matlab lecture 7 – calculus 微积分 ref: symbolic...

MATLABLecture 2 SchoolofMathematicalSciencesXiamenUniversity http://gdjpkc.xmu.edu.cn

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MATLABLecture2SolvingLinearSystemsofEquations

Ref:MATLABMathematicsMatricesandLinearAlgebra

SolvingLinearSystemsofEquationsl Vocabulary:

coefficientmatrix linearSystemsofEquationsrowelementary transpositions basisbackslash leastsquaressolutionnonsingularmatrix particularsolutionhomogeneoussystem solutionspacelinearhomogeneousequationnonhomogeneoussystemlinearlyindependent pseudoinverserational componentdeterminant rankoverdeterminedsystemunderdeterminedsystemorthonormalbasis null

l Someoperationsandfunctions . \ rank det inv rref null

l Applicationonsolvinglinearsystemsofequations Ax=b,Aisanmnmatrix Review: b=0

Theoryr(A)=n, thereisonlyoneexactsolutionzero.r(A)


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MATLAB

NonsingularCoefficientMatrix(canbeverifiedbydetorrankfunctions)

>> A=pascal(3) %ObtainaPascalmatrix>>u=[314]>>x=A\u %Tryx=pinv(A)*ux=inv(A*A)*A*ux=

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SingularCoefficientMatrix

If A is singular, the solution to AX=B either does not exist, or is not unique. The backslashoperator,A\B, issuesawarningifAisnearlysingularandraisesanerrorconditionif itdetectsexactsingularity.

IfAissingularandAX=bhasasolution,youcanfindaparticularsolutionthatisnotunique,bytyping>>P=pinv(A)*b

pinv(A)isaMoorePenrosepseudoinverseofA.TheMoorePenrosepseudoinverseisamatrixBofthesamedimensionsasA'satisfyingfourconditions:A*B*A=AB*A*B=BA*BisHermitianB*AisHermitianThecomputationisbasedonsvd(A).

If AX = b does not have an exact solution, pinv(A) returns a leastsquares solution.priv(A)=(AA)1A

Forexample,>> A=[1 3 7

1 4 41 10 18]

issingular,asyoucanverifybytyping>>det(A)

ans=0


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UnderdeterminedSystems

>>R=fix(10*rand(2,4)) % Obtainanintegermatrixby rand and fixfunctionR=

6 8 7 33 5 4 1

>>b=fix(10*rand(2,1))b=

12

>>formatrat %displaythesolutioninrationalformat>>x0=R\b %findsabasicsolution,whichhasatmostmnonzerocomponentsx0=

05/70

11/7>>Z=null(R,'r') %find anorthonormalbasis() forthenullspaceof RZ=

1/2 7/61/2 1/21 00 1

ItcanbeconfirmedthatR*Ziszeroandthatanyvectorxwhere>>symsk1k2k %Definethreesymbolvariables>>k=[k1k2]>>x=x0+Z*k %Generalsolutions

ColumnFullRank Systems>>x=A\b>>x=pinv(A)*b>>x=inv(A'*A)*A'*btheoreticallycomputesthesameleastsquaressolutionx,althoughthebackslashoperatordoesitfaster.

Adoesnothavefullrankx=A\b %abasicsolutionithasatmostrnonzerocomponents,whereristherankofA.x=pinv(A)*b %theminimalnormsolutionbecauseitminimizesnorm(x).x=inv(A'*A)*A'*b %failsbecauseA'*Aissingular.

Example>> A=fix(10*rand(3))>>b=fix(10*rand(3,1))>> ABar=[A b]


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>>d=rref(ABar) %Thelastcolumnofdisthesolutionofsystem

d=

1 0 0 1/80 1 0 73/480 0 1 119/48

MATLAB Lecture 3 - Linear space.pdf


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MATLABLecture3LinearSpace



Vector/linearspace/ linearrelationlinearcombination linearexpression/representationlinearly dependence/correlation linearlydependentlinearly independence linearly independentLinearspace linearspanningdimension linearsubspacemaximallinearlyindependentsubsetscalar spanbasis

factorization symmetricmatrixproduct triangularmatrixtranspose uppertriangularmatrixlowertriangularmatrix diagonalmatrixpermutation orthogonalmatrixunitarymatrix

l Someoperationsandfunctionsrank rref rrefmovie null

l Applicationon linearspace Review:

Let 1 2, ,..., nx x x bevectorsinvectorspaceV.Asumof theform 1 1 2 2 ... n nk x k x k x + + + ,

where 1 2, ,..., nk k k arescalars,iscalledalinearcombinationof 1 2, ,..., nx x x .Thesetof

all linear combinations of 1 2, ,..., nx x x is called the span of 1 2, ,..., nx x x . The span of

1 2, ,..., nx x x willbedenotedby 1 2Span( , ,..., )nx x x .

Thevectors 1 2, ,..., nx x x inavectorspace V aresaidtobe linearlyindependentif

1 1 2 2 ... n nk x k x k x + + +

impliesthatallthescalars 1 2, ,..., nk k k mustequal0.


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Iftheonlywaythelinearcombination 1 1 2 2 ... n nk x k x k x + + + canequalthezerovector

isforallscalars 1 2, ,..., nk k k tobe0,then 1 2, ,..., nx x x arelinearlyindependent.

The vectors 1 2, ,..., nx x x in a vector spaceV are said to be linearly dependent if there

existscalars 1 2, ,..., nk k k notallzerosuchthat

1 1 2 2 ... n nk x k x k x + + + .

If there are nontrivial choices of scalars for which the linear combination

1 1 2 2 ... n nk x k x k x + + + equals the zero vector, then 1 2, ,..., nx x x are linearly

dependent.

A vector x is said to canbe written as a linear combination of 1 2, ,..., nx x x if there

existscalars 1 2, ,..., nk k k suchthat 1 1 2 2 ... n nx k x k x k x = + + + .

Thevectors 1 2, ,..., nx x x inavector spaceVare said tobeabasis ofV if 1 2, ,..., nx x x

are linearly independent and for any vector Vx can bewritten (uniquely) asa linear

combinationof 1 2, ,..., nx x x .TheniscalledasthedimensionofV,andwillbedenotedby

dim(V) .

LetSdenotesasetofvectors 1 2, ,..., nx x x inavectorspaceV. 1 2, , ..., ri i ix x x isasubsetof

S.Ifforany Sx , x canbewrittenuniquelyasalinearcombinationof1 2, , ...,

ri i ix x x ,

1 2, , ...,

ri i ix x x willbecalledasoneofthemaximallinearlyindependentsubsetsof S.

MATLABsupposethat 1V Rn =

linearrelation

>> A=fix(10*rand(3,3)) %ObtainasetofvectorswhicharethecolumnsofA>>r=rank(A)>>ifr==3

disp ThecolumnsofAarelinearlyindependent.else

disp ThecolumnsofAarelinearlydependent.end

linearcombination


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>> A=fix(10*rand(3,2)) %ObtainasetofvectorswhicharethecolumnsofA>>x=fix(10*rand (3,1))>>r0=rank(A)>>r1=rank([Ax])>>ifr0==r1

disp xcanbewrittenbythecolumnsofA.else

disp xcannotbewrittenbythecolumnsofA.end

basisofavectorspace

>> A=fix(10*rand(3,4)) %ObtainasetofvectorswhicharethecolumnsofA>>B=rref(A) % ObservethecolumnsofBwemayobtainthebasisofthelinear

spanningspaceofcolumnsofA>>B=rrefmovie (A) %Movieofthecomputationofthereducedrowechelonform

l *Advancedstudyforlinearsystem Cholesky,LU,andQRFactorizations ()lu qr chol tic toc TheoremAllthreeofthesefactorizationsmakeuseoftriangularmatriceswherealltheelementseitherabove or below the diagonal are zero. Systems of linear equations involving triangularmatricesareeasilyandquicklysolvedusingeitherforwardorbacksubstitution.

