- If you have homework to submit up to section 2.2, please do so now.
- Section 2.3 homework is due Friday.
- Section 2.1 homework is returned:
- On #2, you also need to assert (or show) continuity for the IVT.
- We derived Newton's method graphically, as a root-finding method; we use tangent lines to get successively closer to a root (we hope....).
- When we solved for our next approximation, we realized that we
were basically doing FPI
\[
x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}
\]
Furthermore, this FPI function has 0 derivative at the fixed point, making it a quadratic (or better!) method.
- But (and this is a serious but): Newton's method requires a
derivative calculation, and that can be impossible or expensive.
- So, as a quick cheat, we can spare ourselves a derivative
calculation at the cost of slower convergence (with \(\alpha =
\phi \approx 1.618\)), with
\[
x_{n+1}=x_n-\frac{f(x_n)(x_n-x_{n-1})}{(f(x_n)-f(x_{n-1}))}
\]
Note that this requires two starting values; but, although the convergence is slower, we only need to compute one additional function evaluation at each step: we can reuse everthing else.
And one question was whether we could use Aitkens, and perhaps Steffensen, to speed this up....
- I also mentioned that one downside of these methods is that, if we start with a real value, we can't find any complex roots. So that's where we go today!
- We'll spend the class period with Octave, and attempt to recreate
the beautiful images of section 2.5.
Our author says "Second, the patterns of colors are extremely intricate, even fractal in nature." Yes, we're producing fractals: "a world within a world...."
- The idea is this:
- When a function has multiple roots, it's of interest to see to which root particular values of \(x_0\) will converge (if indeed we get convergence from a particular initial value).
- And, since we're looking for complex roots as well as real ones, we search from starting values in the complex plane.
- So we start from a grid of values in the complex plane, and we iterate for a certain number ("maxits") of iterations, and see if either we're converging (and then to which root), or diverging.
- Now: let's take a look at the example on p. 79. We compare three
methods:
- Newton's method
- Seeded Secant method
- Steffensen's method
- In particular, however, I want to try to improve
on our author's idea: he uses a rather poor fixed
point iteration function to "test" Steffensen's
method: what if we use Steffensen's with Newton's FPI
function? What will we see? [Spoiler alert:
- For this exercise, you may use my version of
Steffensen's method in Octave (modified from Brin's).
You might model your approach with Steffensen's after that for Newton's, since they use the same inputs....
- Then, if we have time, we'll check in on p. 81: but I'm struggling
to reproduce that one:
Let's talk through the complex analysis part of this.
- Octave project for 2.5
- Author's octave code for these complex fractal convergence diagrams
- Our Textbook: Tea Time Numerical Analysis (Leon Q. Brin)
- 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....