CS 170: Computing for the Sciences and Mathematics

1. CS 170:Computing for the Sciences and Mathematics Euler’s Method

2. Administrivia • Last time (in P265) • Basics of Algorithms and computation • Today • More Maple • Euler’s method • HW #6 Due! • HW #7 assigned

3. Euler’s method • Simplest simulation technique for solving differential equation • Intuitive • Some other methods faster and more accurate • Error on order of ∆t • Cut ∆tin half  cut error by half

4. Euler’s Method • Euler’s Method is a simulation technique. • Example: unconstrained growth dP/dt = 0.1P with P0 = 100 • P(t) = P(t - ∆t) + growth(t)∆t (new = old + change) • growth(t) is dP/dt = 0.1P(t - ∆t) (change = r*Pold)

5. Example • dP/dt = 0.1P with P0 = 100 and ∆t= 8

6. Approximating Unconstrained Growth initialize simulationLength, population, growthRate, ∆t numIterations  simulationLength / ∆t fori going from 1 to numIterations do the following: growth  growthRate * population population  population + growth * ∆t t  t + ∆t

7. Approximating Unconstrained Growth (Better) initialize simulationLength, population, growthRate, ∆t numIterations  simulationLength / ∆t fori going from 1 to numIterations do the following: growth  growthRate * population population  population + growth * ∆t t  i*∆t

8. Error • Analytical solution: P = 100 e0.10t • After 100 time units: P = 2,202,647 • Smaller ∆t better estimation • ∆t = 1 estimate of P = 1,378,061 • ∆t = 0.25 estimate of P = 1,947,808 • ∆t = 0.01 estimate of P = 2,191,668 • ∆t = 0.005 estimate of P = 2,197,149

9. HOMEWORK! • Homework 7 • Maple Tutorial 2 • Complete the worksheet’s questions and turn it in • Also complete Module 5.2’s Project 1 on page 167. • Due 11/1/2010 • Thursday’s Class in P115