TheCholesky factorizationexpressesasymmetricmatrixastheproductofatriangularmatrixanditstranspose A R R = ,where R isanuppertriangularmatrix.

LUfactorization,orGaussianelimination,expressesanysquarematrixAastheproductofapermutationofalowertriangularmatrixandanuppertriangularmatrix A LU = whereLisa permutation of a lower triangular matrix with ones on its diagonal and U is an uppertriangularmatrix.

Theorthogonal,orQR, factorizationexpressesanyrectangularmatrixastheproductofanorthogonalorunitarymatrixandanuppertriangularmatrix.Acolumnpermutationmayalso

beinvolved. A QR = or AP QR = whereQisorthogonalorunitary,Risuppertriangular,

andP isapermutation.

Anorthogonalmatrix,oramatrixwithorthonormalcolumns,isarealmatrixwhosecolumns

allhaveunitlengthandareperpendiculartoeachother.IfQisorthogonal,then Q Q I = .

Forcomplexmatrices,thecorrespondingtermisunitary.


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MATLAB Ax=bCholesky>> A =fix(10*rand(3))>>d =linspace(1,50,3) %Generatesarowvectorof3linearlyequally spacedpoints

between 1and 50>>Diag_d =diag(d) %Generateadiagonalmatrixwiththe inputarguments>>A= A*Diag_d*(A) %Generateasymmetricmatrix>>b=fix(10*rand(3,1))>>R=chol(A)>>x=R\(R'\b)

LU>> A =fix(10*rand(3))>>b=fix(10*rand(3,1))>>[L,U]=lu(A)>>x=U\(L\b)

QR>> A =fix(10*rand(3))>>b=fix(10*rand(3,1))>>[Q,R]= qr(a)>>x=R\(Q\b)

>> A =fix(10*rand(100))>>d=linspace(1,50,100) %Generatesarowvectorof 100linearly equally spaced

pointsbetween 1and 50>>Diag_d =diag(d) %Generateadiagonalmatrixwiththe inputarguments>>A= A*Diag_d*(A)>>b=fix(10*rand(100,1))>>ticR=chol(A)x_chol=R\(R\b)t_chol=toc %Computethetimespent>>tic[L,U]=lu(A)x_LU=U\(L\b)t_LU=toc>>tic[Q,R]=qr(A)x_QR=R\(Q\b)t_QR=toc

MATLAB Lecture 4 - Eigenvalue.pdf


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MATLABLecture4Eigenvalues



coefficients characteristicpolynomialroot characteristicrootseigenvalues eigenvectoridentitymatrix decompositiondefectivematrixorthogonalmatrix unitarymatrixnonsymmetric symmetricroundofferror diagonalizeconjugatetranspose associatematrixdefectivematrixnotdiagonalizablesingularvaluedecompositionSchurdecomposition Schur similarmatrixgeometricmultiplicity algebraicmultiplicity

l Somefunctionseig polv roots schur svd

l Eigenvalue&*singularvaluedecomposition EigenvaluesAneigenvalueandeigenvectorofasquarematrixAareascalar andanonzerovectorvthatsatisfy

Av v =EigenvalueDecompositionWith the eigenvalues on the diagonal of a diagonal matrix and the correspondingeigenvectorsformingthecolumnsofamatrixV,wehave

AV V = If Visnonsingular,thisbecomestheeigenvaluedecomposition

1A V V =

Agoodexampleisprovidedbythecoefficientmatrixoftheordinarydifferentialequation>> A=[0 6 1

6 2 165 20 10]

>>lambda=eig(A) %producesacolumnvectorcontainingtheeigenvalues.lambda=


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3.07102.4645+17.6008i2.464517.6008i

>>[V,D]=eig(A) %computestheeigenvectorsandstorestheeigenvaluesinadiagonalmatrix.

V=0.8326 0.2003 0.1394i 0.2003+0.1394i0.3553 0.2110 0.6447i 0.2110 +0.6447i0.4248 0.6930 0.6930

D=3.0710 0 0

0 2.4645+17.6008i 00 0 2.464517.6008i

Thefirsteigenvectorisrealandtheothertwovectorsarecomplexconjugatesofeachother.AllthreevectorsarenormalizedtohaveEuclideanlength,norm(v,2),equaltoone.ThematrixV*D*inv(V),whichcanbewrittenmoresuccinctlyasV*D/V,iswithinroundofferrorofA.And,inv(V)*A*V,orV\A*V,iswithinroundofferrorofD.

DefectiveMatricesSomematricesdonothaveaneigenvectordecomposition.Thesematricesaredefective,ornotdiagonalizable.Forexample,>> A=[6 12 19

9 20 334 9 15]

>>[V,D]=eig(A)V=

0.4741 0.4082 0.40820.8127 0.8165 0.81650.3386 0.4082 0.4082

D=1.0000 0 0

0 1.0000 00 0 1.0000

Thereisadoubleeigenvalueat 1 = .ThesecondandthirdcolumnsofVarethesame.Forthismatrix,afullsetoflinearlyindependenteigenvectorsdoesnotexist.TheoptionalSymbolicMathToolboxextendsthecapabilitiesofMATLABbyconnectingtoMaple, apowerful computeralgebra system.Oneof the functionsprovidedby the toolboxcomputes the Jordan Canonical Form. This is appropriate for matrices like our example,whichis3by3andhasexactlyknown,integerelements.>>[X,J]=jordan(A)X=

1.7500 1.5000 2.75003.0000 3.0000 3.00001.2500 1.5000 1.2500


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J=1 0 00 1 10 0 1

The Jordan Canonical Form is an important theoretical concept, but it is not a reliablecomputationaltoolforlargermatrices,orformatriceswhoseelementsaresubjecttoroundofferrorsandotheruncertainties.

SchurDecompositioninMATLABMatrixComputationsThe MATLAB advanced matrix computations do not require eigenvalue decompositions.Theyarebased,instead,ontheSchurdecomposition

'A USU =whereU is anorthogonalmatrixand S is ablock upper triangularmatrixwith1by1and2by2blocks on the diagonal.Theeigenvaluesare revealedby the diagonalelementsandblocksofS,while thecolumnsofU provideabasiswithmuchbetternumericalpropertiesthanasetofeigenvectors.TheSchurdecompositionofourdefectiveexampleis>>[U,S]=schur(A)U=

0.4741 0.6648 0.57740.8127 0.0782 0.57740.3386 0.7430 0.5774

S=1.0000 20.7846 44.6948

0 1.0000 0.60960 0 1.0000

Thedoubleeigenvalueiscontainedinthelower2by2blockofS.

*SingularValueDecompositionAsingularvalueandcorrespondingsingularvectorsofarectangularmatrixAareascalar andapairofvectorsuand vthatsatisfy

, 'Av u A u v = =

With the singular values on the diagonal of a diagonal matrix and the correspondingsingularvectorsformingthecolumnsoftwoorthogonalmatricesUand V,wehave

, 'AV U A U V = =

Since Uand Vareorthogonal,thisbecomesthesingularvaluedecomposition'A U V =

ThefullsingularvaluedecompositionofanmbynmatrixinvolvesanmbymU,anmbyn ,andan nbynV.Inotherwords,Uand Varebothsquareand isthesamesizeasA.IfA has many more rows than columns, the resultingU can be quite large, but most of itscolumns are multiplied by zeros in . In this situation, the economy sized decompositionsavesbothtimeandstoragebyproducingan mbynU,an nbyn andthesame V.


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The eigenvalue decomposition is the appropriate tool for analyzing a matrix when itrepresentsamapping fromavector space into itself, as itdoes for an ordinary differentialequation. On the other hand, the singular value decomposition is the appropriate tool foranalyzing a mapping from one vector space into another vector space, possibly with adifferent dimension. Most systems of simultaneous linear equations fall into this secondcategory.

