Day 6, Mat360: Numerical Analysis

Last time

Next time

Today:

Announcements:
- If you still have homework to submit up to section 1.3, please do so now.
- How about if your 2.1 homework is due Wednesday, and your 2.2 homework due Friday? Two different methods of root-solving, and both involve creating a bit of code -- you were asked to do it in Octave, but you can do it in other software, if desired -- and then checking that it works. Please scan it and email, or drop it off at my office.
- On Wednesday we'll cover 2.3 -- please read that section in advance. It includes a description of some neat tricks to accelerate convergence.
- Irony: this article appeared to day in the Guardian: The big idea: what if every little thing you do changes history? We like to pretend that momentous events have big causes, but science says otherwise:
  "Chaos theory is a definitively established scientific truth about how complex systems are sensitive to tiny changes - that small flukes can have enormous effects. It's not really a theory; it's been proved over and over again. It's why we can't predict the weather more than a week in advance. If our calculations are off by even a tiny amount, all bets are off."
Questions on anything?
Let's recall what we did last time:
- We talked about fixed point iteration, and the power of that cosine button!
- A little about when to expect convergence to a fixed point, when you can't expect it, and the possible chaos that can ensue (literally!).
Today:
- We're going to first wrap up the discussion of fixed point (Danny had an idea for the farksine function that we'll explore a little), the idea being that if we want to find \(\arcsin(0.7)\), we seek \(x\) such that \[ \sin(x)=0.7 \] That is, we seek a root of \[ \sin(x)-0.7=0 \] Alternatively we might consider using fixed point iteration, seeking a value \(x\) such that \(f(x)=x\), in which case the function \[ f(x)=x - \sin(x) + 0.7 \] works; but Danny suggested an alternative (and our text discusses the various options for an example function): \[ f(x)=x - (\sin(x) + 0.7)/a \] where \(a\) is some number (e.g. 2). Perfectly legitimate. Let's look at the consequences of that choice.
- First of all, I didn't show you my fake arcsin (farksine) using FPI. That's a good place to start. We'll need to load FPI first....
- There was also a question about when the chaotic "logistic equation" splits from a two-cycle into a four-cycle, and I can explain how to figure out where that happens that pretty easily.
  - The first bifurcation point: \(a=3\)
  - Compositions of maps lead to higher order bifurcation points.
  - Period three implies chaos..., and we can find a three-cycle in this system.
Now let's have a look at some of the homework problems. I'll want to look at a couple, and you'll probably want to look at a couple, too!
- 1.2:
  - Some Mathematica solutions
  - I want to look at #14, in particular; I have a little objection to part d....
  - Also #27: what's the upshot?
  How about you look over "my homework", and fix anything that you want? Then get it to me by Wednesday.
- 1.3: What have you got?
  I sense some real trouble on this one, and want to help you get over any rough spots; if you want to continue to work on this, get it to me by Wednesday.
- 2.1: What have you got? I think that you'll find both of these first Chapter 2 homeworks a little easier than the 1.2 and 1.3 homeworks
- 2.2: What have you got?

Links:

Our author's cobwebbing animation.
fixed point in Mathematica
Check out how my bisection works, and how it is applied to arcsine....
Various sins....

Website maintained by Andy Long. Comments appreciated.

Is it 0.31456656061611776?

Is it 0, 1, or something else?