计算方法Matlab 课后习题

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Math6111-Numericalmethods

Coursework2

Due:14December2011

Youmustsubmitelectronically(viaBlackboard)acopyoftheMatlabm- lesrequiredbythetasksbelow.Thedeadlineis12:00Wednesday,December14th2011.

Penaltiesforlatesubmission:10%ofthetotalmarksfortheassignmentperworkingdayaftertheassignmentisdue(whereaworkingdayisa24hourperiodstartingformoriginalhandindeadline,skippingdaysonwhichtheUniversityisclosed),forupto5workingdays.Nomarkswillbeobtainedforsubmissionsthatarelaterthan5workingdays.

AssessmentCriteria:Assessmentwillprimarilybebasedonthesuccessfulsolutionoftheproblemsandyourmethodofsolution,whichshouldbejusti ed.Youwillalsobeassessedontheclarityofyourcode.PleaseensurethatallofitiswritteningoodMatlabstyle,isproperlycommentedandisrobust;e.g.,allinputargumentsshouldbeexplainedandchecked.Figuresproducedshouldbeclearlypresentedwithlabels,titles,etc.

Plagiarism:Yourareencouragedtodiscussyourworkwithyourcolleaguesingeneralterms,butyoursubmittedworkMUSTbeyourown.

1Photorefractiveliquidcrystalcells

Aphotorefractiveliquidcrystalcellis

formedbytheinteractionofaphotoconductinglayerwithastandardliquidcrystalcell,asshownbyFigure1.Thetransmissionofopticalbeamsthroughthecellwillinduceanexchangeofenergy,observedasachangeinamplitudes,fromtheincidentbeamsintoadiscretesetofNoutputbeamswhereNisde nedbytheseparationintheinputbeams.Thisprocessisknownasopticalcoupling.FurtherdetailsaregiveninapaperbyDalyetal,AppliedPhysicsB.95:3589–596(2009).Figure1:Photorefractiveliquidcrystalcell,aliquidcrystallayerissandwichedbetweentwoalignmentlayersPI,apolymer,andPVK,aphotoconductor.Theinteractionofthephotoconductorwiththeliquidcrystalcellcausesenergyfromtheincidentbeamstobecoupledintoadiscretesetofoutputbeams.

Yourtaskisto ndtheamplitudesofthedifferentwavesastheypropagatethroughthephotorefractiveliquidcrystalcell.YoumayuseanyinbuiltMatlabtool,butmustjustifyyourchoiceofalgorithmindetail,andcheckthatitworks.

2Opticalcouplingequations

Wewritetheopticalamplitudesy(z)asavector eldofsizeN(i.e.,foreverylocationz,y(z)isavectorofsizeN).Wewillonlyconsideracoordinaterangeof0≤z≤5.Foreachcomponentofy(z),the

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variationintheamplitudeasthewavepassesthroughtheliquidcrystalcellisgivenby

dy1(z)=ik1y1(z)+if(z)[y2(z)],dz

dyn(z)=iknyn(z)+if(z)[yn 1(z)+yn+1(z)],dz

dyN(z)=ikNyN(z)+if(z)[yN 1(z)],dz

√n=2,...,N 1,wherei=andthevectorkwithcomponentsknisde nedas

nπkn=10sin.N+1(2)

Iff(z)=C,aconstant,thenequation(1)canbesolvedasaneigenvalueproblem.WewritethesolutionasalinearcombinationofNvector eldsy(k)(z),tobedetermined,as

y(z)=A(k)y(k)(z)(3)

k

andthenusethesubstitution

y(k)=v(k)eiλ(k)z(4)

toobtainequationsfortheconstantvectorsv(k).Thebehaviourofthewavesdescribedbyequation(1)nowbecomestheeigenvalueproblem

λ(k)v(k)=Mv(k),(5)

wherethematrixMistridiagonalwiththediagonalentriesbeingthevectorkandthe rstoff-diagonalentriesbeingC.Forexample,whenN=2wehave

π (k) (k)v10sinC v1λ(k)1=(6)2π(k)(k).C10sinv2v2

Henceλ(k)aretheeigenvaluesandv(k)theeigenvectorsofthematrixM.

