Day 07 in Mat360

Today:

• Announcements:
  • Your third homework set is due.

  • Your second homework set is returned: p. 14, #3; p. 20, #1,2,5-7; p. 28, #1, 4, 5, 8, 14.
    • It was an "ouch" moment for a few of you. Please do come and see me to ask about homework before it's due. I'm happy to talk about them!

    • I graded #3, and two each of p. 20 and p. 28, randomly chosen:
      • p. 14, #3: Katie (part a.) and Michael W. (part b.)
      • p. 20, #5 and 7: Joey or Alyssa (#5); Zach or Michael M. or Madison (#7)
      • p. 28, #1 and 8: table (see Mathematica); #8a (Mean Value Theorem) and Trey (#8b)

  • Your second homework set (for the most part) in Mathematica

• Let's wrap up Section 2.5: Numerical Instability

  This section features several interesting examples of functions, some of them tremendously important, which are also extraordinarily sensitive to errors.

  One of the main points of the section is that a "solution" to a problem may be technically correct, analytically correct, and yet poorly designed to produce good results in general.

  An excellent example is the quadratic formula. Many of you have memorized it as

  

  We can imagine situations, however, for which this calculation may be dangerous. What do you notice?

  Question: an old trick from your past may be used to improve things: can you think of how to change this formula, to make it less sensitive?

  • #6, p. 68
  • #7, p. 69 (part of your homework): Quadratic Breakdown
  The authors say "You might also wonder whether we seriously expect everyone to know all these trick. We do not...." They're just pointing out some serious issues that you should be aware of on p. 65.

• As we begin Chapter 3, we seek to solve

  The most important equation in the world (Physicist Charles Shirkey, BGSU):

  $f(x)=0$

  How would you do it?

  • Today I want to lay out the problems and issues, and conditions under which we can solve this. Maybe even a strategy for solving some cases.

    Then we'll see what our authors have to suggest. They have essentially three methods to propose. We'll look at those, and then one more -- Muller's method

  • Some questions:
    • Does a solution exist?
    • What kind of solutions are we allowing?
    • If a solution exists, is it unique? If not, which one(s) do you want?
    • We need to think about the form of $f$:
      • Do we have an expression for $f$?
      • Is $f$ of a known class, with solutions with known features (e.g. polynomials have a known number of zeros; sines and cosines have an infinite number).
      • Can we simplify $f$? Perhaps as a product?
      • Is $f$ continuous (connected)?
      • Is $f$ differentiable (smooth)?
      • Is $f$ monotonic (increasing or decreasing)?

    • Can we graph $f$?
    • Can we show that $f$ changes sign?
    • If $f$ is continuous and changes sign, then we are assured of the existence of a root (by the Intermediate Value Theorem).
    This is a very good start.

    Let's talk about two methods today:

    1. bisection
    2. Newton's Method (Newton-Raphson)

    Before we do, however, let's check out our authors' discussion on page 76. There are some comments that are especially pertinent.

    • Bisection: relies on continuity and a change of sign.

      As we approach bisection we want to be aware of the issues illustrated in figures 3.2 and 3.3, p. 77.

      Notice the comment on page 79: it's interesting, but important:

      new approximation = current approximation + correction

    • Newton's method: relies on differentiability and a good guess.

      I'd like to go through the geometric sense of Newton's method, and derive the formula on p. 82.

      If we have time, we'll then see how the Babylonian method for finding square roots is an application of Newton's Method....

  • Links:
    • Mathematica notebook for bisection (from this source)

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