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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
• 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