If A is square, symmetric, and positive definite, then its eigenvalue and singular valuedecompositionsarethesame.But,asAdepartsfromsymmetryandpositivedefiniteness,thedifference between the two decompositions increases. In particular, the singular valuedecomposition of a realmatrix is always real, but the eigenvalue decomposition of a real,nonsymmetricmatrixmightbecomplex.>> A=[9 4

6 82 7]

>>[U,S,V]=svd(A) %thefullsingularvaluedecompositionU=

0.6105 0.7174 0.33550.6646 0.2336 0.70980.4308 0.6563 0.6194

S=14.9359 0

0 5.18830 0

V=0.6925 0.72140.7214 0.6925

YoucanverifythatU*S*V'isequalto Atowithinroundofferror.

>>[U,S,V]=svd(A,0) %theeconomysizedecompositionisonlyslightlysmaller.U=

0.6105 0.71740.6646 0.23360.4308 0.6563

S=14.9359 0

0 5.1883V=

0.6925 0.72140.7214 0.6925

Again,U*S*V'isequalto Atowithinroundofferror.

MATLAB>>A=round(10*randn(3))


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A=

4 3 1217 11 01 12 3

>>p_A=poly(A) %computesthecoefficientsofthecharacteristicpolynomialof Aans=1.0e+003*0.0010 0.0120 0.0380 2.0310

>>r=roots(p_A)r=16.8781

2.4391+10.6951i2.4391 10.6951i

Weusuallyuse eig tocomputetheeigenvaluesofamatrixdirectly.

MATLAB Lecture 5 - Symbol Computation.pdf

MATLABLecture5 SchoolofMathematicalSciencesXiamenUniversity http://gdjpkc.xmu.edu.cn

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MATLABLecture5SymbolComputation

Ref: Symbolic Math ToolboxGetting Started

Symbolic Math ToolboxUsing the Symbolic Math ToolboxTheseversionsof theSymbolicMathToolboxesaredesigned toworkwithMATLAB6orgreaterandMaple8.

l Vocabulary:datatype toolboxsymbol symbolcomputationsymbolicobject numericalvaluevariable representstore stringexpression equationsubstitute functionabstract indeterminatecirculantmatrix nested/horner collecting expandfactor factoringalgebraic trigonometricidentities transformsimplification

l Somefunctionssym syms simple simplify collect expand horner factor *findsym*sum *double *eval

l SymbolicMathToolbox SymbolicObjectsThe SymbolicMath Toolbox defines a newMATLAB data type called a symbolic object.(SeeProgrammingandDataTypesintheonlineMATLABdocumentationforanintroductiontoMATLABclassesandobjects.)Internally,asymbolicobjectisadatastructurethatstoresastring representation of the symbol. TheSymbolicMathToolbox uses symbolic objects torepresent symbolic variables, expressions, andmatrices.Theactualcomputations involvingsymbolic objects are performed primarily byMaple, mathematical software developed byWaterlooMaple,Inc.

The following example illustrates the difference between a standardMATLAB data type,suchasdouble,andthecorrespondingsymbolicobject.TheMATLABcommand>>sqrt(2) %returnsafloatingpointdecimalnumber:ans=


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1.4142>>a=sqrt(sym(2)) %usingsymbolicnotationforthesquarerootoperationa=2^(1/2)MATLAB gives the result 2^(1/2), which means 21/2, without actually calculating anumerical value. MATLAB records this symbolic expression in the string that represents2^(1/2).You can always obtain the numerical value of a symbolic object with the doublecommand:>>double(a)ans=

1.4142Noticethattheresultisindented,whichtellsyouithasdatatypedouble.Symbolicresultsarenotindented.

Whenyoucreateafractioninvolvingsymbolicobjects,MATLABrecordsthenumeratoranddenominator.Forexample:>>sym(2)/sym(5)+sym(1)/sym(3) %performsarithmeticonsymbolicobjectsans=11/15>>2/5+1/3 %givestheanswerasadecimalfractionans=

0.7333

TheSymbolicMathToolboxenablesyoutoperformavarietyofsymboliccalculationsthatariseinmathematicsandscience. CreatingSymbolicVariablesandExpressions>>x=sym('x') %constructsymbolicvariable>>a=sym('alpha+2') %constructsymbolicexpression>>rho=sym('(1+sqrt(5))/2') %useasymbolicvariabletorepresentthegoldenratio>>f1=rho^2rho 1f1=(1/2+1/2*5^(1/2))^23/21/2*5^(1/2)>>eval(f1) %evaluatef1ans=

0

>>f2=sym('a*x^2+b*x+c') % assignsthesymbolicexpressiontothevariablef2

Toperformsymbolicmathoperations(e.g.,integration,differentiation,substitution,etc.)onf2,youneedtocreatethevariablesexplicitly.Abetteralternativeistoenterthecommands

>>a=sym('a')b=sym('b')c=sym('c')x=sym('x')orsimply>>symsabcx


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>>f2=sym('a*x^2+b*x+c')or>>f2=a*x^2+b*x+c

Ingeneral,youcanusesymorsymstocreatesymbolicvariables.Werecommendyouusesymsbecauseitrequireslesstyping.Ifyouwanttogenerationansymbolicequation, thecodef2=a*x^2+b*x+c=0is illegal.The correctcommandisf2=sym('a*x^2+b*x+c=0')or f2=sym('a*x^2+b*x+c=0').

Note Tocreateasymbolicexpressionthatisaconstant,youmustusethe symcommand.>>s_5=sym('5')s_5/3 %createasymbolic5ans=5/3>>a_5=5 a_5/3 %createanumber5ans=

1.6667

Ifyousetavariableequaltoasymbolicexpression,andthenapplythesymscommandtothevariable,MATLABremovesthepreviouslydefinedexpressionfromthevariable.>>symsab>>f3=a+b %createasymbolicvariablef3,whosevalueisasymbolicexpressiona+bf3=a+b>>symsf3 %removesthepreviouslydefinedexpressionfromthevariable.>>f3f3=f3

ThesubsCommandYoucansubstituteanumericalvalueforasymbolicvariableusingthe subscommand.to>>f4=2*x^2 3*x+1 %createa char variablef4,equivalenttosymf4

f4=2*x^23*x+1>>subs(f4,1) %substitutethevaluex=1inf4,i.e.2*1^23*1+1ans=

0*Note To substitute a matrix A into the symbolic expression f, use the commandpolyvalm(sym2poly(f), A), which replaces all occurrences of x by A, and replaces theconstanttermoffwiththeconstanttimesanidentitymatrix.>> A=[1234]A =

1 23 4

>>polyvalm(sym2poly(f4), A)???Function'sym2poly'isnotdefinedforvaluesofclass'char'.


