Day 18 in Mat360

Today:

• Announcements
  • Return of Exam I, part A
    • #1: When evaluating using either method, every calculation must incorporate the method. The most "forgotten" calculation was the .
    • #2: This one went pretty well, although I'd hoped for a little more explanation for your choices. The best response actually included the complete solution: the real line can be divided up into five different regions, and starting from each of which the convergence is qualitatively different (the fifth, not mentioned on the key, is x=0).
    • #3: You can see the roots (-2, -1, 0, 1, 2), and they're all simple. That means that the sign changes between roots. In one single iteration, you'll find only a single root bracketed by two guesses.
    • #4: I was truly surprised that folks couldn't give bounds on the approximation of f by a quadratic Taylor polynomial. I even gave the bounding formula, and the function one needs to bound! (Some skipped over the word "quadratic", and considered higher order Taylor polynomials.)
    • #5: Part a: show that in each case; Part b: check derivatives of each.
    • #6a: No one who attempted part a got it.
      1. You definitely should know that for all x>0; you can use that, and a few variables changes and inequalities later show the result.
      2. You could turn to the Taylor polynomial and error, to show that the convergence is of order .
      3. In terms of other evidence, you might look at a graph of the function.
    • #6b: part b was a homework problem from 2.4. It requires assuming that , and then simply differentiating.

  • If you'll come in and see me, to discuss your exam, I'll give back 50% of the points you missed. Visit the key first, however.

  • May I have your exams, part b?

  • section 3.1 worksheet (don't forget to read the introduction to chapter 3, either: it gives you some good insight).

    There will be a quiz on Wednesday over the reading from this section.

  • Homework from Muller is due Friday.

• Section 2.6: Muller's method
  • section 2.6 worksheet
  • section 2.6 summary
  • About polynomials:
    • Polynomials and their Roots, and values
    • evaluating polynomials well -- Horner's method
    • A neat trick for Newton's method
  • Deflation to handle successive roots (causes increased rounding errors)
  • Using quadratics rather than linear functions to find roots
  • Don't forget linear algebra!
  • Deflation to handle multiple roots (causes increased rounding errors)

    Let's see it at work:
  • Motivating Lagrange polynomials (section 3.1) from solving for "Muller quadratics".

Links:

• Who was Lagrange?