Lecture note Data visualization - Chapter 30

This chapter presents the following content: Cubic spline interpolation, multidimensional interpolation, curve fitting, linear regression, polynomial regression, the polyval function, the interactive fitting tools, basic curve fitting, curve fitting toolbox, numerical integration.
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Lecture note Data visualization - Chapter 30Lecture30RecapCubicSplineInterpolationMultidimensionalInterpolationCurveFitting LinearRegression PolynomialRegression ThePolyvalFunction TheInteractiveFittingTools BasicCurveFittingCurveFittingToolBox NumericalIntegration ExampleHere’sanotherexample,usinga functionhandleandananonymous function,insteadofdefiningthe functioninsidesinglequoteFirstdefineananonymous functionforathirdorder polynomial fun_handle=@(x)x.^3+20*x.^2 5Nowplotthefunction,toseehow itbehaves.TheeasiestapproachisExampleContinued….SolvingDifferentialEquationNumericallyContinued….Eachsolverrequiresthefollowingthreeinputsasa minimum: Afunctionhandletoafunctionthatdescribesthefirst orderdifferentialequationorsystemofdifferential equationsintermsoftandy Thetimespanofinterest AninitialconditionforeachequationinthesystemThesolversallreturnanarrayoftandyvalues: [t,y]=odesolver(function_handle,[initial_time,final_time], [initial_cond_array])FunctionHandleInputContinued….SolvingtheProblemContinued….Continued….Whentheinputfunctionorsystemoffunctionsisstored inanMfile,thesyntaxisslightlydifferentThehandleforanexistingMfileisdefinedas @m_file_nameTosolvethesystemofequationsdescribedintwofunwe usethecommand ode45(@twofun,[1,1],[1,1])Thetimespanofinterestisfrom1to1,andtheinitial conditionsareboth1SolvingHigherOrderDifferentialEquationsContinued….NowallweneedtodoiscreateanMfilefunctiontouse inoneoftheodesolversThefunctionshouldhavetwoinputs,whicharetypically calledtandyThevariabletistheindependentvariable,andthe variableyisanarrayofdependentvariablesInthisexampley(1)correspondstotheyusedinthe handformulation,andy(2)correspondstozThefunctioncontainingthesystemofequationsshould looklikethis: functiondydt=twoeq(t,y)Continued….OncethesystemofequationsisdefinedinafunctionM fileitisavailabletouseasinputtoanodesolverForexample:iftherangeoftimeisdefinedas1to+1 andtheinitialconditionsaredefinedasy=0andz=0, thenthecommandbecomes ode45(@twoeq,[1,1],[0,0])whichgivestheresultsAproblemwherethestartingvaluesareknowniscalled aninitialvalueproblemBoundaryValueProblemsbvp4cfunctionisusedtosolveboundaryvalueproblemsThebvp4cfunctionrequiresthreeinputs: Afunctionhandletothesystemofode’stobesolved Afunctionhandletoafunctionthatsolvesfortheresidual valuesofthefunction AsetofguessesfortheinitialconditionsContinued….Thefirstfunctionhandleisexactlythesameasweused fortheodesolversetoffunctionsItshouldcontaintheequationsforthederivativesof interestandtheresultsmustbeacolumnvectorTosolvetheproblemaguessismadefortheinitialvalue ofallthederivatives,thentheprogramcheckstoseehow itdidbycomparingthecalculatedboundaryvalueswith theactualvaluesForexample,if: att=1,y=0and att=1,y=3