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 Intuitive Some other methods faster and more accurate Error on order of ∆t  Cut ∆t in half  cut error by half

Euler’s Method t n = t 0 + n  t P n = P n-1 + f(t n-1, P n-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 (t n-1, P n-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 2 nd 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 = *(.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( ) = 14 Corrected estimate of P 1 is (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 * population Y  population + growth * ∆t t  i*∆t population  population+ 0.5*( growth + growthRate*Y) estimating next point (Euler) averaging two slopes

Error With P(8) = 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  tEstimated PRelative error 1.02,168, ,193, ,200, 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.  P 1 = P 0 +  1,  1 = rate*P 0 *  t  P 2 = P 1 +  2,  2 = rate*P 1 *  t  P 3 = P 2 +  3,  3 = rate*P 2 *  t   4 = rate*P 3 *  t P1 = (1/6)*(  1 + 2*  2 + 2*  3 +  4 ) error: O(  t 4 )

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  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 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

HOMEWORK! Homework 8  READ “Seeing Around Corners”  around-corners/2471/ around-corners/2471/  Answer reflection questions – to be posted on class site  Due THURSDAY 11/4/2010 Thursday’s Class in HERE (P265)