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....

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Updated on 01/29/2024 19:48:30

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