COMS3101ProgrammingLanguages:MATLAB

Lecture4

Fall2013Instructor:IliaVovsha

hCp://www.cs.columbia.edu/~vovsha/coms3101/matlab

LectureOutline

  Review:EHW#1,2

  PloQng(figures)  Datastructures:cellarrays,structs,strings,handles  PracWcalmath:formulaWngandsolvingproblems

  NextLecture:opWmizaWon  NextLecture:FinalprojectoverviewandopWons  NextLecture:advancedfuncWonality

4.2

UsefulRemarks

  Avoidmakingrepeatedcallstobuilt‐infuncWons

  InsteadofaccessinganelementmanyWmes,justdefineanduseavariable:t=M(i,j)

  Someprogramminglanguagesallowyoutoincrementvariablesusingtheoperators{++,‐‐}.ThisisnotanopWoninMATLAB

  FuncWonsthatreturnaBooleanvariable(0/1)couldbeusedasthecondiWon:“ifisprime(N)”insteadof“ifisprime(N)==1”

  CanusevectorsandmatricesincondiWonstoo

4.3

UsefulRemarks

  SoluWonsareposted(average:80+)  MulWplymatrixMbyascalarC:

  Answer:M*C %M.*Cisredundant

  Noneedtoincrementtheloopvariableinsidethefor‐loop  Ontheotherhand,incremenWngisoienessenWalinwhile‐

loops

  Ifyoudefinedavariableforonepurpose,donotredefineitforanotherbelow.Thiscreatesunnecessaryconfusion

4.4

SwitchStatement–Usage

  When?•  Morethan2‐3choices•  Choiceisbasedonacommonexpression

  How?switchswitch_expr %Expressionisascalarorastring

casecase_expr %Match‘case_expr’

statement(s)

case{case_expr1,case_expr2} %Matchanycaseinthearray

statement(s)

otherwise

statement(s)

end

4.5

SwitchStatement–Usage

  Brackets{}arenotrequiredunlessyouwishtoexecutethesamecodeformorethanonecase

  ‘break’statementsareredundant.UnlikeC++forexample,thereisno‘fallingthrough’

  ‘otherwise’isopWonal

4.6

ASillyGame

  Considerthefollowing‘game’:1.  ChooseanyposiWveinteger(1,2,…),callitX2.  IfXisanevenintegeryoudivideXby2

3.  IfXisanoddintegeryoumulWplyXby3andadd1

4.  ConWnuethisprocedureunWlatsomepointyouget1asyournextinteger.Atthatpointstop

  Claim:thegameendsforeveryposiWveinteger  Canyouprovetheclaim?Tryit!

4.7

CollatzConjecture

  DecepWvelysimple

  ProposedbyCollatzin1937,sWllunsolved(!)  “MathemaWcsisnotyetripeforsuchproblems”–PaulErdos

  Thisconjecture/problemhasmanyversionsandnames

  WecanuseMATLABtogeneratesomeplotsandperhapsobtainsomeinsight

4.8

CollatzConjecture–Example

  Problem:writeafuncWonthatgeneratesthesequencefromN(inputparameter)to1,accordingtothegame,andthenploteachpair(sequence#,value)onafigure

  Forexample,ifX=3,thesequenceis{3,10,5,16,8,4,2,1}.Soyoushouldplotthepoints(1,3),(2,10),(3,5)…(8,1)

4.9

CollatzConjecture–Example

  SoluWon:1.Y=N; %IniWalize

2.whileN>1 %orN~=13. ifmod(N,2)==0 %Computesequence

4. N=N/2;

5. else

6. N=(N*3)+1;

7. end

8. Y(end+1)=N;

9.end

10.LY=length(Y);

11.X=1:LY;

12.plot(X,Y,'r.',X,Y,'b:’); %PlotredpointsanddoCedblueline

4.10

BasicPloQng–Figures

  Problem:•  plot(x,y);plot(x,z);replaces1stplotwith2nd

  SoluWon:‘figure’command•  plot(x,y);figure;plot(x,z);

•  figure(1);figurewithhandle#1

  Closefigures:•  Specificfigure:close1•  Allfigures:closeall

  MATLABstoresahandletoeachfigure

4.11

BasicPloQng–Figures

  MulWpleplots:•  plot(x,y);holdon;plot(x,z);holdoff;

  MulWpleplots,samefigure:•  subplot()command

•  figure(1);subplot(2,2,1);

4.12

BasicPloQng–Appearance

  ManyopWons,canmodifyplotsusingtheGUI

  Commands:Wtle(),xlabel(),ylabel(),axis(),legend()  Example:1.figure(1);

2.Wtle(‘test’);3.xlabel(‘quanWty’);ylabel(‘price’);

4.axis([15110]); %AXIS([XMINXMAXYMINYMAX])5.gridon; %Showgridlines

6.xlim([13]); %Changex‐axislimits

4.13

BasicPloQng–Figures

  YouhavecreatedmulWplefigures.Whichoneisthe‘currentfigure’?

