Day 1, Mat360: Numerical Analysis

• Thanks for signing up. I really like numerical analysis, and I hope that you will, too.

• Introductions (note cards):
  1. Name (that you want me to call you)
  2. Hometown
  3. What is your calling? ("a calling has to do with one's larger purpose, personhood, deepest values, and the gift one wishes to give the world.... A calling is about the use one makes of a career." David Orr, Earth in Mind)
  4. What is your dream job?
  5. What is something special about you?

• 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. Some of the reading is challenging. Please bring difficult parts to our attention during 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 (group) projects for which you will be creating Mathematica notebooks to demonstrate some known techniques in Numerical Analysis to solve some interesting problems.

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 quite old (by numerical analysis standards), 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, keep an eye out for out-dated Mathematica, and update as necessary. It will be useful if we keep a website with any updates of this kind. The first time I used this text, we were somewhat amazed to discover that there weren't many!

• 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. Our author says this about the objective of numerical analysis (p. 6): "The objective of numerical analysis is to create reasonably general algorithms for the numerical solution of problems in continuous mathematics."

• 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?)

• 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.
  • Let me use some now. I want you to each enter the Mathematica from one section of the appendix into a Mathematica notebook as part of your first assignment. We'll assign those now.
  • Exams will include a numerical component, which you may solve using your preferred software (although I hope that you'll use Mathematica).

• Let's dig in to Chapter 1, to get you started on your first homework problems (p. 11, #1-7).

  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:

  • "Historically, numerical methods were regarded as an alternative to analytical methods...." (p. 7)
  • "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".

  S&K suggest that, rather than focus on "how many correct digits" a number possesses, that we focus rather on

  |computed answer - exact answer| $\le$ tolerance

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

  If we have a little time left, let's work on this question: "How do we represent $\pi$ in a computer with only a finite number of storage locations?" (What's the problem?)