Day 10 in Mat360

Today:

• Announcements:
  • A website glitch....

  • Mathematica for the algorithm Zach mentioned last time

  • Your fifth homework is due today.

  • Your Mathematica projects are due (again) next Tuesday. Your sixth homework assignment is also due Tuesday.

    Java people: I've got Eclipse installed, but I'm still learning. Would it be possible for you to give me executable versions of your code, that I could run under windows (or unix or mac)?

  • Our first exam is scheduled for next Thursday. It will cover up through Newton's method.

• Root-finding: Let's review what we discussed last time, and push on.
  • Bisection:
    • relies on continuity and a change of sign.

    • We tore apart our textbook's naive implementation of Bisection on p. 72 (and appreciated some of the good features, e.g. the computation of New Approximation = Current Approximation + Correction

      (See the example immediately following the naive bisection algorithm for details.)

      On the downside, however, we noted that it

      • didn't check for a change of sign;
      • didn't check to see if a root was obtained (so it might find a root, and then ignore it and go on looking);
      • didn't have a stopping point other than machine precision;
      • didn't check that function values were heading toward zero.
      In terms of improvements, we talked about how, since we're already computing these function values, we might
      • store them and plot the progression later, and
      • compute derivative approximations as well, to use in attempting to determine if we have a discontinuity rather than a root.

    • One interesting example we saw is that we can use bisection to define division, using only multiplication, addition, and subtraction!

      It involved first finding the appropriate function whose root is the quotient we seek. If we need to do the quotient , we define and compute it as

      

      where a and b are chosen to bracket the root. [What might be a good convention for those?]

    • We talked about adding a slope approximation into the improved bisection. How would we do that in an intelligent way?

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

    • So an immediate disadvantage is that you need a derivative.

    • We derived the iterative form of Newton's method (the formula on p. 82), by using the linearization (which I suggested is one of the most important tools we mathematicians have).

    • When you're guaranteed convergence (p. 84)

    • Now I want to look at a couple of things:
      • Figures 3.8-3.11 -- for how things can go awry
      • The examples on p. 82
      • #3, p. 89

    • Let's talk about order of convergence (p. 86). How can we derive that result? Taylor Series! This leads to Definition 3.3, and Newton's quadratic convergence.

      You might think that bisection is linear -- however we can show that

  • Secant Method

    This is Newton's method, where we approximate the derivative: in particular, we approximate the tangent line with a secant line.

    Let's see if we can derive the formula.

• Links:
  • Mathematica notebook for bisection