ToreconstructthesolutionyweneedtheweightingA(k)ofeachterminthesplitgivenbyequation(3),whichisrelatedtothechoiceofinitialdatay(0)≡y0.Iftheeigenvectorsv(k)formthecolumnsofthematrixVthentheweightsA(k)canbefoundasthecomponentsofthevectorAsatisfyingthelinearsystem

VA=y0.

√(7)ForthesimplecasewhereN=2theeigenvaluesareλ(k)=10±Cwitheigenvectorsv=(1,±1)T.

TWiththeinitialconditionsy1(0)=Randy2(0)=StheweightingsareA=1

((R+S),(R S)),

whichgivestheanalyticsolution

√1y1(z)1(10√2+C)iz11e+(R S)e(102 C)iz.(8)=(R+S)1 1y2(z)22

Inarealphotorefractiveliquidcrystalcellf(z)=C.Ifthisisthecasethenequation(1)mustbesolvedasaninitialvalueproblem.Inparticularwewillalwaysusetheinitialconditiongivenbythesincfunction,

yn(0)=sinc(n nin),(9)

whereninisgivenbytheMATLABcommand:floor(N/2).

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3Tasks

1.Solveequation(1)usingMATLAB’seigfunctionforN=2andcomparewiththeanalyticso-lution,equation(8),forR=1,S=0andC=1.Yourcodeshouldproduceasingle g-ureoutputpidedintothreesubplots,oneshowingtheanalyticsolutionfortheopticalintensityIn(z)=|yn(z)|2,onethesolutioncalculatedusingMATLAB’seigfunctionandthe naloneshowingtheabsolutedifferencebetweenthetwoasafunctionofthecoordinatez.

2.Solveequation(1)forN=4,10,20andf(z)=1usingbothaneigenvaluemethodandyourchoiceofIVPsolvingmethod.ForeachvalueofNplottheopticalintensityIn(z)asafunctionofthecoordinatezforn=nincomputedusingboththeeigenvaluemethodandtheIVPsolution,andthedifferencebetweenthetwo.YoushouldinadditionshowhowthedifferenceconvergesastheaccuracyofyourIVPsolutionisimprovedinanappropriateway.

http://inganappropriatemethodsolveequation(1)forf(z)=1+αsin(2πz/5)exp( (z 5/2)2)withN=4andαintherange[0,1/2].Comparetheresultsagainstusingtheaveragevalue,f(z)=1.Youshouldproduceasingle gurewithtwosubplots,oneshowingthemaximumdifferencebetweentheamplitudesy(z)computedbythetwomethodsfordifferentvaluesofα,theothershowingthemaximumdifferenceoftheintensityI(5)http://mentontheaccuracyofthef(z)=1approximation;inwhatsense,ifatall,isitaccurate?

3.1Summaryandassessmentcriteria

YoushouldsubmittheMatlabcodeelectronicallyasnotedabove.

Theprimaryassessmentcriteriawillbethechoiceandjusti cationofthealgorithmimplemented.Thecorrectnessofthealgorithmsmustbeclear,andmustbedemonstratedbythoroughtestingwhichshouldbecarefullyexplained.Thejusti cationsofyouralgorithmsmaybemanygrounds-exampleswouldincludeef ciency,conceptualorcomputationalsimplicity,ortheeaseofimplementationinMatlab-butmustbedetailedforeachstageofyourchosenmethod.Thejusti cationshouldbegivenincommentswithintheMatlabcode,preferablyatthestartofthemasterscript.

Theclarityandrobustnessofyourcodeandtheclarityofyouroutputwillalsobeassessed,asincoursework1.Thecommentswillbeexaminedcloselyforjusti cationsofth …… 此处隐藏：3743字，全部文档内容请下载后查看。喜欢就下载吧 ……