- I'm going to postpone your new assignment, which was due next Tuesday --
in favor of having you revise this homework assignment. I'll
then take the average of your two grades.
Distribution of homework grades shows dangerous trend. It's almost a binary world: some of you get it, and some of you don't.
You need only "correct" those which cost you points; however I'd suggest that you also correct those which are not quite right! Figure out what you're missing.
- Your first exam is next Thursday, 2/16. It will cover only
chapters one and two (and not bisection, as previously stated).
In addition to repairing this homework, I suggest that you go back and correct your previous ones, and ask questions on Tuesday.
- Your fourth homework is returned. I'm going to have several of you
help me today, as we look over some of the issues which have
slipped under the radar for some.
- Section 2.4: #3, 5, 12
- #3: Try real examples, and see! Use the toy
computer. Everything IEEE can be modelled with the toy
computer. Consider intervals. (Andy)
By the way, you need only consider positive values of $x$ and $y$. But you might ask yourself how things change if you make $x$ negative....
- #5: machine numbers have successors and predecessors. Real numbers don't!
- #12: The answers should be close to 1/3. But they have to be machine numbers -- exactly representable -- in four digits. (Alyssa)
- #3: Try real examples, and see! Use the toy
computer. Everything IEEE can be modelled with the toy
computer. Consider intervals. (Andy)
- Section 2.5: #1, 2, 3, 7, 12, 13
- #1: Maximize the errors. How can you make the error as big as possible? A few examples aren't enough -- we need analysis. (Andy)
- #2: Demonstrates the fact that machine addition is not associative (nor commutative)! (Katie)
- #3:
- #7: Follow the advice on p. 65: you have to use two different formulas! (Lauren)
- #12: Why would $\cos(x)\cos(\delta)-\sin(x)\sin(\delta)-\cos(x)$ be better? [Hint: it wouldn't!] (Joey)
- #13: See the discussion of "sensitive" versus "insensitive" sub-problems on p. 63. While they did not ask explicitly for an example, a mere assertion about what values would be bad is not nearly as good as producing an example -- (Michael W.)
- Section 2.4: #3, 5, 12
- Let's look at them in this order:
- I'll make a few comments about those no one will present.
- Alyssa and Katie
- Joey and Lauren
- Michael W. and I (2.5 #1)
- Then we'll look at 2.4, #3.
- Bisection:
- If we have time, 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?
- If we have time, 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
- Newton's method:
- 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).
- Last time we derived the formula on p. 82.
- Now that we've got the formula, we should discuss the two
examples on p. 82:
- Division without division...
- Babylonian roots
- 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?
- An examination of what can go wrong (figures on p. 85)
- That Newton's method won't work for
- Exercise #3, p. 89
- That Newton's method won't work for
- 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).
- 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...."