Day 27 in Mat360

Today:

• Announcements:

  • Homework returned
    1. p. 264, #1
       • Expanding and simplifying Taylor series
    2. p. 264, #3
       • Fit one cubic to four points.
    3. p. 281, #1
       • They ask you to do composite Simpson's rule (see the footnote on the problem).
    4. p. 281, #2
       • Recognize as
       • Then integrate, and use that as an estimate
    5. p. 281, #15: If the rule is , then
       • halving h for p=1 halves the error;
       • halving h for p=2 cuts the error by 1/4;
       • halving h for p cuts the error by ;
    6. p. 293, #1
       • If it's quadratic, the error term (which involves higher derivatives than two) disappears, and they become equal.
    7. p. 293, #4a
       • The terms are not the same.
    8. p. 293, #5
       • Wow, did you guys make this easy problem into a hard problem. ...

  • I've not finished regrading your tests yet, or your assaults on my adaptive quadrature code. I'll have those Thursday, including a Key for the 2nd exam.

  • Your Bezier curve project is due next time. How is that coming? I'd like to be able to show off your signatures in class, so please send your files (Mathematica, or images) ahead of time, if you can.

  • Today we'll see how to replace the Taylor methods, which require analytical derivatives, with Runge-Kutta methods.

  • Thursday I'll assign a small Mathematica component to the final, covering integration/ODEs. It will be a relatively routine comparison of solvers for integration and ODEs on a couple of problems.

  • Thursday we'll also have a review, and you might look over your exams and homeworks for problems you'd like to discuss. There's also a brief "Chapter 9" review you might look over. We'll start with that.

  • 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.

• Review: Taylor methods
  • Euler's method is an example of a Taylor method -- it's the first in a series (we might call it "Taylor-1"). So it's our introduction to Taylor methods, which we can think of as generalizations of Euler's method.

    In particular, we generalize the formula on p. 323,

    

    to the more general formula

    

    Note the parameter in the function: will generally be a function of .

  • As we saw last time, Euler can be derived as the tangent line approximation to solution of the ODE. We could just as easily pass a quadratic through the point that interpolates the slope and the curvature (2nd derivative) as well (Figure 8.19, p. 340). This is the idea behind Taylor-2.

    And so on! That's the idea of the higher-order Taylor methods.

    Let's have a look at a derivation with a slightly different emphasis from that in our text (our authors want to avoid partial derivatives, but I don't!).

    Examples:

    1. Computer problem #1, p. 344 (I'm not sure why the authors suggest the step-size control that they do -- they seem to be off by a power of , and a derivative.)
    2. A modification of that code to do other examples, e.g. Example 8.10, p. 340

• Now we go about eliminating the derivatives, creating Runge-Kutta Methods
  • 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
    Taylor-2
    Runge Midpoint
    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