- Test 2:
- Test revisions returned. A couple of comments:
- Taylor series
- Use one variable -- x or h -- but not both!
- bounding -- where did the
come from?
- Hermite
- Make sure that your interpolators interpolate. Also that
they're quadratics. You're fitting three points -- generally a
quadratic is necessary.
- Complete (and use!) the divided difference table...
- You all don't seem to be getting Horner's... Or expanding about 4.
- Remember that Muller's is just a quadratic form of secant: we find the root of a quadratic to get our next estimate (and have to make a choice).
- You shouldn't automatically use the alternative form of the quadratic formula as your default now -- remember that each form had its use.
- Sometimes you can solve a problem without doing a calculation (the secant problem following Muller).
- Root-finding
- Remember that Newton's method is a fixed-point method -- it just converges better.
- Muller is roughly 2 (about 1.84)
- More root-finding
- Bisection needs a symmetric interval to find 0.
- Why didn't anyone, that I know of, actually try
Newton's method on an initial value near 0? You'd have
seen exactly what happens.
A lot of you said it will cycle -- it won't.
There are three ways to see that things will go awry at the origin:
- Ratio of errors (
)
- calculation of AEC (infinite)
- calculation of |g'(0)| (2)
- Ratio of errors (
- Taylor series
- You broke my poor, innocent Adaptive
Quadrature code! You dirty rats....
Here are some example problems:
- Many identified calculation at points involving 1/0.
- Andres
- Victoria
- Trent and Vicheth identified discontinuous functions as problems.
- Your Bezier curve project is due today (by Midnight tonight!)
- Today I want to
- say a few final words RE Runge-Kutta methods, and
- talk a little about the final, and
- summarize the course, and allow for questions.
- I've assigned a small Mathematica component to the final,
covering integration/ODEs. It is a relatively routine comparison
of solvers for integration and ODEs on a couple of problems.
You may discuss this with others (but produce your own work). It's more to prepare you for the final.
- Thursday, 1:00-3:00 pm
- I'm also willing to do a review on Wednesday night of finals week (before our final on Thursday). If you're interested, let me know.
- You may have a one-page (8.5x11, inches!) cheat sheet of whatever you'd like.
- The assignment will count for 20% of the final.
- 40% will be old stuff, and 40% will be new stuff.
- If your grade on the "old stuff" portion of the final is better than the lowest of your hourly exams, it will replace that grade (i.e. it will count twice).
- We start with an illustration of how we might improve on Euler's
method, by replacing it by Runge's midpoint. So check out 8.5.1,
p. 346, and Figure 8.20 in particular.
- Alternatively we might replace it with Runge's trapezoidal
method (Figure 8.21).
Let's compare other estimates on Example 8.11:
Euler's 1.5 Taylor-2 1.57 Runge Midpoint 1.5765 Runga Trapezoidal 1.583 - But ideally, we'll figure out how to replace a method (such as
Taylor-2) with something that gives us the same order, but without
computing derivatives. That's the job of RK-2.
- The derivation involves Taylor series expansions of functions of
two variables.
- How do we think about Taylor series geometrically for functions of two variables?
- You might be thinking "why not just do two half-steps of Euler?" I
mean, we're doing more work to get an RK-2. Why do you think that RK-2
is better than simply doing Euler twice as often? It's about the same
amount of work....
- Many folks go with an RK-4, with step-size control. I am trying to
adapt the step-size control we used with Taylor to the RK-4 in some
sensible way (but not having much luck yet). If we have a little time
left, maybe we can talk about that issue....
- Applications
- General RK-4 (no step-size control)
- The primary fact of life is that calculators are almost always
wrong; but they're so close that we don't care. "All models are wrong;
some models are useful."
- We're going to make errors: there are those we can control, and
those we can't. We should do everything to minimize those we can
control, and to remain mindful for those we can't control.
- What does your calculator do? It
- performs the basic arithmetic operations,
- graphs functions,
- computes derivatives,
- computes integrals,
- solves equations (e.g. finds roots), and
- may even solve ODEs.
These are among the things we learned to do in this course.
- An approximation without an estimate of the error made is a poor
thing.
- Start with representation of numbers. Numbers in the computer are
approximations, not exact. Very few (relatively) of the reals are in
there (and in this picture we can see what kind of error we're making
in representing any real number):
One problem is that computers speak base 2, and we speak base 10; and only rarely do these both end up being machine numbers. So one or both of us is in error.
Rounding or chopping occurs, and we need to be careful in how we do that.
- Algorithms for even some of the most common operations you've
always known may not be ideal ("ill-conditioned", or inefficient).
- Quadratic formula
- Evaluation of a polynomial should be carried out via Horner's method.
- Representation of function is similar: sine is not in your
calculator; a Taylor series representation of sine is. This is so important because we replace complicated
functions with the operations of addition, subtraction, multiplication,
and division -- those operations which we can program computers to do.
It turned out that Taylor series were useful tools in many places across the semester. One thing that Taylor's theorem does is allow us to bound errors.
Errors also compete: one of my favorite parts of the course is discovering that the step-size for integration should be made small, but only so small: if you make it too small, the round-off errors from all the additions you do start to add up and overwhelm the advantage achieved by reducing the truncation error.
There's an optimal value, given the form of
and the machine you're using.
Sometimes errors can even cooperate! This is the lesson of Simpson's rule: wherever you have two estimates, you have a third; and if you're clever, you can add them so that their errors cancel:
- We can use polynomials for capture the behavior of a series of
points (splines), a function (Taylor), or even for design (e.g. Bezier).
- The most important polynomial is the linear. We base a lot of
methods off of the tangent line, for example:
- Newton's method for root-finding
- Euler's method for solving ODEs.
- Methods involving the computation of derivatives may be improved
(or faked) by replacing analytic derivatives by numerical
versions. We perform a rather precise balancing act, similar to what
happens in Simpson's rule. And so
Newton's method is replaced by secant (and then Muller's), or
Higher-order Taylor methods for ODEs are replaced by RK methods.
- Each of these is an iterative scheme: we compute
and hope that we have improved.
Newton's method is an example of one of these "fixed point" schemes, where you know you're done when
- Some schemes are recursive, e.g. adaptive quadrature. Our adaptive quadrature code was also useful for intelligent choice of points for plotting a function (the points being joined by a spline to make a nice smooth curve).