chaos. state-of-the-art calculator,1974 (about $400)
TRANSCRIPT
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ChaosChaos
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State-of-the-artcalculator,1974
(about $400)
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State-of-the-artcalculator, 2013
(about $40)
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How does the `solve’ function work?
Research (looking in the manual) shows thatit employs something called `the secant method’.
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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.
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x
f(x)
Find the point(s) at which f(x)=0
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x0
f(x0)
First guess: x0
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x0
f(x0)
Second guess: x1
x1
f(x1)
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Draw the secant and locate x2
x2
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Draw another secant and locate x3
x2 x3x1
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Does this always work?
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Showing the success ofthe secant method formany different pairs ofinitial guesses:
x0
x1Colour this pointaccording to howlong it takes toget to the right answer.
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Complex Numbers
What is the solution to
x2 = -1?
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Complex Numbers
-i 0 i
-i
0
i
-1 0 1
0 0 0
1 0 -1
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Complex Numbers
i (0.5+i)
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Complex Numbers
Now the equation
x3 - 1 = 0
has 3 roots:
x=1, x=0.5+√3i/2, x=0.5-√3i/2
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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.
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Complex Numbers
We flatten the tesseract by one of several strategies:
1. Let x0 be 0, choose x1 freely.
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Strategy 2:
Choose x0 freely, let x1 be very close to x0.
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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
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Newton’s Method
x0
f(x0)
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Newton’s Method
x0 x1
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Newton’s Method
x0 x1
f(x1)
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Newton’s Method
x2
x1
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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.
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One more equation to solve by Newton’smethod:
(x+1)(x-1)(x+ß)=0
…where ß is our first guess.
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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.
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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.
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When can chaos arise?
Trying to get two non-linear programs to converge:
x
y