Math-387: Honours Numerical Analysis, Winter 2020

Introduction

This is an applied mathematics course in honours numerical analysis. All three words in the course title are relevant. While we will introduce lots of numerical algorithms, the main focus of the course will be the derivation and analysis of the methods. We will address the questions: what is a good method?, and why?, and how can you compare different methods? Error analysis and stability will be a recurring theme in the course.

Math-387 is the honours version of Math-317, but the courses differ significantly, since the audiences and their backgrounds are different. The major version of the course focuses much more on the algorithms and less on their analysis.

This course is intended for honours students. It is taught once every two years in the Winter semester of even numbered years. It alternates with Math-397 which is taught in the Winter of Odd numbered years. In some programs (eg Honours Applied mathematics) it is compulsory to take at least one of math387/397.

Topics

We will discuss numerical root-finding, interpolation and polynomial approximation, and numerical differentiation and integration. We shall also study initial and boundary value problems. We will introduce the finite difference method for linear second order PDEs, and draw attention to the special characteristics of elliptic, parabolic and hyperbolic solvers. Error analysis and stability will be a recurring theme in the course.

Syllabus

Error analysis. Numerical solutions of equations by iteration. Interpolation. Numerical differentiation and integration. Introduction to numerical solutions of differential equations.

Audience/Prerequisites

The course is intended for honours students with an interest in numerical analysis. The class will combine theory and practice, with the emphasis towards theory. Nevertheless we will also use matlab to perform computations.

The prerequisites are

• MATH 315 or MATH 325 (Differential Equations),
• MATH 242 (Analysis I),
• COMP 202 (Programming - but see below)

• MATH 243 or MATH 255 (Analysis II).

References

There is not a required textbook for this course. I intend to give a complete treatment in lectures, so do not feel that you need to rush out and buy any book straightaway.

The main books that I will use for the course are

• Numerical Mathematics, 2nd Ed. by Quarteroni, Sacco & Saleri. Springer, ISBN#: 978-3540346586. The best feature of this book is that is freely (and legally) available online from campus. Access it here. However, I find this book to be very abstract; numerical analysis as a pure mathematician would do it. Some of you might like it! I will not follow it particularly closely.
• An Introduction to Numerical Analysis, by Süli & Mayers. Cambridge university Press, ISBN #: 978-0-521-00794-8. This book is available online as an e-book; follow the link from the McGill library catalogue. This is the first time that I will use this book (Prof Tsogtgerel used it in 2018). It seems to be at just the right level for this class, however there are a couple of topics that it does not cover.

• Numerical Analysis, 9th Ed. by R.L. Burden and J.D. Faires, ISBN#: 0-534-39200-8. This is a standard text, used for many years in the majors math317 course, for which it is well suited. However, it is too low a level for our course, with too much emphasis on algorithms and not on their analysis. You will find the basic information in it, but no proofs.
• Afternotes in Numerical Analysis by G.W. Stewart, also available online.
• An Introduction to Numerical Analysis, by K.E. Atkinson. This is an older text, not (legally) available online, and horrendously expensive to buy as a print to order copy. However, you will find it in the library and it is a well written book.

Assessment

Assignments will be contain both theoretical and computing questions, which will help illustrate concepts learnt in class. You may use any computing language for your assignments, though Matlab is recommended and preferred. The midterm and final exam will be theoretical only.

Computing Stuff

Computing skills will be assumed in this course (COMP202 prereq). I will supply some matlab assistance and examples particularly early in the semester, but you will be largely expected to learn matlab yourselves. There are many good online guides that can be found using google. But, here are

• The official matlab website has a Getting Started Guide (2.3MB pdf) but if you don't want to download all 200 pages at once, its also possible to access an online version at the Matlab documentation page.
• If you want like something shorter the MATLAB Notes written by David Griffiths of the University of Dundee, is very comprehensive and just 48 pages.
• For a more advanced treatment of matlab see The Matlab guide by D.J. Higham and N.J. Higham, Second Ed, SIAM 2005, which is also available on-line from campus for free as a siam e-book.
• Amongst the many other works available online is Numerical Computing with MATLAB by Cleve Moler (the original creator of matlab).

McGill has a site license for matlab, and in principle it can be found or installed on any computer belonging to the university. Students can also download and install for free on your own computer from Matlab. Anyone on the course can get an account on the computers on the top floor of Trottier and use matlab there, however these computers run linux and are not popular with all students. The library computers run matlab under windows, which seems to suit more students.

MyCourses

This class will be run using MyCourses and you will need to be registered to receive e-mail announcements, and to receive and submit assignments. This page is only intended to provide information to students considering registering and will not be updated after the course commences.

Further Information

• The first lecture will be on Tuesday 7th January 2:30-4:00 in BH-1104.
• Course Outline