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>>symsf4 %define f4asansymbolicvariable>>f4=sym('2*x^23*x+1') % assignsthesymbolicexpressiontothevariablef4or>>symsxf4=2*x^23*x+1f4=2*x^23*x+1>>polyvalm(sym2poly(f4), A) %computematrixpolynomial2*A^23*A+Ians=

12 1421 33

When your expression contains more than one variable, you can specify the variable forwhichyouwanttomakethesubstitution.>>symsxy>>f5=x^2*y+5*x*sqrt(y)>>subs(f5,x,3) %substitutethevaluex=3inthesymbolicexpressionans=9*y+15*y^(1/2)>>subs(f5,y,3) %tosubstitutey=3ans=3*x^2+5*x*3^(1/2)

CreatingaSymbolicMatrixA circulant matrix has the property that each row is obtained from the previous one bycyclicallypermutingtheentriesonestepforward.Youcan,usingthecommands>>symsabc>> A=[abcbcacab] %createthecirculantmatrixAwhoseelementsarea,b,andcA=[a,b,c][b,c,a][c,a,b]SinceAiscirculant,thesumovereachrowandcolumnisthesame.Tocheckthisforthefirstrowandsecondcolumn,enterthecommand>>sum(A(1,:))ans=a+b+c>>sum(A(1,:))==sum(A(:,2)) %Thisisalogicaltesttocheckwhetherthesumof

thefirstrow isequaltothatofthesecondrow.ans=

1Itimpliesthatthesumofthefirstrowisequaltothatofthesecondrow.Iftheansweris0,itmeansthesumofthefirstrowisnotequaltothatofthesecondrow.>>symsalphabeta>> A(2,3)=beta % replacethe(2,3)entryofAwithbeta


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>> A=subs(A,b,alpha) %replace thevariablebwithalphaA=[ a,alpha, c][alpha, c, beta][ c, a,alpha]Fromthisexample,youcanseethatusingsymbolicobjects isverysimilar tousingregularMATLABnumericobjects.

Simplificationspretty>>clear>>symsx>>f=x^36*x^2+11*x6>>g=(x1)*(x2)*(x3)>>h=6+(11+(6+x)*x)*x>>pretty(f),pretty(g),pretty(h) %generate theirprettyprintedforms3 2x6x +11x6(x1)(x 2)(x 3)6+(11+(6+x)x)x

collectcollect(f) views f as a polynomial in its symbolic variable, say x, and collects all thecoefficientswiththesamepowerofx.Asecondargumentcanspecifythevariableinwhichtocollecttermsifthereismorethanonecandidate.f collect(f)(x1)*(x2)*(x3) x^36*x^2+11*x6x*(x*(x6)+11)6 x^36*x^2+11*x6(1+x)*t+x*t 2*x*t+t

expandexpand(f) distributesproductsoversumsandappliesotheridentitiesinvolvingfunctionsofsums.f expand(f)a*(x+y) a*x+a*y(x1)*(x2)*(x3) x^36*x^2+11*x6x*(x*(x6)+11)6 x^36*x^2+11*x6exp(a+b) exp(a)*exp(b)cos(x+y) cos(x)*cos(y)sin(x)*sin(y)cos(3*acos(x)) 4*x^33*x

hornerhorner(f) transformsasymbolicpolynomialfintoitsHorner,ornested,representation.f horner(f)


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x^36*x^2+11*x6 6+(11+(6+x)*x)*x1.1+2.2*x+3.3*x^2 11/10+(11/5+33/10*x)*x

factorIffisapolynomialwithrationalcoefficients,thestatementfactor(f) expressesfasaproductofpolynomialsoflowerdegreewithrationalcoefficients.Iffcannotbefactoredovertherationalnumbers,theresultisfitself.f factor(f)x^36*x^2+11*x6 (x1)*(x2)*(x3)x^36*x^2+11*x5 x^36*x^2+11*x5x^6+1 (x^2+1)*(x^4x^2+1)

Here is anotherexample involving factor. It factorspolynomialsof the formx^n+1.Thiscode>>symsx>>n=(1:9)'>>p=x.^n+1>>f=factor(p)>>[p,f]returns a matrix with the polynomials in its first column and their factored forms in itssecond.[ x+1, x+1][ x^2+1, x^2+1][ x^3+1, (x+1)*(x^2x+1)][ x^4+1, x^4+1][ x^5+1, (x+1)*(x^4x^3+x^2x+1)][ x^6+1, (x^2+1)*(x^4x^2+1)][ x^7+1, (x+1)*(1x+x^2x^3+x^4x^5+x^6)][ x^8+1, x^8+1][ x^9+1, (x+1)*(x^2x+1)*(x^6x^3+1)]

simplifyThesimplifyfunctionisapowerful,generalpurposetoolthatappliesanumberofalgebraicidentitiesinvolvingsums,integralpowers,squarerootsandotherfractionalpowers,aswellasanumberoffunctionalidentitiesinvolvingtrig functions,exponentialandlogfunctions,Besselfunctions,hypergeometricfunctions,andthegammafunction.f simplify(f)x*(x*(x6)+11)6 x^36*x^2+11*x6(1x^2)/(1x) x+1(1/a^3+6/a^2+12/a+8)^(1/3) ((2*a+1)^3/a^3)^(1/3)symsxypositivelog(x*y) log(x)+log(y)exp(x)*exp(y) exp(x+y)cos(x)^2+sin(x)^2 1


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simpleThesimplefunctionhastheunorthodoxmathematicalgoaloffindingasimplificationofanexpression thathas the fewestnumberof characters.Ofcourse, there is littlemathematicaljustificationforclaimingthatoneexpressionis"simpler"thananotherjustbecauseitsASCIIrepresentationisshorter,butthisoftenprovessatisfactoryinpractice.

Thesimplefunctionachievesitsgoalbyindependentlyapplyingsimplify,collect,factor,andothersimplificationfunctionstoanexpressionandkeepingtrackofthelengthsoftheresults.Thesimplefunctionthenreturnstheshortestresult.

simple(f) displayseachtrialsimplificationandthesimplificationfunctionthatproduceditintheMATLABcommandwindow.Thesimplefunctionthenreturnstheshortestresult.>>simple(cos(x)^2+sin(x)^2) %displaysthefollowingalternativesimplificationsin

theMATLABcommandwindow:simplify:1radsimp:cos(x)^2+sin(x)^2combine(trig):1factor:cos(x)^2+sin(x)^2expand:cos(x)^2+sin(x)^2combine:1convert(exp):(1/2*exp(i*x)+1/2/exp(i*x))^21/4*(exp(i*x)1/exp(i*x))^2convert(sincos):cos(x)^2+sin(x)^2convert(tan):(1tan(1/2*x)^2)^2/(1+tan(1/2*x)^2)^2+4*tan(1/2*x)^2/(1+tan(1/2*x)^2)^2collect(x):cos(x)^2+sin(x)^2andreturnsans=1

f=simple(f) simplyreturnstheshortestexpressionfound.

>>f=simple(cos(x)^2+sin(x)^2)f=


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1

[F,how]=simple(f) returnstheshortestresultinthefirstvariableandthesimplificationmethodusedtoachievetheresultinthesecondvariable.

>>[f,how]=simple(cos(x)^2+sin(x)^2)f=1how=combine

The simple function sometimes improves on the result returned by simplify, one of thesimplifications that it tries. Forexample,whenapplied to the examplesgiven for simplify,simplereturnsasimpler(oratleastshorter)resultintwocases.f simplify(f) simple(f)(1/a^3+6/a^2+12/a+8)^(1/3) ((2*a+1)^3/a^3)^(1/3) (2*a+1)/asymsxypositivelog(x*y) log(x)+log(y) log(x*y)

Insomecases,itisadvantageoustoapplysimpletwicetoobtaintheeffectof twodifferentsimplificationfunctions.>>f=(1/a^3+6/a^2+12/a+8)^(1/3)>>Simple_f=simple(f)Simple_f=(2*a+1)/a>>Simple_Simple_f=simple(Simple_f)Simple_Simple_f=2+1/aor>>Simple_Simple_f =simple(simple(f))Simple_Simple_f=2+1/a

The simple function is particularly effective on expressions involving trigonometricfunctions.f simple(f)cos(x)^2+sin(x)^2 12*cos(x)^2sin(x)^2 3*cos(x)^21cos(x)^2sin(x)^2 cos(2*x)cos(x)+(sin(x)^2)^(1/2) cos(x)+i*sin(x)cos(x)+i*sin(x) exp(i*x)cos(3*acos(x)) 4*x^33*x

*ThefindsymCommandTo determine what symbolic variables are present in an expression, use the findsym


Lec59

command.>>symsabntxz>>c=1>>ff=x^ngg =sin(a*t+b+c) %giventhesymbolicexpressionsfandg>>findsym(ff) %findthesymbolicvariablesinffans=n,x>>findsym(gg) %findthesymbolicvariablesinggans=a,b,t

*CreatingAbstractFunctions>>f=sym('f(x)') %createanabstract(i.e.,indeterminate)functionThenfactslikeandcanbemanipulatedbythetoolboxcommands.>>df=(subs(f,'x','x+h')f)/'h' %toconstructthefirstdifferenceratioor>>symsxh>>df=(subs(f,x,x+h)f)/hdf=(f(x+h)f(x))/hThisapplicationofsymisusefulwhencomputingFourier,Laplace,andztransforms.

