Day 09 in Mat360

Today:

• Announcements:
  • The fifth homework assignment is due next time (that's a change): today we'll take a look at some hints, and insights into why we're looking at some of those problems.

  • I've posted a new assignment, which will be due next Tuesday -- on roots.

  • I haven't finished grading your first programming assignments (you java folk -- I haven't gotten started on yours yet, but the comments today may be meaningful, nonetheless). I've also got a python guy....

    I'm testing your projects with some of these numbers:

    I should have been more specific in my instructions. Let's look at two extremes:

    • If Mathematica had a function called "CNFOBTA[x,b_in,b_out]" (convert number from one base to another), I would hope that you wouldn't just say "Use CNFOBTA."

      So what is legal? BaseForm? IntegerDigits? Those are half of CNFOBTA, so my answer is NO, these are not legal.

    • On the other extreme, are you allowed to use Mathematica's multiplication, division, addition, and subtraction, or are you required to design and implement your own computer in order to do the conversion?

      You may use logs, and the standard mathematical operations. You may also use string operations, or almost anything else -- but you can't use functions designed to convert from one base to another!

    So -- for those of you who used a canned converter (e.g. BaseForm, IntegerDigits), I'm asking you to go back and write those:

    your job is to create the converter (not just invoke one).

    Please try again. If you used BaseForm (or equivalent) and IntegerDigits, you need to write a procedure to do what they're doing.

    Let's take a look at IntegerDigits. How do we use well-worn functions to get the integer digits in a base-10 representation of a number?

    So you might be thinking "544890 -> {5,4,4,8,9,0}. How do we do that?

    If you need more help, please see me. We can sit down and I'll help you figure out HOW, so that all you need to do is to make it happen.

• Chapter 3: root finding

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

  

  Summary so far:

  • 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 :
    • Do we have an expression for ?
    • Can we simplify ? Perhaps as a product?
    • Is continuous (connected)?
    • Is differentiable (smooth)?
    • Is monotonic (increasing or decreasing)?

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

  We'll take another pass through the first two methods today, which we started discussing last time.

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

    • Let's take a look at the algorithm that the authors propose for bisection, on p. 79.
      1. What trouble do you see on the horizon, if one carelessly implements the algorithm as given?

      2. The strategy for a stopping criterion is interesting. What do you think of this:

         "The need for an error tolerance is avoided by iterating until an answer of machine precision is obtained."

      3. How can we improve this algorithm?

    • Notice the comment on page 79: it's interesting, but important:
      new approximation = current approximation + correction

      Check that example calculation.

    • One way we can improve the algorithm is to incorporate some useful features into the process. What can we add, so that we don't waste calculations, and that the user will appreciate?

    • Exercise #2, p. 80

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

    2. Let's start by deriving the formula on p. 82.

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

    4. One needs to think carefully about the form of the function that one uses in Newton's method.

    5. An examination of what can go wrong (figures on p. 85)
       • That Newton's method won't work for
       • Exercise #3, p. 89

  • Order of conversion. 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)
  • Mathematica demonstration of the next step using Newton's method