ChaosChaos

State-of-the-artcalculator,1974

(about $400)

State-of-the-artcalculator, 2013

(about $40)

How does the `solve’ function work?

Research (looking in the manual) shows thatit employs something called `the secant method’.

Using the secant method to solve f(x)=x3-1=0:

Guess a solution x0

Is it right?Guess a second solution x1

Is it right?Construct a third guess:

x2 =x1 - (x0-x1)/(f(x0)-f(x1))

(This is where the secant through the first twopoints cuts the x axis)

Repeat indefinitely.

x

f(x)

Find the point(s) at which f(x)=0

x0

f(x0)

First guess: x0

x0

f(x0)

Second guess: x1

x1

f(x1)

Draw the secant and locate x2

x2

Draw another secant and locate x3

x2 x3x1

Does this always work?

Showing the success ofthe secant method formany different pairs ofinitial guesses:

x0

x1Colour this pointaccording to howlong it takes toget to the right answer.

Complex Numbers

What is the solution to

x2 = -1?

Complex Numbers

-i 0 i

-i

0

i

-1 0 1

0 0 0

1 0 -1

Complex Numbers

i (0.5+i)

Complex Numbers

Now the equation

x3 - 1 = 0

has 3 roots:

x=1, x=0.5+√3i/2, x=0.5-√3i/2

Complex Numbers

The secant method doesn’t take us to the complex roots unless our initial guesses are complex.

But now our initial two guesses have four components.

Complex Numbers

We flatten the tesseract by one of several strategies:

1. Let x0 be 0, choose x1 freely.

Strategy 2:

Choose x0 freely, let x1 be very close to x0.

Newton’s Method

To find the roots of f(x) = 0, construct the series {xi}, where

xi+1 = xi – f(xi)/f/(xi)

(and x0 is a random guess)

Example: f(x) = x3 -1, so f/(x) = 2x2

x0 = 2, so x1 = 2 –(23-1)/(2*22) = 2 – 7/8 = 1.125

and x2 = 1.125-(1.1253-1)/(2*1.1252) = 0.9575

Newton’s Method

x0

f(x0)

Newton’s Method

x0 x1

Newton’s Method

x0 x1

f(x1)

Newton’s Method

x2

x1

Apply Newton’s method to

z3-1=0

which in the complex plane has threeroots.

Let the x and y axes represent the real andimaginary components of the initial guess.

Colour them according to which root theyreach, and when.

One more equation to solve by Newton’smethod:

(x+1)(x-1)(x+ß)=0

…where ß is our first guess.

We recognise the Mandelbrot set, which can alsobe generated by a simpler process:

Repeat the calculation

zn = z2n-1+z0

until zn > 2 or you give up. Colour in the complex point z=x+iy according to how long this took.

Characteristics of Chaos

Two ingredients-- non-linearity and feedback --can give rise to chaos.

Chaos is governed by deterministic rules, yetproduces results that can be very hard to predict.

Images of chaotic processes can display a highlevel of order, characterised by self-similarity.

When can chaos arise?

Trying to get two non-linear programs to converge:

x

y