Welcome to MATH3Q03

Fall 2015
Time & Place: 9:30-10:20 on Mondays & Thursdays and 10:30-11:20 on Tuesdays in HH/217

Instructor: Dr. Bartosz Protas
Office: HH 326, Ext. 24116
Office hours: 1:30-2:30pm on Tuesdays and 11:30-12:30 on Thursdays, or by appointment

Teaching Assistant: Athinthra Krishnaswamy Sethurajan


- Results and solutions of Homework Assignment #5 are already posted (see links on the left).

- Results and solutions of Homework Assignment #4 are already posted (see links on the left).

- Homework Assignment #5 is already posted and is due by December 10.

- Results of Test #2 are already posted together with solutions (see links on the left). If you have any questions concerning your mark, please contact Athinthra, our TA.

- Course evaluations are already open and will close on December 8. Please see here for additional information.

- Results and solutions of Homework Assignment #3 are already posted (see links on the left).

- Homework Assignment #4 is already posted and is due by November 26.

- Homework Assignment #3 is already posted and is due by November 12.

- Results and solutions of Homework Assignment #2 are already posted (see links on the left).

- Homework Assignment #2 is already posted and is due by October 29.

- Graduate School Information Session will be held on Tuesday, October 20th - more information here.

- Announcement of a study about your Fall Break experience.

- Results of Test #1 are already posted together with solutions (see links on the left). If you have any questions concerning your mark, please contact Athinthra, our TA.

- Test #1 will take place at 9:30am on Thursday, October 8. It will be 50 minutes long and will cover the material discussed during the lectures up to (including) Monday, October 5. Please bring calculators and a few sheets of paper.

- Homework Assignment #1 is already posted (see link on the left)

- A first set of lecture notes has already been posted (see the link in the menu on the left)

Information about the Final Exam:

- The exam will take place at 9:00am on Monday, December 21; information about the location will be provided by Registrar's Office.

- The questions on the final exam will test students' understanding of the theory and their ability to solve simple problems (students will be required to outline key analytic aspects of the solutions to such problems without any MATLAB programming). The emphasis will be on understanding basic concepts rather than on details.

- Topics covered after Test #2 will be weighted more heavily and will account for approximately 62% of the exam score (this is meant to ensure that all topics covered in the course are weighted approximately equally in the final grade). However, topics involving complex-plane methods in error analysis will not be emphasized (recognizing that not all students have the required prerequisite knowledge).

- Some questions may involve comparison between different sections of the material.

- Students may wish to review the problems from the tests and homework assignments.

- All answers must be entered in the booklets provided using permanent ink.

- The exam paper will contain a formula sheet a copy of which is available here.

- Only the McMaster Standard Calculator Casio FX991MS is allowed.

Outline of the Course:

In this course we will study key questions of numerical analysis such as approximation of functions and approximate differentiation and integration. We will see how various problems arising in calculus (both in single and in multiple variables) can be solved approximately, but with controlled accuracy, using computer algorithm. In addition to proving theorems about various numerical methods, we will develop, analyze and implement actual computational algorithms using MATLAB. We will also show how computational techniques can be used to illustrate and verify different results of mathematical analysis. As a highlight of the course, we will introduce Chebfun which is a MATLAB toolbox for performing hybrid numerical-symbolic computations with very high accuracy.

Topics: [the actual order may be different; characters in brackets represent the reference ("I" means that the material will be provided by the instructor]

1) Introduction & Review of the Background Material
     a) basic definitions [GP],
     b) introduction to MATLAB and Chebfun [GP,T],
     c) properties of polynomials [GP,T],
     d) solution of systems of equations: linear & nonlinear [GP].
2) Interpolation
     a) Vandermonde, Lagrange & Newton interpolation [GP],
     b) error analysis: Runge phenomenon [GP, T, I],
     c) Chebyshev interpolation [T],
     d) trigonometric interpolation [GP].
3) Approximation
     a) best approximations and orthogonal projections [T,I],
     b) systems of orthogonal polynomials [T],
     c) finding best approximations [GP,T].
4) Numerical Differentiation and Integration
     a) derivatives via finite differences, error analysis [GP],
     b) Richardson extrapolation [GP],
     c) numerical quadratures, error analysis [GP],
     d) spectral differentiation [T],
     e) Gaussian quadratures [T].
5) Special Topics
     a) relation between interpolation and approximation [T],
     b) collocation vs. Galerkin methods for differential equations [I].

