Day 09 in Mat360

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Announcements:
- I've posted a new assignment, which will be due next Tuesday -- on roots.
- Your first exam is Thursday, 2/16. It will cover through bisection.
- Your fourth homework is due.
- Your third 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 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").
  - 1, p. 34:
    - Pros:
      - 12 has more divisors (1,2,3,4,6,12) than 10.
      - with base 12 you can express most common fractions without repeating decimals.
      - useful when we are expressing time, eggs, months
      - Babylonians like it.
    - Cons:
      - We only have 10 fingers.
      - It would cost a lot of money and time to change from one base to another.
      - Requires two new digits. Suggestions?
  - 3: the authors showed that terminating decimal expansions (base 10) do not always have terminating binary expansions (p. 35 -- 5.2 does not have a terminating base 2 representation).
    But because 2 divides 10.... Consider terminating binary decimal ${\frac{1}{{2^i}}}$ for any integer ${i>0}$ .
    And because the sum of terminating decimals is terminating....
    Patrick had the best reason I saw.
    Some of you commented on other bases. It's possible, for example, that a terminating decimal in base 3 will not have a terminating decimal expansion in base 10. Can you give an example?
  - 8:
    - You need to first interpret the fraction (and some of you had trouble with that).
    - Then you'll need the geometric series (that is, the Taylor series for ${\frac{{1}}{{1-x}}}$ ).
  - 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. But you were also told to use floor or ceiling, so you should do that.
  - 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....
    Lauren can 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.
    Joey? Michael W.?
  - 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 a variety of answers, 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).
    Michael M.
Chapter 3: root finding
- 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 ${f(x){=}x^{\frac{1}{3}}}$
    - Exercise #3, p. 89
- 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...."
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