Day 08 in Mat360

Today:

• Announcements:
  • Last time we talked about some of the important tools that you need to put in your numerical analysis toolbox:
    • The limit definition of the derivative
    • Geometric series
    • Taylor series in general
    • The triangle inequality
    • The Mean Value Theorem (MVT)
    • Properties of logs and exponentials
    Make sure that they're there.

  • Your fourth homework is returned.
    • Many of you seemed to have questions, which you didn't ask. Ask me. You can ask in class. I have office hours. You can email me for hints. I'm here to help -- really!

    • Some of you still aren't showing your work; aren't explaining. Answers just magically appear.

      I need some rationale -- if you think it's obvious, all the better -- it should be easy to explain.

    • In this homework you were to actually begin writing some algorithms (sometimes just short "functions").

    • p. 42, #1: You're to find a function for e, given x -- e(x); some of you included s, as thought it were known. We can find s, too. We can write s(x).

      Some of you assumed x was positive.

      Part of the problem here is unfamiliarity with logs.

    • p. 42, #3: Explain. Some of you merely presented some that have known machine representation. That's a good start, but why aren't others representable?

      What do you say? Fill in the details....

      Can someone give a number theoretic answer?

    • p. 42, #5: folks took this several ways --
      1. the smallest integer that can't be represented in IEEE single-precision;
      2. the smallest positive real number the format can store;
      3. or the smallest integer that can't be represented exactly (without error).
      The authors meant the last. The point is this: when are we making errors, and when aren't we? We need to know.
    • p. 48, #1bcd: some of you appear to think round-to-even means essentially "truncate to even". No, you're still rounding.

      On c. how do we know whether that 0 in the ones place is significant?

    • p. 48, #2: I got answers all over the place, many expressed as just an interval (again, without explanation). This shows why it is so important to explain.

    • p. 48, #5: You are to explain how to use this function nint to round decimal (non-integer) numbers to the nth decimal place. You need to produce a function: round(x,n).

    • p. 48, #6: Now actually create nint. So I should see some computable function, nint(x).

  • Your first programming assignment is due today.

  • The fifth homework assignment is due next time.

• As we begin Chapter 3, we seek to solve

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

  

  How would you do it?

  Summary of some thoughts from 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 :
    • 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.

  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.....

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