Day 15 in Mat360

Today:

• Announcements:
  • Your test corrections are awaiting grading, and
  • your homework is still awaiting grading... sorry about that. I want to apologize: I had three exams to grade this weekend, and they took priority. I'll have these for you on Thursday.
  • Another homework is due Thursday, about Muller's method.
  • Your first project will be due in three weeks, 3/21.

    Since you have groups of three, I suggest that each of you choose one problem to lead on, and then gang up on the fourth.

    Speaking of groups...:(

• Before we head on into interpolation, which I've indicated we need to do, I do have to say a little more about fixed point iteration. So let's go back to Day 14 for a bit....

• Chapter 5: Interpolation
  • A general overview of the difference between
    • interpolation
    • extrapolation
    • approximation/estimation

    Let's think about the difference in the context of a dataset of points, .

    Our authors begin by distinguishing between extrapolation and interpolation. What is the difference, in their minds?

    Look at the first page of the chapter (p. 169): how would you solve those two problems?

    • $f(0.25)$
    • $f(0.50)$

    You might notice that Muller's method is discussed in the beginning of this section. (I hope that you noticed!) The graphics on page 170 (Figures 5.1 and 5.2) shows how to think of the Secant and Muller's method as an interpolation problem, and then a simple root-finding method.

    Question: Which root would be chosen next in Muller's?

    The authors make a rather bold claim (p. 171): that "it might be fair to say that interpolation is the most important topic in this book...."

    While classes of interpolation functions -- interpolants -- are described (e.g. polynomials, sines and cosines -- Fourier series --, exponentials, rational functions), our authors end by saying that we'll focus on polynomials, and we start with Taylor series polynomials.

  • Taylor series:
    • The thing to focus on is the Theorem 5.1, p. 172. Let's take a look at that.

      You have to be careful that you're not deceived by the form of the expression to the right. Every function is not polynomial -- that's not what it's saying. All the interesting stuff is buried in that Greek letter "xi".

    • An informal way of thinking about this is given just below the theorem:
      = truncated Taylor series + remainder term

    • Some interesting applications of Taylor series follow. Let's check them out.
      1. Example 5.1: Taylor series approximate functions (exponentials in this case -- see Figure 5.3)

         An important observation is made about the computation of these polynomials. It is the introduction of Horner's rule (or method).

         This is generally how all polynomials should be evaluated.

      2. Example 5.2: Approximating square roots.

      3. The derivation of Newton's method

      4. Notice that we've already covered the example of using Taylor series in the demonstration of the order of convergence for fixed point iteration.

  • The next topic of interest in the early going is Big-oh notation.
    • Perhaps the best way to introduce this is through an example, such as they give: the expansion of the geometric series
      

    • Our authors use an approach which I am not particularly fond of in this section, and I may sideline a little of this discussion. Their use of containment, on p. 176, is logical but certainly a little unconventional.

    • I will prefer that you think of Big-oh in terms of limits (see the final comment just before "Review question" on p. 178).
    • An important example: order of convergence the Secant Method

      Even though the order of convergence is less, the routine may run in about the same time as Newton's (depends on the problem, of course).

  • Section 5.2: Existence and Uniqueness
    • The interpolation problem is seen in this section as a problem from linear algebra. Let's check out the formulation of it that way.

    • Theorem 5.2: There exists a unique polynomial of degree that interpolates data at distinct abscissas.

    • The proof is from linear algebra. Let's take a look at it.