MATLAB Lecture 1 - Matrices & Arrays.pdf


Lec11

MATLABLecture1_MatricesandArrays

MATLAB GettingStarted MatricesandArraysMATLAB GettingStarted Expressions

MATLAB Mathematics MatricesandLinearAlgebra MatricesinMATLAB

l Vocabulary:addition substrationmultiplication divisionleftdivision powercomplexconjugatetranspose explicitlist, elemententer/input matrix(pl.matrices)array vectorvariable assignmentstatementexpression datarow columndimension

l UsingMATLABasaScatchpadOperators Expressionsusefamiliararithmeticoperatorsandprecedencerules.

>>pi*0.1^2ans=

0.0314>>area=pi*0.1^2 %theareaofacirclewithradius0.1

area=

0.0314


Lec12

>>volume=area*0.5%thevolumeofacylinderwithradius0.1andlength0.5

volume=

0.0157>>(12+2*(74))/3^2

ans=

2

>>S=11/2+1/31/4+1/51/6...+1/71/8

S=

0.6345>>z1=3+4i

z1=

3.0000+4.0000i>>z2=1+2*i>>z3=2*exp(i*pi/6)>>z=z1*z2/z3MATLAB Getting Started Matrices and Arrays Expressions Operators

l EntermatricesinMATLAB Enteranexplicitlistofelements. Loadmatricesfromexternaldatafiles.

Generatematricesusingbuiltinfunctions.

CreatematriceswithyourownfunctionsinMfiles.Array:acollectionofdatavaluesorganizedintorowsandcolumnsVector:anarraywithonlyonedimensionMatrix:anarraywithtwoormoredimensionsSize:specifiedbythenumberofrowsandthenumberofcolumnsinthe

array,withthenumberofrowsmentionedfirst.

>>[123456789] %createanarrayswith9elements>>[1,2,34,5,67,8,9] %createa33matrix

ans=

123

456

789>>[123

456789]

ans=

123

456

789

MATLAB Getting Started Matrices and Arrays Matrices and Magic Square


Lec13

Entering Matrices

1. InitializingVariablesinMATLABVar =expression

Assigndatatothevariableinanassignmentstatement(). Inputdataintothevariablefromthekeyboard.

Readdatafromafile.Someruleson variable: Beginwithaletter Followedbyanycombinationofletters,numbersandtheunderscore(_)character Onlythefirst31charactersaresignificant CasesensitiveGoodprogrammingpractice

Easytoremember(e.g.name,NAME,Namearedifferent)Useonlylowercaseletter(e.g.nameisthebest)

>>aa=[123456789] %createanarrayswith9elements>>a=[123456789] %createa33matrix

a=

123

456

789

>>b=[1,2,34,5,67,8,9] >>c=[1234]>>d=[a(1,2),7+3]MATLAB Getting Started Matrices and Arrays Expressions Variables

2. InitializingwithShortcutExpressionsbyColonOperator ( )First:incr :last

>>a1=1:9%obtainarowvectorcontainingtheintegersfrom1to9

>>a2=1:2:10>>a3=10:3:0>>a4=pi:0.3:pi>>suba1=a(1:2,3)%obtainamatrixsubA1,whichisapartition

ofmatrixAwiththefirst2elementsofthe3rdcolumnofA>>suba2=a(1:2,:)>>suba3=a(:,2)>>C=a(:,[2,3,1])%foreachoftherowsofmatrixA,reorder

theelementsintheorder2,3,1>>D=a(:,end)MATLAB GettingStarted MatricesandArrays MatricesandMagicSquare

TheColonOperator


Lec14

3. ArrayandMatrixOperations

Operation MATLABForm CommentsArrayAddition a+b Arrayadditionandmatrixadditionareidentical.ArraySubtraction ab Arraysubtractionandmatrixsubtrationareidentical.ArrayMultiplication

a.*b Elementbyelement multiplication of a and b. Botharraymustbethesameshape,oroneofthemmustbeascalar.

MatrixMultiplication

a*b Matrix multiplication of a and b. The number ofcolumnsinamustequaltothenumberofrowsinb, oroneofthemmustbeascalar..

Array RightDivision

a./b Elementbyelement division of a and b: a(i,j)/b(i,j).Both arrays must be the same shape, or one of themmustbeascalar.

Array LeftDivision

a.\b Elementbyelement division of a and b:b(i,j)/b(i,j).Both arrays must be the same shape, or one of themmustbeascalar.

Matrix RightDivision

a/bMatrixdivisiondefinedby 1ab .

Matrix LeftDivision

a\bMatrixdivisiondefinedby 1a b .

ArrayExponentiation

a.^a Elementbyelement exponentiation of a and b:

( , )( , )b i ja i j .Botharraysmustbethesameshape,orone

ofthemmustbeascalar.

4. *InitializingwithBuiltinFunctions

clock,date

sin,cos,tan,cot,sec,csc,asin,acos

x.^a,sqrt,exp,log,log10,log2

sbs,sign,round,fix,floor,ceil,sum,prod,max,min,mean

sort,rand,randneye,diag


Lec15

MATLAB Mathematics MatricesandLinearAlgebra MatricesinMATLAB

5. *InitializingVariableswithKeyboardInputmy_val=input(Enteraninputvalue: )

>>age=input(Pleaseenteryourage:)Pleaseenteryourage:20>>profession=input(Pleaseenteryourprofession:,s)Pleaseenteryourprofession:teacher

l Clearing>>clc %ClearingtheCommandWindow.Thisdoesnotclearthe

workspace,butonlyclearstheview.Afterwards,youstill

canusetheuparrowkeytorecallpreviousfunctions.>>clear %Removeitemsfromworkspace,freeingupsystemmemory.

MATLAB Lecture 6 - Polynomial.pdf


Lec61

MATLABLecture6Polynomial

Ref: MATLABMathematicsPolynomials and Interpolationl Vocabulary:

polynomialroot arithmeticoperationmultiply dividederivative differentiationevaluation partialfractionexpansion convolutionproduct deconvolution quotient remaindermultipleroots direct term transferfunction

l Somefunctionsconv deconv poly polyder polyval polyvalm roots * residue *polyfit

l Polynomials RepresentingPolynomialsMATLAB represents polynomials as row vectors containing coefficients ordered bydescendingpowers.

>>p=[10 25] %represents 3 2 5x x

p=1 0 2 5

>>sym_p=poly2sym(p) %representsapolynomialinsymformsym_p =x^32*x5

CreatePolynomials

>>p=[10 25] %representsapolynomail 3 2 5x x

p=1 0 2 5

>>r =[0,1, 1]poly(r) %generateapolynomial ( 1)( 1)x x x + ,whoserootsare0,1,1

ans=1 0 1 0

>>a =[1234]poly(a) %generatethecharacteristicpolynomialsofmatrix

1 23 4

,i.e. 21 2

5 23 4

=


Lec62

ans=1.0000 5.0000 2.0000

PolynomialEvaluation>>polyval(p,5) %evaluatesapolynomialataspecifiedvalue,say5.ans=

110>>subs(sym_p,5) %substitutethesymvariable xinsym_pwith5ans=

110

>>X=[245103715]>> Y=polyvalm(p,X) %createasquarematrixXandevaluatethepolynomialpatX

3 2 5Y X X I =

Y=377 179 439111 81 136490 253 639

PolynomialRoots>>r=roots(p) %calculatestherootsofapolynomialpr=

2.09461.0473+ 1.1359i1.0473 1.1359i

Polynomial ArithmeticoperationAddition>>p2=[02 13]add_p=p+p2 %calculatessumoftwopolynomialspandp2.

