- Your fifth homework is returned. Here's a link to the solutions.
I graded 12 (let's look at soln - part a), 2 (Andres), and 7, and gave a score for "completeness of attempts," for a total of 8 points.
I'll post my solutions on-line tonight.
- Your Mathematica projects are due (again) today.
- I thought you had a homework due -- but I see that in the website
meltdown, the due date got lost. Is that a problem?
- Our first exam is scheduled for this Thursday. It will cover up
through Newton's method. Some thoughts:
- Mechanics: there will be some short computations (bring a
calculator), some short answer, and some analysis of algorithms.
- Do your reading. Some of you have not read your book. I assume
that you've read all the way up to Newton's method.
I'll ask some questions about some of the ideas we've discussed in class, that our author has addressed.
- Big picture: we're making errors. Some errors we have control
over, and others we don't. We have to deal with a finite collection of
numbers, rather than all the real numbers. This introduces errors.
- You should be comfortable with the use of different bases, and
changing between bases (in particular 2 and 10).
- The "toy computer" (Figure 2.3, p. 41) should be completely
understood. Can you do computations on the toy computer? What
does it give for .7*1.2?
- Last time we laid out everything we needed for an implementation
of "Bivision" -- using bisection to replace division.
You should examine this implementation, looking for problems, improvements, and for understanding of the process.
- We talked about adding a slope approximation into the improved
bisection (I've since included it, but used a naive
slope calculation -- a straight finite difference). How
would we do that in the most intelligent way? How would
you change the algorithm so that it would alert the
user to the possibility of a discontinuity rather than
a root?
- You might think about how Newton's method could be used for the same process.
- Mechanics: there will be some short computations (bring a
calculator), some short answer, and some analysis of algorithms.
- We'll start with Newton's method, and then migrate to the Secant
method.
Last time, at the end of the hour, Tyler asked the right question:
how do we know which function to choose when we're using Newton's method? The question can be generalized: how do we know which function to use when we're using the quadratic formula(s), for example....
One strategy is the condition number, for unary functions. That involves derivative, if you recall (and I hope that you do, for the test!). It was called the "relative derivative".
Similarly, we'll have choices when it comes to chosing an iteration function, and it's the derivative that will help us decide between them.
- Newton's method:
- We've derived the iterative form of Newton's method (the
formula on p. 82), by using the linearization (which I suggested
is one of the most important tools we mathematicians have).
- Last time we saw that if we start in the vicinity of a
root, close enough, hen you're guaranteed convergence (provided
the derivative is well-behaved).
- We also talked about how things can go awry. Ordinarily,
however, things go well.
-
Newton's method is an example of a process called "fixed point
iteration". Let's look at a couple of examples of that. In particular,
look at the importance of the derivative at the root to see why one
solution is achieved, whereas another is not:
Now here is how a problem looks using Newton's method:
Newton's method as Fixed Point Iteration
- Let's talk about order of convergence (p. 86).
Definition 3.3 (p. 87): If a method converges to root
, and if
then the method is convergent of order
, with asymptotic error constant
.
We can show that Newton's convergence is quadratic (with Taylor polynomials).
Question: What would you expect for bisection? You might think that it is linear, but interesting things can happen....
Let's look at the convergence of a general fixed point method. This will answer Tyler's question about "which function to choose".
- We've derived the iterative form of Newton's method (the
formula on p. 82), by using the linearization (which I suggested
is one of the most important tools we mathematicians have).
- Secant Method
This is Newton's method, where we approximate the derivative: in particular, we approximate the tangent line with a secant line, using a finite difference approximation to the derivative.
Let's see if we can derive the formula.
One thing that we notice right away is that this method requires two approximations to start -- to prime the pump.