This page focuses on the course 18.330 Introduction to Numerical Analysis as it was taught by Professor Laurent Demanet in Spring 2012.
This is a lecture-based course on basic techniques for the efficient numerical solution of problems in science and engineering. It is a centerpiece of MIT’s undergraduate offerings in applied mathematics.
Course Goals for Students
- Familiarity with numerical discretization of objects learned in calculus, such as integrals, derivatives, and differential equations
- An idea of the size of the errors involved in numerical discretization
- Familiarity with Fourier analysis and its numerical implementation
Possibilities for Further Study/Careers
The skills and knowledge taught in this course are fundamentally important for many scientists and engineers who go on to do simulations or data analysis in industry, as well as research in computational mathematics or computational engineering.
Below, Prof. Laurent Demanet describes various aspects of teaching 18.330 Introduction to Numerical Analysis.
This class had a mathematical edge, in contrast to what can be found in most textbooks. I constantly questioned my students to get them to go through the steps of figuring out what the answer should be before they heard it from me. I frequently linked to background material the students should know, as needed, even if it meant occasionally slowing down the expected pace of the class.
Balancing Fundamental Content with More Exciting Material
It is important to find the right tradeoff between important, enabling, but “soft” concepts (e.g., interpolation, quadrature, root-finding), and more interesting, intellectually substantive, “hard” material (e.g., Fourier series, sampling, and aliasing). Some old-fashioned topics such as Newton divided differences and Newton-Cotes formulas were deliberately trimmed out of this course to make more room for Fourier analysis.
I found it useful to show targeted numerical experiments with a clear message (live demos are good), in order for the students to gain a clear intuitive understanding of what the various numerical methods are accomplishing.
- 18.330 can be applied toward a Bachelor of Science in Mathematics, but is not required.
- Every spring
Breakdown by Year
Mostly sophomores and juniors in equal proportion; a few seniors
Breakdown by Major
1/3 math majors; 1/3 engineering majors; 1/3 from other sciences
Typical Student Background
- Good quantitative instincts
- Familiarity with calculus and elementary linear algebra, but not necessarily mathematical proofs
Ideal Class Size
Having more than 40 students tends to hinder class participation and interaction, while having fewer than 10 students carries the risk of the group not being lively enough to engage in a good discussion.
How Student Time Was Spent
During an average week, students were expected to spend 12 hours on the course, roughly divided as follows:
- Two class sessions per week; 1.5 hours per class
- 26 class sessions total
- Concepts were often introduced with examples and motivation.
- In-class time was primarily spent on blackboard lectures. Some key mathematical proofs were presented, but the course did not cover the notes in their entirety.
- About one-eighth of class time was spent on numerical examples presented with MATLAB™ via the LCD projector.
Out of Class
- Problem Sets
- Review of course material, such as lecture notes
- Exam preparation