  Answer:lastfigureyouclickedon  BeCeranswer:usegcf()togetthehandle  Usefigure(#)tomake‘#’thecurrenthandle  Commands:gcf(),gca(),clf()  Set‘object’properWesusingtheset()command:

•  set(gca,'XTick',[123]) %Toset‘Wcks’onthex‐axis

  Get‘object’properWesusingtheget()command

4.14

Saving&LoadingFigures

  MATLABhasaspecialfigureformat:‘.fig’

  Useopenfig(‘name.fig’)toopenasavedfigure  Savingfigures:useprintcommand–  GeneralForm

–  print -dformat filename –  Example –  print –depsc ‘figure.eps’

  ‘eps’isaformatthatstoresyourimageinavectorizedway,whichavoidsqualitylossaierrescaling.It’sparWcularlyusefulwhenusedwithinLaTeX

4.15

Exercise(InClass)

  Simpleplot

  WriteafuncWonthathasoneinputparameterM,amatrixwith3columns.Columns{1,2}ofMarethe{x,y}coordinatesofpointsintheplane.Column3istheclasstowhichthepointbelongs.Thevaluesofcolumn3areoneoftwounknownintegers

  YourfuncWonshouldplotallpoints,pointsinoneclassshouldbedotsinred,pointsinthe2ndclassshouldbesquaresingreen.Adjustyouraxesproperlysothatthefurthestpointsarenotontheedge

4.16

DataStructures–Structs

  Whenacompositedatastructureisrequired,usea‘struct’(structurearray)•  MulWplefields,differentdatastructuresforeachfield

•  SimilarinformtoC++classes•  Use‘dot’toaccessfields

  IniWalizaWon:•  S=struct([]) %Emptystruct,nofields

•  S=struct(‘f1’,v1,‘f2’,v2) %Structwithtwofields•  S.field1=2.5; %Createdirectly

4.17

DataStructures–Structs

  FuncWonality:•  isstruct(S); %CheckifSisastruct•  S=se�ield(S,‘field1’,5); %Setafieldtoavalue

•  isfield(S,‘field1’); %Checkiffieldexists

  Example:

1.S=struct(‘vec’,[1,2,3],‘mat’,rand(3));2.my_field=‘mat’;3.isfield(S,my_field);

4.18

Structs–Note

  WhenmulWplevariablesaresavedina.matfile,andthenloadedintoasinglevariable,theyaresavedasfieldsofastruct

  Example:1.savefile1.matXYZ;2.S=load(‘file1’);

3.isstruct(S); %Structwith3fields:S.X,S.Y,S.Z

4.19

DataStructures–CellArrays

  Cellarrayisan‘arrayofmatrices’.•  Eachelementofacellarraycanbeascalar/vector/matrix.•  Whyisituseful?Recallthe1stexercise(loadinga

sequenceoffiles)

  IniWalizaWon:•  C=cell(N) %NbyNarrayofemptymatrices•  C=cell(N,M) %NbyMarrayofemptymatrices

•  Usebrackets{}toaccesselements

•  Rulesforregulararraysapply•  C={[1][1,2,3];rand(3),rand(3)}; %Createdirectly

4.20

CelltoStructConversion

  Commands:cell2struct(),struct2cell()•  S=cell2struct(C,my_fields,my_dim)•  ‘my_fields’isacellarrayofstrings

  Example:1.my_fd={‘num’,‘name’,‘naWon’};2.my_arr={8,‘Iniesta’,‘Spain’};

3.my_st=cell2struct(my_arr,my_fd,2);%Youare‘folding’thedimension,sizemustmatch

4.21

Strings–FuncWons

  MATLABisnotrecommendedasatoolformanipulaWngstrings.However,thefuncWonalityisavailable

  Checkstring:•  isleCer(str),isspace(str)

  Convertstring:•  lower(str),upper(str)•  str2num(str),num2str(num)