Primary Reference:

     [GP] M. Grasselli and D. Pelinovsky, Numerical Mathematics, Jones and Bartlett, (2008).

Supplemental Reference:

     [T] N. Trefethen, Approximation Theory and Approximation Practice, SIAM, (2013) (we will focus primarily on the first chapters, six of which are available free of charge on the Author's webpage; all chapters can be generated from the source files provided)


All computational examples will be presented using MATLAB and Chebfun. While MATLAB is available on the computers in most of the computer labs on the campus, students are encouraged to purchase The Student Edition of MATLAB to be able to work with MATLAB at home. Chebfun can be downloaded free of charge from http://www.chebfun.org/download/.
During the lectures we will provide introduction to Chebfun, however, students are expected to have already some familiarity with MATLAB.


Advanced Calculus (MATH 2A03 or 2X03) and Introduction to Numerical Analysis (MATH 2T03).


Five homework assignments will be posted on the course website on the dates indicated in the table below. The assignments will be due by midnight on Thursday the following week. Solutions of the assignments should be submitted by e-mail to math3q03_AT_math_DOT_mcmaster_DOT_ca using the template provided. Late submissions will not be accepted. Only four best assignments are counted towards the final mark. The assignments and solutions will be posted on the course webpage.

Homework Post & Due Dates (tentative):


Post Date

Due Date

HW 1

September 24

October 1

HW 2

October 22

October 29

HW 3

November 5

November 12

HW 4

November 19

November 26

HW 5

December 3

December 10

Mid-Term Tests:

There will be two in-class tests planned tentatively on October 8 and November 19. They will last 50 minutes and will cover analytical issues only (no programming). Only the McMaster standard calculator Casio fx-991 will be allowed during the tests.

Final Exam:

The course will be completed by a 2.5 hour final examination. The date and location of the final exam will be announced by the Registrar's office in mid-term. The exam will cover all course material.

Marking Scheme:

     a) Final exam (3 hrs) - 40%
     b) Two Test (50 min) - 20% (10% each)
     c) Four best homework assignments - 40% (10% each)

Relief for Missed Work:

In the event of an absence for medical or other reasons, students should review and follow the Academic Regulation in the Undergraduate Calendar Requests for Relief for Missed Academic Term Work. Please note these regulations have changed beginning Fall 2015.

Academic Integrity:

You are expected to exhibit honesty and use ethical behaviour in all aspects of the learning process. Academic credentials you earn are rooted in principles of honesty and academic integrity.

Academic dishonesty is to knowingly act or fail to act in a way that results or could result in unearned academic credit or advantage. This behaviour can result in serious consequences, e.g., the grade of zero on an assignment, loss of credit with a notation on the transcript (notation reads: "Grade of F assigned for academic dishonesty"), and/or suspension or expulsion from the university.

It is your responsibility to understand what constitutes academic dishonesty. For information on the various types of academic dishonesty please refer to the Academic Integrity Policy,. The following illustrates only three forms of academic dishonesty:
     1) Plagiarism, e.g., the submission of work that is not one's own or for which other credit has been obtained.
     2) Improper collaboration in group work.
     3) Copying or using unauthorized aids in tests and examinations.

Important Notice:

The instructor and university reserve the right to modify elements of the course during the term. The university may change the dates and deadlines for any or all courses in extreme circumstances. If either type of modification becomes necessary, reasonable notice and communication with the students will be given with explanation and the opportunity to comment on changes. It is the responsibility of the student to check their McMaster email and course websites weekly during the term and to note any changes.