Day 11, Mat360: Numerical Analysis

Today:

• Announcements:
  • Our first exam is next week (Wednesday), and will cover everything through 2.6, which we'll start today.
  • Section 2.3 homework is returned.
  • Your homework for section 2.6 will make up the "take-home" portion of your exam. Please work on this on your own, although you can ask me questions.

    Homework pp. 93--, #1-5, 6, 8, 11, 13, 14, 16, 22 (use your Muller code), 23

• Questions on anything?

• Let's recall what we did last time:
  • There was a last nod to the fractal convergence stuff:

    I could stare at that all day!

    

  • Then we got started on Section 2.6: finding all roots of polynomials

• Today: We finish off Chapter Two, root finding, with Section 2.6: special attention to finding roots of polynomials. That has been traditionally an interesting and challenging problem, starting with the quadratic formula. We give that formula a little more attention today.

• Now on to Section 2.6: Mistakes, he's made a few.... The most serious is in his formula for the result of synthetic division, p. 87.

  That one I've got to let him know about!:)

  \[ a_{n}(\cdots a_{n}(a_{n}(a_{n}t+a_{n-1})+a_{n-2})+\cdots+a_{1}) \] becomes \[ a_0+t(a_{1} + t(a_{2}+t(a_{3}+t(a_{n}+\cdots+t(a_{n-1}+t(a_{n}))\cdots)) \]

  Let me show you how that comes about....

• Turns out we can just keep going using synthetic division, and get the derivative at \(t\) as well! Very cool, and easy.

  We can even get the 2nd derivative at \(t\) as well, in case that's useful! Let's see....

• His discussion of the proper way to compute roots of the quadratic (Crumpet 18, p. 90) is part of the code you'll be writing.

• Exercises (we'll use these to motivate discussion):
  • #7 and 9 -- Horner's method (synthetic division), and Newton's using synthetic division
  • Then we'll check out a couple more that I attacked with my Mathematica code. One of the more interesting ones involves Fibonacci....

• Muller's method generalizes the secant method, which uses two points and a straight line to probe the x-axis for a better estimate of the root: it uses a quadratic, and a parabola.

  (In the exercises you'll extend this idea to a cubic!)

  I'll gripe a little about our author's choice for how he represents the algorithm on pp. 91-92 (he's taking some abuse in this section!:). My feeling is that this would be a good time to introduce divided differences. If you look at the formulas on p. 91, you'll see that we re-use some material -- and any time you re-use material, you should avoid re-calculating.

  Divided differences are what we use to approximate derivatives, say: \[ DD(x_1,x_0) \equiv \frac{f(x_1)-f(x_0)}{x_1-x_0} \]

  Of course, once you have divided differences, you can take divided differences of them (essentially higher-order derivatives):

  \[ DD(x_2,x_1,x_0) \equiv \frac{DD(x_2,x_1)-DD(x_1,x_0)}{x_2-x_0} \]

  This allows us to write \[ a=DD(p_1,p_2,p_0)=\frac{DD(p_1,p_2)-DD(p_2,p_0)}{p_0-p_1} \] and \[ b=DD(p_1,p_2)-(p_1-p_2)a \]

• Usually after one uses deflation to find roots, one wants to "polish" the roots -- to make sure that errors haven't crept in... Well they will creep in, but we want to minimize them, of course.

  This is sort of a "2.7" thing, however....