CS 170: Computing for the Sciences and MathematicsAdministriviaCS 170 Exam 2CS 170 Exam 2 – Study SuggestionsSlide 5Computational BackgroundComputational Tools: MapleEuler’s methodEuler’s MethodExampleRunge-Kutta 2 methodConcept of methodSlide 13Example: Cellular AutomataMr. von Neumann’s NeighborhoodExample: DiffusionExam 2 Review InfoCS 170:Computing for the Sciences and MathematicsAdministriviaLast timeLab TimeTodayHW#9 DueMonday is Exam 2In this roomGuest Host: Dr. Paul WagnerCS 170 Exam 2Monday, November 15th 4:00pm – 5:25pmFormat:Written/Short AnswerOpen textbook, Open notesCalculators allowedNo computers, cell phones, PDAs, etc.If you can use it to connect to outside (or someone else) Length < 75 minutesCS 170 Exam 2 – Study SuggestionsTextbookYes, its open bookIf you have to spend 20 minutes finding it, the book doesn’t really help!Book exercisesSlides and notes from lectureHomeworks!!Computational BackgroundAlgorithmsSeries of steps followed to accomplish some result. A sequence of steps taking place over time to calculate a result.Algorithms have only a handful of state-mutating manipulationsCore ManipulationsAssignment to variablesSequenceSelectionIterationComputational Tools: MapleMaple is a general-purpose commercial computer algebra systemImplements the basic components of a languageAlso includes powerful libraries for mathematical functionsIn math, used for many direct solution methodsSymbolic differentiation and integrationIn computational science, can also be used to implement numerical simulations.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 halfEuler’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)ExampledP/dt = 0.1P with P0 = 100 and ∆t = 8Runge-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.Error proportional to (∆t)2Concept 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Slope of chord ≈ average of slopes of tangents at P(0) and P(8)Example: Cellular AutomataStructureGrid of positionsInitial valuesRules to update at each timestepoften very simpleNew = Old + “Change”This “Change” could entail a diff. EQ, a constant value, or some set of logical rulesMr. 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 cellsExample: DiffusionDiffusion: movement of some property over timeFire SimulationMovement of fire through a forestSimulation varied parameters to affect the resultsHeat DiffusionDistribution of heat in an area over