Here thematrixdimensionsmustagree.add_p=

1 2 3 2>>p3=poly2sym(p2)add_p_sym=sym_p+p3 %sym_ppulsesp3anddisplaythe

resultin symform.add_p_sym=x^33*x2+2*x^2>>sym2poly(add_p_sym) %returnsarowvectorcontainingthecoefficients

ofthesymbolicpolynomialPans=

1 2 3 2Subtraction (Omit.Itissimilartoaddition)Multiplication (Correspondtotheoperationsconvolution)>>a=[12]b=[20 1]c=conv(a,b)poly2sym(c)


Lec63

%compute theproductof 2( 2)(2 1)x x +

ans=2*x^3+4*x^2x2Division (Correspondtotheoperationsconvolutionanddeconvolution)>>[q,r]=deconv(c,a) %dividingc by aisquotientqandremainderrq=

2 0 1r=

0 0 0 0

PolynomialDerivatives

>>q=polyder(p) %computesthederivativeofpolynomial 3( 2 5)x x

q=3 0 2

>>a=[135]b=[246]c=polyder(a,b) %computesthederivativeofthe

productoftwopolynomials 2 2[( 3 5)(2 4 6)]x x x x + + + +

c=8 30 56 38

>>[q,d]=polyder(a,b) %computesthederivativeofthequotientoftwopolynomials

2

2

( 3 5) ( )(2 4 6) ( )x x q xx x d x

+ + = + +

q=2 8 2

d=4 16 40 48 36

*PartialFractionExpansionresidue finds the partial fraction expansion of the ratio of two polynomials. This isparticularly useful for applications that represent systems in transfer function form. Forpolynomialsband a,iftherearenomultipleroots,

1 2

1 2

( ) ...( )

ns

n

rr rb x ka x x p x p x p

= + + + +

whererisacolumnvectorofresidues,pisacolumnvectorofpolelocations,andkisarowvectorofdirectterms.

>>b=[48]a=[168] [r,p,k]=residue(b,a) % 24 8 12 86 8 4 2x

x x x x

= + + + + +

r=128


Lec64

p=42

k=[ ]

Giventhreeinputarguments(r,p,andk),residueconvertsbacktopolynomialform.

>>[b2, a2]=residue(r,p,k) % 212 8 4 84 2 6 8

xx x x x

+ = + + + +

b2=4 8

a2=1 6 8

PolynomialFunctionSummaryFunction Descriptionconv Multiplypolynomials.deconv Dividepolynomials.poly Polynomialwithspecifiedroots.polyder Polynomialderivative.polyval Polynomialevaluation.polyvalm Matrixpolynomialevaluation.roots Findpolynomialroots.residue Partialfractionexpansion(residues).polyfit Polynomialcurvefitting.

MATLAB Lecture 7 - Calculus.pdf


Lec71

MATLABLecture7Calculus

Ref: Symbolic Math ToolboxUsing the Symbolic Math Toolbox

Calculusl Vocabulary:

calculus functioncomposite/compoundfunction inversefunctionlimit derivativedifferentiation differentialquotient,indefinite integral definite integralTaylorseries TaylorexpansionTaylorformula item/termsummation accumulatesingleton dimensionindex/subscript/suffix theNthorderdifference N

l Somefunctionscompose finverse limit diff int symsum taylor *gradient *sum *prod

l CalculusThis section explains how to use the Symbolic Math Toolbox to perform many common

mathematicaloperations.

Functioncompositioncompose(f,g)returnsf(g(y))wheref=f(x)andg=g(y).Herexisthesymbolicvariableoff

asdefinedbyfindsym andyisthesymbolicvariableofgasdefinedbyfindsym.compose (f,g,z) returns f(g(z)) where f = f(x), g = g(y), and x and y are the symbolic

variablesoffandgasdefinedby findsym.compose(f,g,x,z)returns f(g(z))andmakesx theindependentvariableforf.Thatis, if f=

cos(x/t),then compose (f,g,x,z)returnscos(g(z)/t)whereascompose (f,g,t,z)returnscos(x/g(z)).compose (f,g,x,y,z) returns f(g(z)) and makes x the independent variable for f and y the

independent variable for g. For f = cos(x/t) and g = sin(y/u), compose (f,g,x,y,z) returnscos(sin(z/u)/t)whereascompose (f,g,x,u,z)returnscos(sin(y/z)/t).

Examples:>>symsxyztu>>f=1/(1+x^2)g=sin(y)h=x^tp=exp(y/u)

>>compose(f,g) % 21

1 sin y +

>>compose(f,g,t) % 21

1 sin t +


Lec72

>>compose(h,g,x,z) %sint z

>>compose(h,g,t,z) % sin zx

>>compose(h,p,x,y,z) % ( )/ tz ue

>>compose(h,p,t,u,z) %/y zex

NOTICE:theorderofthecodesisimportant.>>clearsymsxyzz=exp(y)y=sin(x)compose(z,y),zans=exp(sin(x))z=exp(y)>>clearsymsxyzy=sin(x)z=exp(y)compose(z,y),z %NOTRECOMMENDans=exp(sin(sin(x)))z=exp(exp(x))>>clearsymsuvyzy=sin(u)z=exp(v)compose(z,y) %RECOMMENDtouse

differentvariablesasindependentvariablefordifferentfunctions

Functionalinverseg=finverse(f)returnsthefunctionalinverseoff.fisascalarsymrepresentingafunctionof

exactlyonesymbolicvariable,say'x'.Thengisascalarsym thatsatisfiesg(f(x))=x.g=finverse(f,v)usesthesymbolicvariablev,wherevisasym,astheindependentvariable.

Then g is a scalar sym that satisfies g(f(v)) = v.Use this formwhen f containsmore than onesymbolicvariable.

Examples:>>finverse(1/tan(x)) %returnsatan(1/x).>>f=x^2+y>>finverse(f,y) %returnsx^2+y.>>finverse(f)Warning:finverse(x^2+y)isnotunique.>Insym.finverseat43ans=(y+x)^(1/2)

Limitofanexpressionlimit(F,x,a)takesthe limitofthesymbolicexpressionFasx a.limit(F,a)usesfindsym(F)astheindependentvariable.limit(F)usesa=0asthelimitpoint.limit(F,x,a,'right')orlimit(F,x,a,'left')specifythedirectionofaonesidedlimit.


Lec73

Examples:>>symsxath

>>limit(sin(x)/x) %evaluate0

sinlimx

xx

ans=1

>>limit((x2)/(x^24),2) %evaluate 222lim4x

xx

ans=1/4

>>limit((1+2*t/x)^(3*x),x,inf) %evaluate32lim 1x

x

tx

+

ans=exp(6*t)

>>limit(1/x,x,0,'right') %evaluate0

1limx x +

ans=Inf

>>limit(1/x,x,0,'left') %evaluate0

1limx x

ans=Inf

>>limit((sin(x+h)sin(x))/h,h,0) %evaluate0

sin( ) sinlimh

x h xh

+

ans=cos(x)>>v=[(1+a/x)^x,exp(x)]

>>limit(v,x,inf,'left') %evaluate lim 1 , limx

x

x x

a ex

+

ans=[exp(a), 0]

Differenceandapproximatederivativediff (X),foravectorX,is[X(2)X(1) X(3)X(2) ... X(n)X(n1)].diff (X),foramatrixX,isthematrixofrowdifferences,[X(2:n,:)X(1:n1,:)].diff(X,N)is theNthorderdifferencealongthefirstnonsingletondimension(denoteitby

dim). If N >= size(X,dim), diff takes successive differences along the next nonsingletondimension.

diff(X,N,DIM)is theNthdifferencefunctionalongdimensionDIM.IfN>=size(X,DIM),diff returnsanemptyarray.


