- Exam 1
- There's a curve: Compute your percentage, and then add 6
percentage points. When I do that, there are 7 As, 4 Bs, 1C,
and 7 Ds. What we notice is two ends.
- Mean: 81; Standard deviation: 12.
- Keys:
- One issue: Problem #1 was interpreted in two different ways.
I'm struggling with what to do -- seems like an honest mistake, and yet....
What do you think?
Here's what I propose: those of you who took "the easy way out" will do an additional homework problem: Carry out problem #1, only replace the thing to compute by
.
Get that to me by Tuesday after break. I'll read off a list of names, and if you're on it, you owe me. Let me know if I've falsely accused you!
- Other issues?
- There's a curve: Compute your percentage, and then add 6
percentage points. When I do that, there are 7 As, 4 Bs, 1C,
and 7 Ds. What we notice is two ends.
- Your homework
is returned. We need to talk about some things....
- p. 77, #1: The problem states "Determine an interval
that contains
and on which
is bounded." No one did that.
That's the interesting part! That's the analysis.
The very boring part is doing bisections, which is why we spent the time in class working up a very nice bisection algorithm for Mathematica -- why not use it? I can do a quadrillion bisections before breakfast....
- p. 80, #1. Ari found the most elegant approach.
- p. 88, #2. See comment above concerning Newton algorithm
already coded up for you in Mathematica.... Why do this
stuff by hand? It's much more fun to do the
analysis.
For example, notice the Fibonacci numbers popping out in Ben's Mathematica code.
Notice how quickly it converges (try
instead of
in Ben's example).
With the code in hand, you can experiment; you can play. And it's in play that you really learn things, not in drudgery.
That being said, it really does pay to do a few iterations of any iterative scheme by hand. That, too, can be instructive. My question to you thus becomes, how many iterations by hand is enough before you'll head for the computer?
- p. 88, #4: Trent found the strategy.
- p. 88, #7: Eden has a good answer on this --
anywhere in the interval! How do we justify this
answer, however?
Defeng agreed, and mentioned "monotonically increasing" on each interval, which was great -- but that's not enough. Why not?
- p. 88, #8: Some of you are graphing! Hooray! That's the right start, anyway... Drew can get us started.
- p. 77, #1: The problem states "Determine an interval
- Another homework is due Thursday.
The answer is this: "Generally? Yes; always? No."
Newton's method can be fooled, and I'll illustrate two ways using an example.
That discussion leads us to
- A general overview of the difference between
- interpolation
- extrapolation
- approximation
Let's think about the difference in the context of a dataset of points,
.
Our authors begin by distinguishing between extrapolation and interpolation. What is the difference, in their minds?
You might notice that Muller's method is discussed in the beginning of this section. (I hope that you noticed!)
The authors make a rather bold claim: that "it might be fair to say that interpolation is the most important topic in this book...."
While classes of interpolation functions -- interpolants -- are described (e.g. polynomials, sines and cosines -- Fourier series --, exponentials, rational functions), our authors end by saying that we'll focus on polynomials, and we start with Taylor series polynomials.
- Taylor series:
- The thing to focus on is the Theorem 5.1, p. 172. Let's
take a look at that.
You have to be careful that you're not deceived by the form of the expression to the right. Every function is not polynomial -- that's not what it's saying. All the interesting stuff is buried in that Greek letter "xi".
- An informal way of thinking about this is given just below
the theorem:
= truncated Taylor series + remainder term
- Some interesting applications of Taylor series
follow. Let's check them out.
- Example 5.1: Taylor series approximate functions
(exponentials in this case -- see Figure 5.3)
An important observation is made about the computation of these polynomials. It is the introduction of Horner's rule (or method).
This is generally how all polynomials should be evaluated.
- Example 5.2: Approximating square roots.
- The derivation of Newton's method
- Notice that we've already covered the example of using Taylor series in the demonstration of the order of convergence for fixed point iteration.
- Example 5.1: Taylor series approximate functions
(exponentials in this case -- see Figure 5.3)
- The thing to focus on is the Theorem 5.1, p. 172. Let's
take a look at that.
- The next topic of interest in the early going is Big-oh notation.
- Perhaps the best way to introduce this is through an
example, such as they give: the expansion of the geometric series
- Our authors use an approach which I am not particularly
fond of in this section, and I may sideline a little of this
discussion. Their use of containment, on p. 176, is logical but
certainly a little unconventional.
- I will prefer that you think of Big-oh in terms of limits (see the final comment just before "Review question" on p. 178).
- An important example: order of convergence the Secant Method
Even though the order of convergence is less, the routine may run in about the same time as Newton's (depends on the problem, of course).
- Perhaps the best way to introduce this is through an
example, such as they give: the expansion of the geometric series
- Section 5.2: Existence and Uniqueness
- The interpolation problem is seen in this section as a
problem from linear algebra. Let's check out the formulation of it that
way.
- Theorem 5.2: There exists a unique polynomial of degree
that interpolates data at
distinct abscissas.
- The proof is from linear algebra. Let's take a look at it.
- The interpolation problem is seen in this section as a
problem from linear algebra. Let's check out the formulation of it that
way.
Website maintained by Andy Long. Comments appreciated.