- You have a homework due Wednesday, over
section 4.4. Any questions on that?
This is your last "homework" to turn in; any other assignments will be part of your final takehome (thinking of 6.2, 6.3).
- Exam corrections?
- Your homework for chapter 6 will be handed with the final as a take-home portion.
- Your sigmacs have not been graded yet. Sorry about that.
- If your DEs are rusty, or you've never seen them, you might want to check out Andy's 'DEs in a Day' page!
- I began by noting that there's a difference between open- and
closed-Newton-Cotes rules -- whether we use the endpoints or not. This
can be a big deal in integration of improper integrals. Midpoint is
open, trapezoidal is closed.
- Simpson's rule is just a weighted average of midpoint and
trapezoidal, with weights cleverly chosen to cancel exactly in the case
of integrating a quadratic (giving us exact answers when integrating
quadratics -- midpoint and trapezoidal give exact answers when
integrating linear functions, and midpoint is generally twice as good
as trapezoidal, having half the error).
It turns out, however, that the degree of precision of Simpson's is even better -- it gets cubics right! That's because the error term is proportional to the fourth derivative.
- If our interval of integration is too big, then we cannot expect
just a few values (one or two or three values, as used in our elementary
rules: midpoint, trapezoidal, or Simpson's) to do a good job of
approximating a definite integral.
- So we split up the integral, by partitioning the interval of
integration, and we glue together our elementary rules into a
"composite" scheme.
- When we're proving convergence of Riemann sums to the integral,
etc. in calculus class, we frequently use equally sized sub-intervals;
but for real work, we let the function inform us about the proper
partition.
This is called "adaptive integration", because we let the partition adapt to the function -- where the function is highly active, we take a finer partition; where it is boring, and relatively straight, we can take very big steps.
- The error formulas for the simple rules guide us: if a method has
order of convergence 2 (the error term is proportional to \(h^2\)),
then by cutting the step-size in half we might expect to cut our error
by a quarter (the square of a half).
Details of the scheme, and the error analysis leading to adaptive quadrature in the case of the trapezoidal method, are on page 171.
- And so we use this idea to deduce an adaptive scheme for
integration, which I have captured here in
Mathematica. You were working at the end of the hour to capture it
in Octave, and I hope that you are making progress on that (it's one of
your homeworks to create such a function).
- I begin with a little overview of a couple of the key points of
section 6.1, which we're going to skip over (to get right to solving
these things).
- What is the general method for solving differential equations?
- Here's an example:
\[
y'=\frac{t}{y}
\]
How do we use the general method in this case?
(Compare to the method of "separation of variables", on page. )
- Notice that there are infinitely many solutions! In order
to pin it down, we'll need to add an initial condition.
- Galileo's pendulum (p. 203), and two ways to approximate a
solution.
"Galileo asserts that the period of a swinging pendulum (the time it takes to swing one way and back) is independent of the amplitude of the swing...."
One way is to approximate the DE!
- Now on to the "Taylor methods": so what's the other way to approximate?
- Use the real DE to approximate the solution
- But rather than continuous "steps", we're going to take
small step sizes, and approximate using the Taylor series expansion:
\[
f(x_0+h) = f(x_0) + hf'(x_0) + O(h^2)
\]
and we're going to start by simply throwing away the \(O(h^2)\)
stuff, resulting in Euler's method in (6.2.2):
\[
y_{i+1}=y_{i}+\dot{y}(t_{i},y_{i})\cdot(t_{i+1}-t_{i}).
\]
or, with equal step-size \(h\),
\[
y_{i+1}=y_{i}+h\cdot\dot{y}(t_{i},y_{i}).
\]
I hope that you realize that we're just shooting a tangent line! :)
- Let's look at some examples:
- Mathematica code for the example of page 209 (pdf).
- Exercises #1-3ac
- Writing second order ODEs as first order systems (p. 212)
- Our Textbook: Tea Time Numerical Analysis (Leon Q. Brin)
- A paper my dad and I wrote about integration techniques: including lots of details!:)
- Some notes
- An interesting
alternative derivation technique. Here, for example, is a
four-point forward difference formula for the derivative, obtained
using the Newton polynomial interpolator (notice, however, that the \(x\)
values in the numerator should actually be \(y\) values -- function values):
But the idea is really swell, and I am delighted by this method!
- An example of cubic splines: how do we intepolate two
points and their slopes using a cubic? We can derive the so-called
"Hermite spline" as a variant of Lagrange interpolation, featuring "get
out of the way" functions!
Here's another picture of how we might build one of these.
- Homework solutions:
- 1.1 (Mathematica)
- 1.2 (Mathematica)
- 1.3 (Mathematica)
- 2.1 (Mathematica)
- 2.2 (Mathematica)
- 2.3 (Mathematica)
- 2.4 (Mathematica)
- 3.2 (Mathematica)
- 3.3 (Mathematica)
- 4.1 (Mathematica)
- My Mathematica code for Section 2.6. Muller code, too.
- My Octave Project for 2.6 homework.
- Octave project for 2.5
- Divided Differences. I use these in my Muller code, and they will prove useful when it comes to interpolating polynomials.
- Author's octave code for these complex fractal convergence diagrams
- Newton's method in Mathematica
- fixed point with Aitken's acceleration in Mathematica
- Some nice notes (and code) snippets on fixed point iteration, including these acceleration techniques.
- fixed point in Mathematica
- Check out how my bisection works, and how it is applied to arcsine....
- Various sins....