Lec74

Examples:>>h=.001x=0:h:pi

>>diff(sin(x.^2))/h %2 2sin( ) sin( ) , : :i i h i h h pih

= ,thereare approximation

to2*cos(x.^2).*x,x=0:h:pi

>>diff((1:10).^2) %evaluate 2 2( 1) , 1, 2,...,10i i i + =

ans=3 5 7 9 11 13 15 17 19

>>X=[375257]diff(X) %thesameasdiff(X,1,1)ans=

1 2 2>>diff(X,1,2)ans=

4 23 2

>>diff(X,2,2) %the2ndorderdifferencealongthedimension2ans=

61

>>diff(X,3,2)ans=

Emptymatrix:2by0>>symsxyy=atan((x+1)/(x1))>>yx=diff(y,x)yx=(1/(x1)(x+1)/(x1)^2)/(1+(x+1)^2/(x1)^2)>>yxx=diff(y,x,2)yxx=(2/(x1)^2+2*(x+1)/(x1)^3)/(1+(x+1)^2/(x1)^2)(1/(x1)(x+1)/(x1)^2)/(1+(x+1)^2/(x1)^2)^2*(2*(x+1)/(x1)^22*(x+1)^2/(x1)^3)>>symsxy>>z=x^4+y^4cos(2*x+3*y)

>>zx=diff(z,x) %compute4 4( cos(2 3 ))x y x y

x + +

zx=4*x^3+2*sin(2*x+3*y)

>>zy=diff(z,y) %computezy

zy=4*y^3+3*sin(2*x+3*y)


Lec75

>>zxx=diff(zx,x) %compute2

2

zx

,equivalentto zxx=diff(z,x,2)

zxx=12*x^2+4*cos(2*x+3*y)

Integrateint(S) is the indefinite integral of S with respect to its symbolic variable as defined by

findsym.Sisasym (matrixorscalar).IfSisaconstant,theintegraliswithrespectto'x'.int(S,v)istheindefiniteintegralofSwithrespecttov.visascalarsym.int(S,a,b)isthedefiniteintegralofSwithrespecttoitssymbolicvariablefromatob.aandb

areeach doubleorsymbolicscalars.int(S,v,a,b)isthedefiniteintegralofSwithrespecttovfromatob.

Examples:>>symsxx1alphaut>> A=[cos(x*t),sin(x*t)sin(x*t),cos(x*t)]

>>int(1/(1+x^2)) %compute 21

1dx

x + withoutconstantitemCans=atan(x)

>>int(sin(alpha*u),alpha) %compute sin( )u d withoutconstantitemCans=1/u*cos(alpha*u)

>>int(x1*log(1+x1),0,1) %compute1

1 1 10ln(1 )x x dx +

ans=1/4>>int(4*x*t,x,2,sin(t))ans=2*t*(sin(t)^24)

>>int([exp(t),exp(alpha*t)]) %compute [ , ]t te dt e dt ans=[ exp(t),1/alpha*exp(alpha*t)]

>>int(A,t) %compute [ cos , sin sin , cos ]xtdt xtdt xtdt xtdt ans=[1/x*sin(x*t), cos(x*t)/x][ cos(x*t)/x,1/x*sin(x*t)]

Taylorseriesexpansiontaylor(f)isthefifthorderMaclaurinpolynomialapproximationtof.


Lec76

Threeadditionalparameterscanbespecified,inalmostanyorder.taylor(f,n)isthe(n1)storderMaclaurinpolynomial.taylor(f,a)istheTaylorpolynomialapproximationaboutpointa.taylor(f,x)usestheindependentvariablexinsteadof findsym(f).

Examples:>>taylor(exp(x)) %evaluatethefirst5itemsof Taylorseriesexpansion at0ans=1x+1/2*x^21/6*x^3+1/24*x^41/120*x^5>>taylor(log(x),6,1) %evaluatethefirst61itemsof Taylorseriesexpansion at1ans=1x+1/2*x^21/6*x^3+1/24*x^41/120*x^5>>taylor(sin(x),6,pi/2) %evaluatethefirst61itemsof Taylorseriesexpansion atpi/2ans=11/2*(x1/2*pi)^2+1/24*(x1/2*pi)^4>>taylor(sin(x)*t,t) %evaluatethefirst5itemsof Taylorseriesexpansion responding

totans=sin(x)*t>>taylor(sin(x)*t,x) %evaluatethefirst5itemsof Taylorseriesexpansion responding

to x

ans=x*t1/6*t*x^3+1/120*t*x^5

SumofelementsS=sum(X)isthesumoftheelementsofthevectorX.If Xisamatrix,Sisarowvectorwiththesumovereachcolumn.IfXisfloatingpoint,thatisdoubleorsingle,Sisaccumulatednatively,thatisin thesame

classasX, andShasthesameclassasX.IfXisnotfloatingpoint,SisaccumulatedindoubleandShasclassdouble.S=sum(X,DIM)sumsalongthedimensionDIM.

Examples:>>X=[012345]>>sum(X) %evaluatearowvectorwiththesumovereachcolumnans=

3 5 7>>sum(X,1) %evaluatearowvectorwiththesumovereachcolumnans=

3 5 7>>sum(X,2) %evaluateacolumnvectorwiththesumovereachrowans=

3


Lec77

12

Symbolicsummationsymsum(S)istheindefinitesummationofSwithrespecttothesymbolicvariabledetermined

by findsym.symsum(S,v)istheindefinitesummationwithrespecttov.symsum(S,a,b)and symsum(S,v,a,b)arethedefinitesummationfromatob.

Examples:>>symskn

>>a1=simple(symsum(k)) %evaluate1

0

k

ii

= a1=1/2*k*(k1)

>>a2=simple(symsum(k,0,n1)) %evaluate1

0

n

kk

= a2=1/2*n*(n1)

>>a3=simple(symsum(k,0,n)) %evaluate0

n

kk

= a3=1/2*n*(n+1)

>>a4=simple(symsum(k^2,0,n)) %evaluate 20

n

kk

= a4=1/6*n*(n+1)*(2*n+1)

>>symsum(k^2,0,10) %evaluate10 2

0kk

= ans=385

>>symsum(k^2,11,10) %evaluate10 2

11kk

= ans=0

>>symsum(1/k^2) %evaluate 211k

i i = ans=Psi(1,k)

>>symsum(1/k^2,1,Inf) %evaluate 211

k k

= ans=1/6*pi^2

MATLAB Lecture 8 - Graphics-Curve.pdf


Lec81

MATLABLecture8GraphicsCurve

Ref: MATLABGraphicsBasic Plotting Commandsl Vocabulary:

plot xaxisxdisplay annotatecurve/curvedline explicitimplicit parameterpolarcoordinates boldfacegrid titlefigure legendradian radius

l Somefunctionsplot ezplot polar plot3 title xlabel ylabel *figure *legend *subplot*gridon *gridoff

l GraphicsMATLABprovidesavarietyoffunctionsfordisplayingvectordataaslineplots,aswellas

functionsforannotatingandprintingthesegraphs. Therepresentationofcurveandsurface TherepresentationofcurveonplaneExplicitscheme:y=f(x)Implicitformula:F(x,y)=0

Parameterrepresentation:( )( )

x x ty y t

= =

Polarcoordinatesrepresentation: ( )f =

Tabulation,saytableof trigonometric function,experimentaldataetc.Figure,saysinecurve,experimentalcurveetc. TherepresentationofcurveinspaceExplicitscheme:z=f(x,y)Implicitformula:F(x,y,z)=0

Parameterrepresentation:

( , )( , )( , )

x x u vy y u vz z u v

= = =

Dataform, { }, { }, { }, 1, 2,..., , 1, 2,...,i j ijx x y y z z i m j n = = = = = .

