28.4 "Numerical Quadrature" and Chapter 29 "The Exponential Function". Advanced: Section 28.5. In section 29.2: the proof that Un(x) is a Cauchy sequence.

Exercises: Basic: 28.1, 28.4, 28.6, 28.7, 29.1, 29.2, 29.9, 29.10, 29.12, 29.13, 29.14, 29.15, 29.16, 29.17, 29.18. Advanced: 28.2, 28.5, 29.3, 29.4, 29.5, 29.6, 29.19

Problems for chapter 29 (updated)

Preparation for the project: In the project you will study one application that you will select from, for example, Chapters 35, 36, or 37. As a preparation for choosing your application, read Chapter 35 "The Crash Model" quickly without going into the details. Then solve Problem 35.2 in the following way:

1. use the matlab program ode23:
>> [t,u]=ode23('crash', [0, 20], [1; kappa*epsilon]);
>> plot(t,u)

Use the skeleton crash.m.

Programming

Goal: To write a matlab function that computes integrals using the midpoint quadrature rule. To compare the error estimate with the true error.

Try matlab's editor. You may like it better than emacs. To open the file funk1.m you type "edit funk1" in the matlab command window.

Exercises:

1. Write a matlab function that computes the integral of a function f over an interval [a,b] using the midpoint quadrature rule.

The program should be written according to the specification in the skeleton midpoint.m.

2. Repeat the experiment in section 28.4 for your midpoint program and with the error estimate (28.9). This means that you have to write a program that computes the error estimate according to the skeleton errorbound.m.

Advanced (for the most ambitious students): write a program that implements an adaptive quadrature rule as in section 28.5.

Individual report

In the beginning of week 4 each student should report (in any suitable form, e.g., a short discussion in front of the computer) to her/his supervisor the following:
1. problem 35.2
2. midpoint.m
3. problem 29.10

/stig


Last modified: Mon Nov 15 00:59:56 MET 1999