- You have a homework due today, as well as your test revisions.
- You have a short homework assignment, due Thursday:
breaking the adaptive quadrature code that we discussed last
time. See your assignment page.
- Your Bezier curve project is due the last day of classes, which is
Thursday of next week.
- I hope that you had a chance to read the introductory section for
ODEs (chapter 8). It's really quite a nice introduction (reminder)
about why we're interested in solving ODEs, how to solve them
analytically in a few cases, and a history of numerical solutions
(dating back to the mid 1700s).
The story of the prediction of Halley's comet is priceless (p. 312).
- We'll be starting off today with Euler's method. In the next two class
sessions we'll see how to extend Euler's method to higher-order Taylor
methods, and then how to eliminate higher derivatives using Runge-Kutta
methods.
You might read through 8.4 for next (higher-order Taylor methods).
- How would you feel about a small Mathematica component to the final, covering integration/ODEs? This would be assigned the last day of class. Otherwise I feel that I should give you one more homework assignment, and I'd hate to do that -- you seem busy enough....
- I'm not going to spend much time on the analytic stuff. You've all
had a course in DEs, so hopefully that part makes sense to you. But you
may be a bit rusty. In that case, you might check out Andy's
"DEs in a Day" page.
I may be making a few references to that page as we go along. Let's start with this one (about what we're actually solving, or attempting to solve).
The upshot is expressed well in the authors' quote that "...the numerical solution values
and
do not lie on the true solution. In fact, they do not even lie on the same solution."
-
Let's pick up then on p. 321. Euler's method is the simplest, sanest
starting point, and it can be derived in two different ways:
- By integrating (p. 321), or
- As a tangent line approximation (Figure 8.10, p. 322) -- which, by the way, is the same as using a Taylor series.
- The method as a recursive algorithm is highlighted on
p. 323 (box equation). Note that the step-size
is also a function of
and can vary. So you might start thinking about adaptive step-size control right away....
- Let's have a look at the code on p. 323.
- An example for the non-autonomous equation
- We can solve this one analytically by separation of variables.
- Let's compare some numerical solutions.
- How should you feel about your numerical solution, if you didn't have the exact solution to compare them to?
- The obvious question is this: "How should we choose the step size(s)?"
- Now, as usual, we want to talk about errors (a function of
step-size). Let's face it: we're clearly making errors.
Start by taking a peak at Figure 8.13, p. 325. It illustrates everything that we want to consider.
Our author distinguishes three types of error: in this figure there are two,
- local discretization error, and
- global discretization error.
I want to prove a result which gives a bound on the error we're making (ignoring rounding error).
Compare this to the result at the bottom of p. 326. They claim that the global discretization error is proportional to
. (We're in the ballpark!)
- However Making
h small may not be a good idea if we add in the rounding error.
The neat thing is that we can arrive at an optimal h, provided we can bound the second derivative and the rounding errors.