## Homepage for Math 451: Intro to Numerical Analysis 2

Time/place: MWF 12-12:50, Eng E206

Instructor: Dan Bates

Office hours: Wednesday, 1-2, Weber 120. I will set just this one formal office hour to leave some bandwidth for individual meetings. If you want/need to meet with me but cannot do Wednesday 1-2, please email me and we'll set up another meeting time.

## Examples

• 20 Jan: Instability: maple, pdf. (Might want to right click on maple worksheet and save.)
• 23 Jan: Forward and central difference approximations: maple, pdf
• 3 Feb: Bertini input file for determining weights and nodes for Gaussian quadrature. Here is an output file with the solutions.

## Assignments

HW 1, tentatively due February 17.

HW 2, tentatively due April 14.

## Project specifications

Specifications (updated 2/13/17).

For in-class presentations, I will record notes throughout your talk and send you an email with comments soon after class. I would be happy to meet to discuss any ideas or concerns coming from my feedback. For your grade, it is subjective, but I will evaluate you on clarity, technical accuracy, and material choice, each on a 10-point scale for a maximum of 30 points. Do not stress heavily about the evaluation part of the feedback; the point of these talks is to gain experience with this skill (and to share technical information), not for me to grade you on a skill that I have never even tried to teach you! Do your best, and I am sure you will be fine.

The rubric for the poster will be similar, to be finalized later.

Latex example: .tex, .pdf, some figure in the sample paper.

## Project ideas

Expository:

• ODE boot camp: Review of the basics of ODEs (from Math 340/345, more or less). What are they? What do solutions look like? Are there always solutions? How many? Examples are good. Techniques for solving aren't so important (since we will develop our own). Taken
• PDE boot camp: Similar to ODE boot camp. What are the basic types? What do we know about them? Nothing more about solving methods. Taken
• Ariane V and similar disasters: Numerical errors sometimes cause trouble. Taken
• Extend a topic from class: Did you find some topic particularly interesting? Go read more about it and present some extension in class. Ditto for topics from Math 450.
• Other numerical methods: We are only covering a handful of types of numerical methods in class. Learn about something completely different!
• Certification/validation/verification: How do you know when a numerical method's output is "correct?"
• Read an academic article: Google Scholar is a decent starting point. Find an article that uses some method(s) from class and report on it.
• Bernoulli numbers: These comes up in both Romberg integration and number theory (!). Give a talk that connects these dots. Taken
• Numerical intergration with limited data available: How do you manage the case that you have only experimental data for certain values (aside from using an interpolant)? Are there schemes to handle this?
• Comparing numerical integration methods: Is there a universally "best" method? When does each method shine? Read up on this and let us know.
• Monte Carlo methods for integration: Basic idea, maybe with VEGAS sampling? Taken
• Numerical methods for solving univariate polynomials. Descartes' rule of signs, Sturm sequences, etc. Taken, maybe room for more
• Survey of numeical methods in software. Pick a software package and several of our methods from class. Show us how to run examples through the software and interpret the results. Taken, maybe room for more
• Optimization methods. I'll have some time to cover some of these in class, but you are welcome to learn about a method and present it. Options include the simplex method (brief overview -- it's a long one!), gradient descent, simulated annealing, etc.
• Differential algebra. Do you really prefer algebra to numerical analysis? If so, maybe read up on this field and explain the core ideas to the class. (I can help a bit with this one as it takes some advanced knowledge).
• Splines, fonts, and/or graphic design. There's a constellation of related concepts here.
• Stability/consistency vs. convergence for multistep methods. Why does the convergence of a multistep method depend on the roots of some polynomial? The answer is out there, just connect the dots for us.

Original work:

• Choose your own adventure: If there is some topic of interest to you that does not appear on this list, please contact me. Applications are a particularly good option here since we will touch very few in class.
• Techniques for simulating propagation of low-amplitude plane waves in highly isotropic and inelastic materials. Taken (student suggestion)
• Benchmarks: Run different software and/or different methods within one piece of software on a few benchmark problems.
• Implementation: Implement some method(s) from class, particularly a method that is not easily found elsewhere.
• Derive your own method: Many of the methods from class come as families of methods or variations on some basic themes. Try building your own! (Be creative; don't just do one more step of Richardson extrapolation and call it original.)
• Comparing numerical integration methods: Is there a universally "best" method? When does each method shine? Implement these methods (or use Matlab or so) and test a range of benchmark problems to come up with advice.
• Vandermonde matrices for higher-order differentiation schemes. Taken
• Extrapolation. Given some data, you can fit a curve to the data. This curve then allows you to recognize patterns and make predictions. Grab some data about your favorite activity/sport/hobby/etc., fit a curve, and predict the next few data points.