4.22

Strings–FuncWons

  OperaWonsonstrings:•  strtok(str,delim),strcmp(str1,str2),strfind(str1,paCern)

  Regularexpressions(helpregexp):conciseandflexiblemeansformatchingstrings

  Example:1.str1=‘one!,notwo’;2.[bef_delim,ai_delim]=strtok(str1,‘!’);

3.idx=strfind(str1,‘n’);4.res1=strcmp(‘hi’,‘HI’);

5.res2=strcmp(‘hi’,lower(‘HI’);

4.23

BuildingaWinningTeam

  Considerthefollowingscenario:youareamanagerthatmustassembleateamofplayers.YouhaveascouWngreportdescribingthesetofskillsofeachplayer,andthesalarydemandsofeach

  Yourgoalistochooseyourteamfromalargepoolofplayers,ensuringthatyourplayerscomplementeachother

  SinceyourownerissWngy,youmustalsominimizeyourtotalplayersalary,whilemaintainingacompeWWveteam

4.24

BWT–ProblemFormulaWon

  Morespecifically:•  YourteamshouldconsistofPplayers•  YouhaveapoolofNplayers(N≥P)tochoosefrom

•  Eachplayer‘p’,demandsasalaryofC(p)

•  ThescouWngreportconsistsofKmarksforeachplayer.Eachmarkisarealvalueintherange[0.0,1.0]indicaWngtheplayer’squalitywithrespecttoaparWcularskill(i.e.0.0is‘noob’,1.0is‘worldclass’)

•  ToensureacompeWWveteam,yourequirethatthetotalqualityofallteamplayersforeveryskill‘k’isatleastsomevalueB(k)

  Givenalltherelevantinfo(playerpool,salarydemands,scouWngreport),howdoyouchooseasetofPplayers,makeyourownerhappy,andsWllhaveacompeWWveteam?

4.25

BWT–ProblemFormulaWon

  AssumpWons:•  Teamsize=P,Poolsize=N,(N≥P).•  C(salary)isanN‐by‐1(column)vectorofposiWverealvalues

•  RisanN‐by‐Kmatrixofrealvaluesintherange[0.0,1.0].Eachrowrepresentsaplayerinthepool.Eachcolumnrepresentsaskill

•  B(totalquality)isa1‐by‐K(row)vectorofposiWverealvalues

  Alltherelevantinfoisgiven.Thatis,{P,N,C,R,B}mustbesuppliedtous

  Note:noguaranteethattheparametersaresetcorrectly

  Note:cannothave‘half’aplayeronateam.Theplayeriseithersignedornot

4.26

BWT–ProblemFormulaWon

  Approach:1.  Collectallparameters{P,N,C,R,B} %Input/loaddata2.  Verifythatparametersaresetcorrectly %Errorchecking

3.  StatetheprobleminmathemaWcalnotaWon:

•  WeareclearlysolvingaconstrainedopWmizaWonproblem

•  WearetryingtominimizealinearobjecWve(minimizethetotalsalary),subjectto:

•  Oneequalityconstraint,ateamshouldhaveexactlyPplayers•  Klinearinequalityconstraints(totalqualityforsomeskillisone

constraint,wehaveKskills)

•  Ourvariablesmustbebinary{0,1},cannotsign‘half’aplayer

4.  Output/verifythesoluWon %Output/savesoluWon

4.27

OpWmizaWon

  “Dothingsbestunderthegivencircumstances”

  ApplicaWonsinalmosteveryfieldimaginable:planning,scheduling,resourceallocaWon,management,trafficcontrol

  OpWmizaWonproblem:•  Makeadecision

•  Express/controlthequalityofthedecisionbytheobjecWvefuncWon

•  TypicallyaminimizaWon/maximizaWontask

•  Express“circumstances”thataffectthedecisionasconstraints

•  Thetypeofopt.prob.isdeterminedbythenatureoftheobjecWvefuncWonandtheconstraints

4.28

OpWmizaWon–GeneralForm

GeneralFrom:

minimize F(x) subjectto: gi(x)≤bi i=1,…,m

  TheproblemischaracterizedbytheobjecWvefuncWonF(x),andtheconstraintsgi(x)

  Thevariableorvectorx,belongstosomedomain/setSspecifiedbytheconstraints

  LinearandQuadraWcprogramsarethemostfrequentproblemsyouarelikelytoencounter

4.29

BWT–ProblemFormulaWon

  MathemaWcalnotaWon:1.  LetXbethevariable/soluWon(column)vector,X∈{0,1}N2.  WewishtominimizetheobjecWvefuncWonCTX

