Day 1, Mat360: Numerical Analysis

• Thanks for signing up.

• I really like numerical analysis, and I hope that you will, too.

• Introductions (4x6 cards)
  1. Name
  2. Where are you from?
  3. Why are you taking numerical analysis?
  4. Something special about yourself.

• The course website will be useful: http://www.nku.edu/~longa/classes/mat360

• Syllabus (logistics, etc.)

• Assignments

• In contrast to my usual policy, you may use a computer running Mathematica (or other course-related software) in class. No phones, however. Please stow those and resist the temptation during class.

• You should do the reading in advance of our discussion of each section in class.

• Your Mathematica (or other software) Projects:

  We will be focusing on Mathematica: because our department has the right to Mathematica and has chosen to emphasize it in coursework, and since you may obtain a personal copy as an NKU students, it seemed sensible to find a text that used Mathematica.

  Hopefully you'll become adept at using Mathematica by the end of the course (if you're not already).

  Your Mathematica projects will be outside projects for which you will be creating Mathematica notebooks to demonstrate some known techniques in Numerical Analysis to solve some simple problems. These may be "book-inspired", or some that I come up with. They may be group projects. We'll see!

Because we're focused on Mathematica here at NKU, it seemed sensible to find a textbook that used Mathematica in an integral way. It was not as easy as I'd hoped, because I also wanted something relatively inexpensive.

I found this text while searching for Mathematica-based texts. I was a little concerned because it's a little old, but I got in touch with one of the authors (Skeel), and discussed whether this text might be appropriate (a decade or so old can be a long time in proprietary software terms). He thought that it should do. So I elected to go with it.

I hope that it will be a good fit. It covered all the topics that I wanted to cover, and I like the parts that I've read so far. I like their approach to things. We'll see!

In particular, we'll need to watch out for out-dated Mathematica, and update as necessary. It will be useful if we keep a website with any updates of this kind. Please alert me, for the moment, until I get this set up.

• What are you comfortable using?
• You are welcome to use other software, but I must be able to run your code, and it's up to you to make sure that I can.

• My word of the course: iteration (do it again, do it again, do it again!)

• Learning Objectives

• What is numerical analysis? In this course students learn to identify the types of problems that require numerical techniques for their solution and see examples of the error propogation that can occur when numerical methds are applied. They accurately approximate the solution of problems that cannot be solved exactly and learn typical techniques for estimating error bounds for the approximations.

  Have you, in your mathematical careers, encountered any problems that cannot be solved exactly?

• Something about your "calculus chops":

  The last time I taught this course, one of the students correctly deduced that this class could be subtitled "How I learned to use (and love) the Taylor Series expansion". Pay especially close attention to that one!

• The major topics we will consider are:
  • Chapters 1 and 2: Numerical issues - numerical representation, algorithms, errors and error propagation
  • Chapter 3: Root finding; solving equations; iteration!
  • Chapter 5: Data fitting; interpolation
  • Chapter 7: Numerical integration and differentiation
  • Chapter 8: Ordinary differential equations (note: this even though some of you may not have had a course in differential equations -- don't fear if you haven't! But, by the way, who has?)
  We'll have about two weeks for each of these topics. Two exams take out a week's worth of class, and we'll have a week of review. That means that I'll have about two more weeks to play with (review, extensions, etc.).

• Software/technology
  • I'll be using some public domain software via web-interfaces in class (such as xlispstat or Octave -- a free MATLAB-like product). These just make it easy for me to do dynamic illustrations via the web, rather than having software installed. You will not need to be able to use this.
  • Exams will include a numerical component, which you will solve using your preferred software.

• Let's dig in to Chapter 1, to get you started on your first homework problems (p. 11, #1-8).

  Section 1.1 is a discussion of some of the reasons for which we are interested in doing numerical analysis. I'll leave that to you.

  I'd like to begin with a few quotes from section 1.2, which I find remarkably astute and important to set the stage:

  • "Generally the numerical analyst does not strive for exactness." (quoting F. B. Hildebrand, p. 7)
  • "...numerical analysis is a response to the challenge of coping with computational error." (p. 8)
  • "...a major theme of numerical analysis [is] the trade-off between accuracy and cost." (p. 9)
  • "The most important idea...in this book is that the exact answer for most problems is either impossible or impractical to get." (p. 10)
  We're going to give the wrong answer (but we know it, and even confess it).

  We're going to strive for perfection, but we need to do some cost-benefit analysis to see how close-to-perfection we're willing to pay for!

  Applied mathematicians are fond of saying that "all models are wrong; some models are useful." That's the spirit to take into numerical analysis.

  Now let's focus in on page 8, because there's some interesting stuff going on. Two kinds of computational errors:

  • Roundoff Errors -- computer limitations
  • Truncation Errors -- mathematical/time/cost limitations

  Now let's talk tolerance versus "correct digits".

  After talking it, we'll try the computational problem on p. 11