Day 13 in Mat360

Today:

• Announcements:
  • Your exams are graded:
    • The exam
    • Key
    • You can fix one of your problems for half the points back (if you fix it up correctly!). Return those to me this Thursday, as well (correction and original page or the exam).

  • Your next assignment is due Thursday.

• Chapter 3: root finding
  • As I told you last time, we're going to tear into root finding methods. Today we're going to
    • get back to bisection and Newton's method
    • look at two different ways to extend Newton's method
    • examine issues of convergence.
    Next time I'll talk about fixed point methods generally, of which Newton's is an example. Then next week we'll turn to interpolation.

    I'll also be assigning you your first project on Thursday. For the first project you can choose your own group of three, if you wish. Those who aren't in a group will then be assigned one.

  • Bisection:
    • I want to return to exercise #2, p. 80. Last time we had come to a rather unsatisfactory point: we could use bisection to do division with only addition and multiplication, but
      • whereas if $c>1$ then we can use $[0,1]$ as the interval, and away we go;
      • our interval was [0,machine max] in the case where $c<1$. That's rather big (but I suppose it will work...:)!

      The question is this: can't we make that interval a little better? In particular,

      What if we knew the machine representation of the number?

    • The same idea can be used for the case of $c>1$, too, to improve on our starting interval.

  • Newton's method:
    1. relies on differentiability of the function $f$ and a good starting guess $x_0$. The derivative is assumed to exist and be available for calculation -- otherwise one uses the secant method (which we'll discuss later).

    2. We've derived the formula on p. 82. Here's one implementation of Newton's method.

    3. Now that we've got the formula, we should discuss the two examples on p. 82:
       1. Division without division...
       2. Babylonian roots

    4. One needs to think carefully about the form of the function that one uses in Newton's method.
       • For bisection we used $f(x)=cx-1$
       • For Newton's we used $f(x)=\frac{1}{x}-c$
       • What if we tried bisection's $f$ with Newton's method, to do division without division? (Exercise #3, p. 89)

    5. An examination of what can go wrong (figures on p. 85).
       • That Newton's method won't work for

  • Extensions/Alterations of Newton's method
    • "discretization" $\longrightarrow$ secant method
    • Another version of "Newton's Method" can be derived in similar fashion to Newton's method, only using a "tangent quadratic".

  • Order of convergence. I want to motivate that result on p. 86, using Taylor series, from which follows the rather astonishing conclusion of our authors: "Each iteration doubles the number of accurate decimal places...."

• Links:
  • The IEEE standard for floating point arithmetic
  • Mathematica notebook for bisection (from this source)
  • naiveBisection.nb
  • Mathematica demonstration of the next step using Newton's method
  • In lisp: