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More Accurate Rate Estimation

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CS 170: Computing for the Sciences and Mathematics More Accurate Rate Estimation * * * * * * * * * * * * Administrivia Last time (in P265) Euler s method for ... – PowerPoint PPT presentation

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Title: More Accurate Rate Estimation


1
CS 170 Computing for the Sciences and Mathematics
  • More Accurate Rate Estimation

2
Administrivia
  • Last time (in P265)
  • Eulers method for computation
  • Today
  • Better Methods
  • Simulation / Automata
  • HW 7 Due!
  • HW 8 assigned

3
Eulers 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

4
Eulers Method
  • tn t0 n ?t
  • Pn Pn-1 f(tn-1, Pn-1) ?t

5
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.

6
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))

7
Concept of method
  • Then, estimate for 2nd point is ?
  • (?t, P(0) slope_of_chord ?t)
  • (8, P(0) slope_of_chord 8)

8
Concept of method
  • Slope of chord average of slopes of tangents at
    P(0) and P(8)

9
EPC
  • How to find the slope of tangent at P(8) when we
    do not know P(8)?
  • Y Eulers estimate for P(8)
  • In this case Y 100 100(.18) 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

10
Predicted and corrected estimation of (8, P(8))
11
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
    growthRateY)

estimating next point (Euler)
averaging two slopes
12
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)

13
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)

14
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 d1, d1 rateP0Dt
  • P2 P1 d2, d2 rateP1Dt
  • P3 P2 d3, d3 rateP2Dt
  • d4 rateP3Dt
  • P1 (1/6)(d1 2d2 2d3 d4)
  • error O(Dt4)

15
CS 170 Computing for the Sciences and Mathematics
  • Simulation

16
Computer simulation
  • Having computer program imitate reality, in order
    to study situations and make decisions
  • Applications?

17
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

18
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

19
Mr. von Neumanns Neighborhood
  • Often in automata simulations, a cells change
    is dictated by the state of its neighborhood
  • Examples
  • Presence of something in the neighborhood
  • temperature values, etc. of neighboring cells

20
Conways 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 cells 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).

21
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/

22
HOMEWORK!
  • Homework 8
  • READ Seeing Around Corners
  • http//www.theatlantic.com/magazine/archive/2002/0
    4/seeing-around-corners/2471/
  • Answer reflection questions to be posted on
    class site
  • Due THURSDAY 11/4/2010
  • Thursdays Class in HERE (P265)
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