- Your next assignment was postponed until next Tuesday.
- Revisions for the previous assignment are due today.
- Our first exam is on Wednesday: it will cover chapters 1 and 2.
- You should understand the "toy machine" on p. 41 really well.
- You should be able to do basic calculations, e.g.
- errors (types, sources, propogation, etc.);
- basic operations (addition, etc.);
- digits and decimals of accuracy;
- base-conversions (especially base 2);
- rounding (truncating, etc.);
- machine numbers, and IEEE representation;
- condition of operations (e.g. catastrophic cancellation, condition of a unary operation)
- You should know a few algorithms or formulas, and why we studied them and what we learned -- e.g. Babylonian square roots, quadratic formula, reworking some to make them better conditioned.
- Some surprising calculations -- e.g. addition in the machine is neither commutative nor associative.
- Approximating functions (e.g. with Taylor polynomials): $e^x$, $\frac{1}{1-x}$. (Do you know $\sin(x)\ and\ \cos(x)$ as well?)
- Pay attention to figures we've studied (p. 17, 19, 23, 34, 41, 45, 55).
- Madison asked me for an "extension" of the toy computer -- Figure 2.3, on page 41:
She wanted to see the next couple of scales.
- 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.
- 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 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).
- We've derived the formula on p. 82. Here's one implementation of Newton's method.
- 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, to do division without division?
- 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 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).
- 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...."