Day 08 in Mat360

Today:

• Announcements:
  • Your third homework set is not graded yet....

  • What questions do you have about your homework?

• 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?

  • Some questions/thoughts we discussed last time:
    • 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.

    Today we talk about two methods:

    1. Bisection
    2. Newton's Method (Newton-Raphson)
    • Bisection: relies on continuity and a change of sign.
      • Check out our authors' discussion on page 76. There are some comments that are especially pertinent.

      • 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

        The example presented is remarkable. They say that (except in base 2) \[ 0.5(a+b) \textrm{ is not as accurate as } a + 0.5(b-a) \]

        And they give an example to demonstrate it: four-digit floating-point arithmetic with the round-to-even rule, where $a=5.001$ and $b=5.003$.

        This has the terrible consequence of actually letting our root out of the box. Once we have a box around our solution, we don't want to let it out!

      • By the way, isn't it interesting that in this case we're actually introducing a difference of two numbers that are very close in value, and calling it better than what appears to be a better conditioned form?

        In this case we're making small changes in a very good approximation ($a$), and we don't want to mess up our good work so far.

      • Notice our old friend from a previous homework rearing its head on p. 80, when our authors insinuate that bisection is fairly slow.

        In fact, however, it's nice to know that you get one extra binary digit with each iteration of bisection. Why is that so?

      • Now I'd like you (with a partner) to implement the algorithm of p. 79 in Mathematica, and use it to find roots of the two functions discussed on pages 75 and 76.

        If you need to, you can cheat and use the Mathematica code link below. Notice, however, that the algorithm as implemented has problems that our authors warn us against! It also doesn't use the same stopping criterion....

        You can also use Mathematica's FindRoot to check your work.

    • 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)
    • Student attempts at Bisection