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More Accurate Rate Estimation

CS 170:Computing for the Sciences and

Mathematics

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Administrivia

Last time (in P265) Eulers method for computation

Today Better Methods

Simulation / Automata

HW #7 Due!

HW #8 assigned

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Eulers method

Simplest simulation technique for solvingdifferential equation

Intuitive Some other methods faster and more accurate

Error on order of t Cut t in half cut error by half

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Eulers Method

tn = t0 + n t

Pn = Pn-1 + f(tn-1, Pn-1) t

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Runge-Kutta 2 method

Euler's Predictor-Corrector (EPC) Method

Better accuracy than Eulers Method

Predict what the next point will be (with Euler)

then correct based on estimated slope.

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Concept of method

Instead of slope of tangent line at (tn-1, Pn-1), wantslope of chord

For t = 8, want slope of chord between

(0, P(0)) and (8, P(8))

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Concept of method

Then, estimate for 2nd point is ?

(t, P(0) + slope_of_chord * t)

(8, P(0) + slope_of_chord * 8)

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Concept of method

Slope of chord average of slopes of tangents atP(0) and P(8)

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EPC

How to find the slope of tangent at P(8) when we donot know P(8)?

Y = Eulers estimate for P(8)

In this case Y = 100+ 100*(.1*8) = 180 Use (8, 180) in derivative formula to obtain estimate

of slope at t = 8

In this case, f(8, 180) = 0.1(180) = 18

Average of slope at 0 and estimate of slope at 8 is 0.5(10 + 18) = 14

Corrected estimate of P1 is 100 + 8(14) = 212

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Predicted and corrected estimation of(8, P(8))

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Runge-Kutta 2 Algorithm

initialize simulationLength,population, growthRate, t

numIterationssimulationLength / t

fori going from 1 to numIterations do the following:growth growthRate * population

Ypopulation + growth * t

t i*t

population population+ 0.5*( growth + growthRate*Y)

estimating next point (Euler)

averaging two slopes

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Error

With P(8) = 15.3193 and Euler estimate = 180,relative error = ?

|(180 - P(8))/P(8)| 19.1%

With EPC estimate = 212, relative error = ? |(212 - P(8))/P(8)| 4.7%

Relative error of Euler's method is O(t)

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EPC at time 100

t Estimated P Relative error

1.0 2,168,841 0.015348

0.5 2,193,824 0.004005

0.25 2,200,396 0.001022

Relative error of EPC method is on order of O((t)2)

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S I M U L A T I O N

CS 170:Computing for the Sciences and

Mathematics

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Computer simulation

Having computer program imitate reality, in order tostudy situations and make decisions

Applications?

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Use simulations if

Not feasible to do actual experiments

Not controllable (Galaxies)

System does not exist

Engineering Cost of actual experiments prohibitive Money

Time

Danger

Want to test alternatives

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Example: Cellular Automata

Structure

Grid of positions

Initial values

Rules to update at each timestep

often very simple

New = Old + Change

This Change could entail a diff. EQ, a constantvalue, or some set of logical rules

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Mr. von Neumanns Neighborhood

Often in automata simulations, acells change is dictated by the state

of its neighborhood Examples: Presence of something in the neighborhood

temperature values, etc. of neighboring cells

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Conways Game of Life

The Rules

For a space that is 'populated': Each cell with one or zero neighbors dies (loneliness)

Each cell with four or more neighbors dies (overpopulation) Each cell with two or three neighbors survives

For a space that is 'empty' or 'unpopulated

Each cell with three neighbors becomes populated

http://www.bitstorm.org/gameoflife/

http://www.bitstorm.org/gameoflife/http://www.bitstorm.org/gameoflife/

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HOMEWORK!

Homework 8 READ Seeing Around Corners

http://www.theatlantic.com/magazine/archive/2002/04/seeing-around-corners/2471/

Answer reflection questions to be posted on class site

Due THURSDAY 11/4/2010

Thursdays Class in HERE (P265)

http://www.theatlantic.com/magazine/archive/2002/04/seeing-around-corners/2471/http://www.theatlantic.com/magazine/archive/2002/04/seeing-around-corners/2471/http://www.theatlantic.com/magazine/archive/2002/04/seeing-around-corners/2471/http://www.theatlantic.com/magazine/archive/2002/04/seeing-around-corners/2471/http://www.theatlantic.com/magazine/archive/2002/04/seeing-around-corners/2471/http://www.theatlantic.com/magazine/archive/2002/04/seeing-around-corners/2471/http://www.theatlantic.com/magazine/archive/2002/04/seeing-around-corners/2471/