More Accurate Rate Estimation

CS 170:Computing for the Sciences

and Mathematics

Administrivia

Last time (in P265) Euler’s method for computation

Today Better Methods Simulation / Automata HW #7 Due! HW #8 assigned

Euler’s method

Simplest simulation technique for solving differential equation

IntuitiveSome other methods faster and more

accurateError on order of ∆t

Cut ∆t in half cut error by half

Euler’s Method

tn = t0 + n t

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

Runge-Kutta 2 method

Euler's Predictor-Corrector (EPC) Method

Better accuracy than Euler’s Method

Predict what the next point will be (with Euler) – then correct based on estimated slope.

Concept of method

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

For ∆t = 8, want slope of chord between (0, P(0)) and (8, P(8))

Concept of method

Then, estimate for 2nd point is ? (∆t, P(0) + slope_of_chord * ∆t) (8, P(0) + slope_of_chord * 8)

Concept of method

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

EPC

How to find the slope of tangent at P(8) when we do not know P(8)?

Y = Euler’s 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

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

Runge-Kutta 2 Algorithm

initialize simulationLength, population, growthRate, ∆t

numIterations simulationLength / ∆t

for i going from 1 to numIterations do the following:growth growthRate * populationY population + growth * ∆tt i*∆tpopulation population+ 0.5*( growth + growthRate*Y)

estimating next point (Euler)

averaging two slopes

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)

EPC at time 100

t Estimated P Relative error1.0 2,168,841 0.0153480.5 2,193,824 0.0040050.25 2,200,396 0.001022

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

Runge-Kutta 4

If you want increased accuracy, you can expand your estimations out to further terms.

base each estimation on the Euler estimation of the previous point. P1 = P0 + 1, 1 = rate*P0*t

P2 = P1 + 2, 2 = rate*P1*t

P3 = P2 + 3, 3 = rate*P2*t

4 = rate*P3*t

P1 = (1/6)*(1 + 2*2 + 2*3 + 4)error: O(t4)

SIMULATION

CS 170:Computing for the Sciences

and Mathematics

Computer simulation

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

Applications?

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

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 constant value, or some set of logical rules

Mr. von Neumann’s Neighborhood

Often in automata simulations, a cell’s “change” is dictated by the state of its neighborhood

Examples: Presence of something in the

neighborhood temperature values, etc. of

neighboring cells

Conway’s Game of Life

The Game of Life, also known simply as Life, is a cellular automaton devised by the British mathematician John Horton Conway in 1970. The “game” takes place on a 2-D grid Each cell’s value is determined by the values of an expanded

neighborhood (including diagonals) from the previous time-step. Initially, each cell is populated (1) or empty (0)

Because of Life's analogies with the rise, fall and alterations of a society of living organisms, it belongs to a growing class of what are called simulation games (games that resemble real life processes).

Conway’s Game of Life

The RulesFor 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/

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

Thursday’s Class in HERE (P265)