Day 17 in Mat360

Today:

• Announcements:
  • Joey had good luck with that Muller's method applied to the function $f(x)=e^x-\frac{1}{x^2}$.

    I'm still looking over your homework. I'll have it for you next time.

    Some of you "complained" about the amount of algebra. That's why you want to use Mathematica! Only five of you included Mathematica files. It would have been a good time to use Mathematica.

  • I put midterm grades in last night. I assumed 1 homework drop (and hadn't included your Muller homework). I also assumed that you all got As on all your projects -- so try to make that happen!:)

  • Speaking of which, your first project will be due 3/21. Groups (one group of four):
    • Alyssa/Katie/Mike W.
    • Joey/Zach/Brooke
    • Kara/Lauren/Leah
    • Madison/Courtney/Trey
    • Patrick/Stefanie/Michael W/Michael M

• Chapter 5: Interpolation
  • While different 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 started last time with Taylor series polynomials.

  • I want to begin today by looking at four different ways of writing the interpolating polynomial that we consider in Muller's method.
    • "The classic": solving a system of equations
    • Lagrange Polynomials
    • Newton's Interpolating Polynomial
    • Horner's Method
    The fourth method (Horner's) is the one that I used to code up my Muller's method, and I want to just showcase Muller's in action to start today.

    Perhaps I'm so enthralled with Muller's method because I recall a line from my first numerical analysis textbook (Conte and de Boor), in which they had this to say about Muller's:

    "A method of recent vintage, expounded by Muller, has been used on computers with remarkable success. This method may be used to find any prescribed number of zeros, real or complex, of an arbitrary function. The method is iterative, converges almost quadratically in the vicinity of a root, does not require the evaluation of the derivative of the function, and obtains both real and complex roots even when these roots are not simple."

    Now the third edition of C&deB was from 1980, but the first edition was written in 1965 -- before we landed on the moon -- so it might be that they meant by "recent vintage" something around that time. In fact, Muller wrote this up in 1956! Perhaps the use of "recent vintage" in textbooks is a mistake....:)

    Question: How would you write the Lagrange interpolator to two points?

  • The next topic of interest in the early going is Big-Oh notation (section 5.1.1).
    • Perhaps the best way to introduce this is through an example, such as they give: the expansion of the geometric series
      $\frac{1}{1{-}h}$

    • 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.

      Notice the increasing powers of $x$. Clearly there's a risk that the coefficients will be of vastly different magnitudes, which could cause severe roundoff errors. This is one of the problems with this formulation.

      Many of you may have used this notion to find the coefficients of your Muller quadratic. The problem is one of solving three equations in three unknowns; and since the equations are linear, we have a linear system to solve. More generally:

    • Theorem 5.2: There exists a unique polynomial of degree $\le{n}$ that interpolates data at $n+1$ distinct abscissas.

    • The proof suggested by the author is from linear algebra. Let's take a slightly different look at it (p. 184). Then we'll look at Exercise #7, p. 185.

• Now let's take another look at Newton's Interpolating Polynomial, relating it to the concept of divided differences (section 5.6.2, p. 212); and we'll have a little more to say about the concept of Horner's rule (just before section 5.6.2).

  We'll finish this off by having a look at the relationship between the divided differences and the derivatives of $f$.

• Now our attention should turn to the error term, the $\xi$ in Taylor's theorem.