3.  Oneequalityconstraint:∑Xi=P

4.  Klinearinequalityconstraints:RTX≥BT

5.  Completeform:

minimize CTX subjectto: RTX≥BT 1TX=P

∀p,Xp∈{0,1}

4.30

MATLABOpWmizaWonToolbox

  Extensivepackage.ManyrouWnes,opWons.PlentyofdocumentaWon(‘help’isnotsufficient)•  Firststep:defineyourproblemclearly,writedownyour

equaWons

•  Secondstep:findtheappropriatesolver(whattypeofproblemareyousolving?)

•  Thirdstep:convertyourproblemtosolverform.ThismightrequirecombiningequaWons,switchingsignofequaWons&objecWve,addingequaWons

•  Fourthstep:setopWons,callsolver,examinethesoluWon

4.31

MATLABOpWmizaWonToolbox

  Findappropriatesolver:•  hCp://www.mathworks.com/help/toolbox/opWm/ug/

bqnk0r0.html

•  Frequentsolvers:linprog(),quadprog(),fmincon()

  Solverformexample:•  Linearprogram

€

minxf T x such that

A ⋅ x ≤ b,Aeq ⋅ x = beq,lb ≤ x ≤ ub

4.32

BWT–ProblemFormulaWon

  MATLABToolboxnotaWon:•  Appropriatesolver:bintprog()•  Why?SoluWonvectorisabinaryintegervector,objecWveislinear

andtheconstraintsarelinear

•  Converttosolverform:

1.  f=C;

2.  A=‐R’; %Changesign,transpose3.  b=‐B’; %Changesign,columnvector

4.  Aeq=ones(1,length(C)); %1TX=P

5.  beq=P; €

minxf T x such that

A ⋅ x ≤ b,Aeq ⋅ x = beq,x is binary

4.33

bintprog()

  Convertedtosolverform,variables{f,A,b,Aeq,beq}

  FuncWoncallopWons(syntax):x=bintprog(f)x=bintprog(f,A,b)x=bintprog(f,A,b,Aeq,beq)x=bintprog(f,A,b,Aeq,beq,x0)x=bintprog(f,A,b,Aeq,Beq,x0,opWons)x=bintprog(problem)[x,fval]=bintprog(...)[x,fval,exi�lag]=bintprog(...)[x,fval,exi�lag,output]=bintprog(...)

4.34

bintprog()

  Syntaxrules(forallsolvers):•  Parameternotpassed,assumeitisempty•  Parameterorderisimportant

•  Toincludeasubsequentparameter,butomitaprecedingone,passanemptyarray[]

•  ‘opWons’isastructspecifyingopWmizaWonmethoddetails.Ignoreit,unlessyouknowathingortwoaboutthefield

•  Insteadofpassingmanyparameters,canpassasinglestruct‘problem’withappropriatefields

•  OutputparametersincludesoluWon(x),valueofobjecWvefuncWonatthesoluWon(fval),flagindicaWngoutcomeofcall(exi�lag),anddetailsabouttheexecuWon(output)

4.35

bintprog()–Examples

  Parameternotpassed,assumeitisempty:•  x=bintprog(f) %MinimizeobjecWvewithoutconstraints

  Includesubsequentparameter,omitprecedingone:•  x=bintprog(f,[],[],Aeq,beq) %Noinequalityconstraints

  Passasinglestruct:1.  problem.f=C;

2.  problem.Aineq=‐R’;3.  problem.solver=‘bintprog’;

4.  x=bintprog(problem);

•  NoWcethatfieldnamesareslightlydifferent

•  Mustsetallfields(settoemptyifdoesn’texist)

4.36

bintprog()–Examples

  Outputparameters: •  ‘x’:thesoluWonvector•  ‘exi�lag’:ifreturns1,problemsolvedsuccessfully

•  ‘output’:astructurewithsoluWondetails.Forexample,output.WmeisexecuWonWme

•  Cannameparametersinanywayyouwish

•  [soln,fval,the_flag,soln_details]=binprog(problem)

•  IfsoluWonvectorisnotwhatitshouldbe,youmustcheckalloutputparameterstodiscovertheproblem.Youshouldstartwiththe‘exi�lag’,thoughthereisnoprescribedapproachtodetectaproblem

4.37