{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 255 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 255 1 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE " Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Ti mes" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "T imes" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 34 " Umas Palavras Sobre \301lgebra Linear" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#7dr%#&xG%$AddG%(AdjointG%3BackwardSubs tituteG%+BandMatrixG%&BasisG%-BezoutMatrixG%/BidiagonalFormG%-Bilinear FormG%5CharacteristicMatrixG%9CharacteristicPolynomialG%'ColumnG%0Colu mnDimensionG%0ColumnOperationG%,ColumnSpaceG%0CompanionMatrixG%0Condit ionNumberG%/ConstantMatrixG%/ConstantVectorG%%CopyG%2CreatePermutation G%-CrossProductG%-DeleteColumnG%*DeleteRowG%,DeterminantG%)DiagonalG%/ DiagonalMatrixG%*DimensionG%+DimensionsG%+DotProductG%6EigenConditionN umbersG%,EigenvaluesG%-EigenvectorsG%&EqualG%2ForwardSubstituteG%.Frob eniusFormG%4GaussianEliminationG%2GenerateEquationsG%/GenerateMatrixG% (GenericG%2GetResultDataTypeG%/GetResultShapeG%5GivensRotationMatrixG% ,GramSchmidtG%-HankelMatrixG%,HermiteFormG%3HermitianTransposeG%/Hesse nbergFormG%.HilbertMatrixG%2HouseholderMatrixG%/IdentityMatrixG%2Inter sectionBasisG%+IsDefiniteG%-IsOrthogonalG%*IsSimilarG%*IsUnitaryG%2Jor danBlockMatrixG%+JordanFormG%(LA_MainG%0LUDecompositionG%-LeastSquares G%,LinearSolveG%$MapG%%Map2G%*MatrixAddG%2MatrixExponentialG%/MatrixFu nctionG%.MatrixInverseG%5MatrixMatrixMultiplyG%+MatrixNormG%,MatrixPow erG%5MatrixScalarMultiplyG%5MatrixVectorMultiplyG%2MinimalPolynomialG% &MinorG%(ModularG%)MultiplyG%,NoUserValueG%%NormG%*NormalizeG%*NullSpa ceG%3OuterProductMatrixG%*PermanentG%&PivotG%*PopovFormG%0QRDecomposit ionG%-RandomMatrixG%-RandomVectorG%%RankG%6RationalCanonicalFormG%6Red ucedRowEchelonFormG%$RowG%-RowDimensionG%-RowOperationG%)RowSpaceG%-Sc alarMatrixG%/ScalarMultiplyG%-ScalarVectorG%*SchurFormG%/SingularValue sG%*SmithFormG%8StronglyConnectedBlocksG%*SubMatrixG%*SubVectorG%)SumB asisG%0SylvesterMatrixG%/ToeplitzMatrixG%&TraceG%*TransposeG%0Tridiago nalFormG%+UnitVectorG%2VandermondeMatrixG%*VectorAddG%,VectorAngleG%5V ectorMatrixMultiplyG%+VectorNormG%5VectorScalarMultiplyG%+ZeroMatrixG% +ZeroVectorG%$ZipG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "# uma matriz 3x3\nA:=Matrix(3,3,[27,99,92,8,29,-31,69,44,67]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\"*!Q:)\\\"-%'MATRIXG6#7%7%\" #F\"#**\"##*7%\"\")\"#H!#J7%\"#p\"#W\"#n%'MatrixG" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 53 "# autovalores e autovetores\nl:=evalf(Eigenv alues(A));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"lG-%'RTABLEG6%\"*GS2 l\"-%'MATRIXG6#7%7#^$$\"+'p'\\_5!\"($!\"$!\")7#^$$!+ayqeZF4$\"*#>\\zE! #<7#^$$\"+'*3uLlF4$\"+330KPF;&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "Autovalores obtidos acima devem ser ra\355zes do \+ polin\364mio caracter\355stico" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Id:=IdentityMatrix(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#I dG-%'RTABLEG6%\"*wc)*[\"-%'MATRIXG6#7%7%\"\"\"\"\"!F/7%F/F.F/7%F/F/F.% 'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "pc:=Determinant (A-lambda*Id);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#pcG,*\"'WsK!\"\"* &\"%T7\"\"\"%'lambdaGF*F**&\"$B\"F*)F+\"\"#F*F**$)F+\"\"$F*F'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "# confere se l's s\343o ra \355zes\nseq(subs(lambda=l[i],pc),i=1..3);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%^$$\"\"!F%$\"+S!=+$=!#8^$F$$!+u\"QXi%!