- Our first exam is next week (Wednesday), and will cover everything through 2.6, which we'll start today.
- Section 2.4 homework is due Wednesday.
- Section 2.2 homework is returned:
- Please explain answers -- "yes" or "no" on continuity or differentiability is not enough!
- Your homework for section 2.6 will make up the "take-home" portion
of your exam. Please work on this on your own, although
you can ask me questions.
Homework pp. 93--, #1-5, 6, 8, 11, 13, 14, 16, 22 (use your Muller code), 23
- We spent the class period trying to recreate the beautiful images
of section 2.5. Fractals: "a world within a world...."
- 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.
- We were fairly successful with the first set of examples.
I'm not going to spend time on Section 2.7: it's basically about "bracketing" the iteration, and makes a lot of sense -- but it's not really novel or particularly interesting. Basically we cobble together some ideas (e.g. bisection + Newton), to prevent failure of convergence (which is very important, of course). Once you've got a root in a box, you really don't want to let it out.
- I want to start, however, with a last nod to the fractal convergence stuff:
- In particular, however, I want to try to improve
on our author's choice of the FPI function for Steffensen's: 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:
- In particular, however, I want to try to improve
on our author's choice of the FPI function for Steffensen's: 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:
- Now on to Section 2.6: Mistakes, he's made a few.... The most
serious is in his formula for the result of synthetic division,
p. 87.
That one I've got to let him know about!:)
\[ a_{n}(\cdots a_{n}(a_{n}(a_{n}t+a_{n-1})+a_{n-2})+\cdots+a_{1}) \] becomes \[ a_0+t(a_{1} + t(a_{2}+t(a_{3}+t(a_{n}+\cdots+t(a_{n-1}+t(a_{n}))\cdots)) \]
Let's work through the example on p. 87.
- Turns out we can just keep going using synthetic division, and get
the derivative at \(t\) as well! Very cool, and easy.
Our job: to write the code to do this, and then add it into Newton's method to compute all the roots of an arbitrary polynomial....
- His discussion of the proper way to compute roots of the quadratic
(Crumpet 18, p. 90) is part of the code you'll be writing.
- Exercises (we'll use these to motivate discussion):
- #7 and 9 -- Horner's method (synthetic division), and Newton's using synthetic division
- Muller's method generalizes the secant method, which uses two
points and a straight line to probe the x-axis for a better
estimate of the root: it uses a quadratic, and a parabola.
(In the exercises you'll extend this idea to a cubic!)
I'll gripe a little about our author's choice for how he represents the algorithm on pp. 91-92 (he's taking some abuse in this section!:). My feeling is that this would be a good time to introduce divided differences. If you look at the formulas on p. 91, you'll see that we re-use some material -- and any time you re-use material, you should avoid re-calculating.
Divided differences are what we use to approximate derivatives, say: \[ DD(x_1,x_0) \equiv \frac{f(x_1)-f(x_0)}{x_1-x_0} \]
Of course, once you have divided differences, you can take divided differences of them (essentially higher-order derivatives):
\[ DD(x_2,x_1,x_0) \equiv \frac{DD(x_2,x_1)-DD(x_1,x_0)}{x_2-x_0} \]
This allows us to write \[ a=DD(p_1,p_2,p_0)=\frac{DD(p_1,p_2)-DD(p_2,p_0)}{p_0-p_1} \] and \[ b=DD(p_1,p_2)-(p_1-p_2)a \]
- Usually after one uses deflation to find roots, one wants to "polish" the roots -- to make sure that errors haven't crept in... Well they will creep in, but we want to minimize them, of course.
- Octave project for 2.5
- My Mathematica code for Section 2.6. Muller code, too.
- 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
- 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....