Day 1, Mat360: Numerical Analysis

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

• I am hearing-impaired (wear hearing aids): if I don't understand your question, please be patient with me; or, if I misunderstand what you've said, just let me know that I'm off track!

  If you think that I'm ignoring you, it's quite likely that I just didn't hear what you said....

• 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?

• I won't be using Canvas.

• We have our own course website, which you will want to bookmark, and visit from time to time: http://www.nku.edu/~longa/classes/mat360

  It has (or will have) our syllabus, assignments, daily agendas, project descriptions, software links, etc. -- everything you need.

• Speaking of which, let's have a look at the Syllabus (logistics, etc.)

• Assignments

• It will be great if you have a computer capable of running Octave (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 Octave (or other software) Projects:

  We will be focusing on Octave, since it jives with our text.

  I may use other software (in particular Mathematica) to demonstrate, or compare.

  Your programming projects will demonstrate some known technique(s) in Numerical Analysis to solve some interesting problem(s). They will be fairly wide open, depending upon your personal interests.

Our text is open and freely available; I am in contact with the author, who is pleased that we are using it. I told him that we would be "battle testing" it, and he is good with that!:)

In particular, I've never used it before; I like the conversational style, and it covers the usual first course material.

Keep an eye out for problems! The author will be pleased if we bring any problems (errors, or difficulties in understanding) to his attention

• 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. One author says this about the objective of numerical analysis: "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 we 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. We 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":

  One of my astute students in numerical analysis once 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! It is the subject of section 1.2.

• I'm hoping that we will be able to make it through the six chapters of our textbook:

  Table of Contents

  1. Preliminaries
  2. Root Finding
  3. Interpolation
  4. Numerical Calculus
  5. More Interpolation (this one will be subject of a project, rather than covered together)
  6. 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
  • Exams may include a numerical component, which you may solve using your preferred software (although I hope that you'll use octave).

  • I'll be using Octave Online in class, and the code provided by the author is already available on-line. For example, his code for section 1.4 is available at this bucket (short-hand link)

• The last time I taught this course I did so using a different textbook (Elementary Numerical Computing with Mathematica (Skeel and Kelper)). I will occasionally make reference to it, because I have a fair amount of material created based on it. In particular, let me reference a few quotes from it to set the stage for us:
  • "Historically, numerical methods were regarded as an alternative to analytical methods...."
  • "Generally the numerical analyst does not strive for exactness." (quoting F. B. Hildebrand)
  • "The most important idea... is that the exact answer for most problems is either impossible or impractical to get." (p. 10)
  • "...a major theme of numerical analysis [is] the trade-off between accuracy and cost."
  We're going to give the wrong answer (but we know it, and even confess it -- but try to quantify just how inaccurate it might be).

  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.

Section 1.1 is entitled "Accuracy"; let's compare accuracy to precision (What's the difference?)....

• Basically we're trying to see how close we're getting to the correct answer (accuracy), and discovering that there are two things in our way: what our author calls
  • Algorithmic error
  • floating-point error

• Before we begin, a couple of questions (and use your technology, as well as your brain!):
  • What's \(\sin(.32)\)?
  • What's \(0^0\)?

  Our calculators and computers make errors, and we'll want to aware of that (and avoid making errors if we can).

• Let's pull up section 1.1 in the text, and check out the structure or style of the text. Notice the organization:
  • The set-up
  • Experiments
  • "Crumpets"
  • Key Concepts
  • Octave notes (examples of how to use it well)
  • Exercises (some of which have "solutions", that is, answers with some explaination; and some of which have merely "Answers")

• So let's take a look at the key concepts:
  • How many distinct numbers can a computer represent?

    Computations are usually done in base 2, and the IEEE standard for single precision and double precision reals are

    	Single	Double
    Base	2	2
    n	24	53
    e	[-126:127]	[-1022:1023]

  • Thus, since we require infinitely many answers, and have only finitely many to choose from, we are stuck: we must make errors (floating-point errors).

    We will make floating-point errors because we're not dealing with the actual numbers, but rather with approximations of varying fidelity.

    The example in the text is illuminating: \(\frac{1}{7}\):

    It illustrates

    • the importance of series
    • an example of truncation error (leading to floating point error)
    • a measure of accuracy (the absolute error, \(|\tilde{p}-p|\))
    • "tea time style"
    • and a quibble....

• Notice the definition of number of significant digits \[ d(\tilde{p}) = \log\left(\frac{|p|}{|\tilde{p}-p|}\right) \equiv \log(RE(\tilde{p})^{-1}) = -\log(RE(\tilde{p})) \] Let's think about why that makes sense....

• Now let's have a look at a little octave: Experiment 2, p. 3, shows us an example of algorithmic error, as well as floating-point error (and compares the two).

  Let's push that example a little further....

• and then we can get started working on some of our homework exercises.