Graphics


Lec82

BasicPlottingCommands PlottingStepsThe process of constructing a basic graph to meet your presentation graphicsrequirementsisoutlinedinthefollowingtable.Thetableshowsseventypicalstepsandsomeexamplecodeforeach.Ifyouareperforminganalysisonly,youmaywanttoviewvariousgraphsjusttoexploreyour data. In this case, steps 1 and 3 may be all you need. If you are creatingpresentation graphics, youmaywant to finetune your graph by positioning it on thepage, setting line styles and colors, adding annotations, and making other suchimprovements.Step TypicalCode1.Prepareyourdata x=0:0.2:12

y1=bessel(1,x)y2=bessel(2,x)y3=bessel(3,x)

2.Selectawindowandpositionaplotregionwithinthewindow

figure(1)subplot(2,2,1)

3.Callelementaryplottingfunction h=plot(x,y1,x,y2,x,y3)4.Selectlineandmarkercharacteristics

set(h,'LineWidth',2,{'LineStyle'},{''':''.'})set(h,{'Color'},{'r''g''b'})

5.Setaxislimits,tickmarks,andgridlines

axis([0120.51])gridon

6. Annotate the graph with axislabels,legend,andtext

xlabel('Time')ylabel('Amplitude')legend(h,'First','Second','Third')title('BesselFunctions')[y,ix]=min(y1)text(x(ix),y,'FirstMin \rightarrow',...'HorizontalAlignment','right')

7.Exportgraph printdepsc tiff r200myplot

Plotfunctionplot(X,Y):plotsvector Y versusvectorX.IfXorYisamatrix,thenthevectorisplottedversustherowsorcolumnsofthematrix,whicheverlineup.It can be used to plot of a curve expressed in explicit scheme or parameterrepresentation.

>>x=[123456]y=[11.52317]>>plot(x,y)


Lec83

>>a=[1:6149162536]>>plot(a)

Plot sin xover [0, 2 ] withtitleandgrid.

>>t=0:pi/100:2*pi>>y=sin(t)>>plot(t,y)>>title(Plotofsinxvsx) %titled Plotofsinxvsx>>xlabel(x)ylabel(sinx) %addstextbesidetheXaxis&Yaxis

onthecurrentaxis>>gridon


Lec84

Plot the curve with sin , cosx a mt y a nt = = over [0, 2 ]t while a=8, m=2 and

n=5.>>t=linspace(0,2*pi) %generatesarowvectorof100linearlyequallyspaced

pointsbetween 0and 2.>>a=8>>m=2>>n=5>>x=a*sin(m*t)>>y=a*cos(n*t)>>plot(x,y,b) %colorblue,linestyle


Lec85

PolarPlotspolar(THETA, RHO) makes a plot using polar coordinates of the angle THETA, inradians,versustheradiusRHO.>>g=0.5>>theta=linspace(0,2*pi,51) %generates51linearlyequallyspacedpoints

between0and2.>>gain=2*g*(1+cos(theta))>>polar(theta,gain,r) %colorred,linestylesolid>>title(\bf Gainversusangle \theta) %boldface,\thetawillbedisplayas

EzplotFunction EasytousefunctionplotterItisusuallyusedtoplotacurveexpressedinimplicitformula.ezplot(FUN)plotsthefunctionFUN(X)overthedefaultdomain 2*pi


Lec86

Plotthefunction 4 4 4x y a + = over[2,2],wherea=2.

>>symsxya=2>>f=x^4+y^4a^4>>ezplot(f,[2,2])

Plot3FunctionPlotlinesandpointsin3Dspace.plot3(x,y,z),wherex,yandzarethreevectorsofthesamelength,plotsalinein3spacethroughthepointswhosecoordinatesaretheelementsofx,yandz.plot3(X,Y,Z),whereX,YandZarethreematricesofthesamesize,plotsseverallinesobtainedfromthecolumnsofX,YandZ.Variouslinetypes,plotsymbolsandcolorsmaybeobtainedwithplot3(X,Y,Z,s)wheres is a 1, 2 or 3 character string made from the characters listed under the plot3command.>>t=0:pi/50:10*pi>>plot3(sin(t),cos(t),t)


Lec87

>>t=0:pi/30:9*pia=20c=3>>x=a*t.*sin(t)y=a*t.*cos(t)z=c*t>>plot3(x,y,z,mo) %colormagenta,linetype circle

PlottingMultiplePlotsontheSameAxes>>x= pi:pi/20:pi>>y1=sin(x)>>y2=cos(x)>>plot(x,y1,b ) %bluecolorsolid()linestyle>>holdon %afterthiscommand,alladditionalplotswillbelaidontopofthe..

previouslyexistingplots.>>plot(x,y2,k) %black colordashed()linestyle>>holdoff %switchesplottingbehaviorbacktothedefaultsituation,inwhich

A newplotreplacesthepreviousone>>legend(sinx,cosx)Or>>x= pi:pi/20:pi>>y1=sin(x)>>y2=cos(x)>>plot(x,y1,b,x,y2,k)>>legend(sinx,cosx)


Lec88

CreateMultipleFigures>>figure(1) %figurewindow1>>x=0:0.05:2>>y1=exp(x) >>plot(x,y1)>>figure(2) %createanotherfigurewindow2>>y2=exp(x)>>plot(x,y2)

Subplots>>subplot(2,1,1) %create21subplotsincurrentfigure,arrangedin2rows

and1column,andselectssubplot1toreceiveallcurrentplottingcommands

>>x= pi:pi/20:pi


Lec89

>>y=sin(x)>>plot(x,y)>>title(1)>>subplot(2,1,2) %create21subplotsincurrentfigure,arrangedin2rows

and1column,andselectssubplot2toreceiveallcurrentplottingcommands

>>x= pi:pi/20:pi>>y=cos(x)>>plot(x,y)>>title(2)

Someexamples>>clearx=pi:pi/10:piy=tan(sin(x))>>y1=sin(tan(x))>>plot(x,y,'rs','linewidth',2,'markeredgecolor','k','markerfacecolor','g','markersize',10)>>holdon>>plot(x,y1,'ko','markeredgecolor','k','markerfacecolor','r','markersize',10)


Lec810

Acurvefittingexample()>>High_boy=[1.55,1.58,1.6,1.64,1.66,1.68,1.7,1.73, 1.78,1.8,1.82,1.85]>>Weight_boy=[60,57,57,65,63, 64,70,65,68,76, 72, 78]>>plot(High_boy,Weight_boy,'b*')>>holdon>>Everage_boy=polyfit(High_boy,Weight_boy,1)>>x=1.5:0.1:1.9boy_line=Everage_boy(1)* x+Everage_boy(2)>>plot(x,boy_line,'g','linewidth',1.2)

AppendixVariouslinetypes,plotsymbolsandcolorsmaybeobtainedwithPLOT(X,Y,S)whereSisacharacterstringmadefromoneelementfromanyorallthefollowing3columns:

b blue . point solidg green o circle : dottedr red x xmark . dashdotc cyan + plus dashedm magenta * star (none) no liney yellow s squarek black d diamondv triangle(down)^ triangle(up) triangle(right)p pentagramh hexagram

You can use a subset of TeX commands embedded in the string to produce special


Lec811

characters such asGreek letters and mathematical symbols. The following table liststhesecharactersand thecharactersequencesusedtodefine them.

matlab lecture 7 – calculus 微积分 ref: symbolic...

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