#9^$F$$\"+7?1#o\" F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 120 "Cada um dos autovalores ca lculados, quando substituimos no polin\364mio caracter\355stico, d\341 um n\372mero pequeno, mas complexo." }}{PARA 0 "" 0 "" {TEXT -1 59 "P arece que a parte imagin\341ria dos lambda's \351 erro num\351rico." } }{PARA 0 "" 0 "" {TEXT -1 40 "Vamos plotar o polin\364mio caracter\355 stico:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "plot(pc,lambda=-5 0..110);" }}{PARA 13 "" 1 "" {GLPLOT2D 546 294 294 {PLOTDATA 2 "6%-%'C URVESG6$7jn7$$!#]\"\"!$\"&1K%F*7$$!3xmmmmFiD[!#;$\"3GsmkX\\\"o;\"!#87$ $!3aLLLLbC^YF0$!3am5A:q7C=F37$$!3')*****\\$3_*\\%F0$!3qJ`!y/?lH%F37$$! 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" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "# pol. car.\npc:=Charac teristicPolynomial(A,lambda);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#pc G,*\"'b5#)\"\"\"*$)%'lambdaG\"\"$F'F'*&\"#nF')F*\"\"#F'!\"\"*&\"%uVF'F *F'F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "plot(pc,lambda=-10 0..100);" }}{PARA 13 "" 1 "" {GLPLOT2D 556 254 254 {PLOTDATA 2 "6%-%'C URVESG6$7W7$$!$+\"\"\"!$!'X:TF*7$$!3GLLLe%G?y*!#;$!3gwBOau7#G$!#77$$!3 cmmm;p0k&*F0$!3[Y%fu/dI[#F37$$!3!4+]P/,WP*F0$!3wK>=fL<:=F37$$!3#QLL3&=\\$F37$$!3%*****\\P8#\\4(F0$\"3K*4YJn$zpVF37$$!3]mm;/siqmF 0$\"3Q'G$y7'=(y^F37$$!3u****\\(y$pZiF0$\"37^&H\\?I$*)eF37$$!3jKLL$yaE \"eF0$\"3!G3CC3g`_'F37$$!3+mmm\">s%HaF0$\"3)y(yw$\"3yxDv#Qh[@)F37$$\"3fVLLezw5VFjs$\"3?&=(zWwM5!)F37$ $\"395++v$Q#\\\")Fjs$\"30-i,s%>]\"yF37$$\"3)QLL$e\"*[H7F0$\"39q:k:o2!f (F37$$\"3a++++dxd;F0$\"3grBZyu'oM(F37$$\"3g+++D0xw?F0$\"30T_I.=x-rF37$ $\"3Q++]i&p@[#F0$\"3#>V-'4:)\\'oF37$$\"3H*****\\2'HKHF0$\"3sX:ORH+/mF3 7$$\"3pmmmmZvOLF0$\"3K,oC:?fwjF37$$\"3m,++]2goPF0$\"3UpqG.k$e9'F37$$\" 3UKL$eR<*fTF0$\"3#ob$)G0V9&fF37$$\"3M-++])Hxe%F0$\"3%Qt$*Q'=IfdF37$$\" 3alm;H!o-*\\F0$\"3>NG>eg.-cF37$$\"3!=++DTO5T&F0$\"3-)>8M'HOmaF37$$\"3< mmm;WTAeF0$\"3%)H!y)[*>jO&F37$$\"3A,+]i!*3`iF0$\"3o9(y%\\;q+`F37$$\"3! fLLL$*zym'F0$\"3A1XZN&Q(z_F37$$\"3uLLL3N1#4(F0$\"3WNN^M!zcI&F37$$\"3sp m;HYt7vF0$\"3'*QQC,p>$Q&F37$$\"39-+++xG**yF0$\"3R1&fx/UP]&F37$$\"3[nmm T6KU$)F0$\"3\"ouWn*4e/dF37$$\"3*\\LLL`v&Q()F0$\"3m\"Qb%G%3]%fF37$$\"3i -+]i`1h\"*F0$\"3eAM:M?&*oiF37$$\"3%H++v.Uac*F0$\"3YC2>y'Q%[mF37$$\"$+ \"F*$\"'bOrF*-%'COLOURG6&%$RGBG$\"*++++\"!\")$F*F*Fg\\l-%+AXESLABELSG6 $%'lambdaGQ!6\"-%%VIEWG6$;F(F\\\\l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 44.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "Ok, agora somente uma raiz real, pois temos um par de com plexos conjugados." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "O que ser\341 que ocorre com uma matriz sim\351trica alea t\363ria?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "A:=Matrix(3,3, [60,99,9,99,16,14,9,14,22]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG -%'RTABLEG6%\"*?*))*[\"-%'MATRIXG6#7%7%\"#g\"#**\"\"*7%F/\"#;\"#97%F0F 3\"#A%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "l:=evalf( Eigenvalues(A));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"lG-%'RTABLEG6% \"*sV%)[\"-%'MATRIXG6#7%7#^$$\"+1D,:9!\"($!\"#!\")7#^$$!+MifujF4$!+;;5 kC!#<7#^$$\"+o6ZC?F4$\"+;;5kWF;&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 101 "Novamente, agora os autovalores devem ser todo s reais. As partes imagin\341rias acima s\343o erro num\351rico." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "# pega parte real\nl[1]:=Re( l[1]);l[2]:=Re(l[2]);l[3]:=Re(l[3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>&%\"lG6#\"\"\"$\"+1D,:9!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&% \"lG6#\"\"#$!+Mifuj!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"lG6#\" \"$$\"+o6ZC?!\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "pc:=Cha racteristicPolynomial(A,lambda);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% #pcG,*\"'5E=\"\"\"*$)%'lambdaG\"\"$F'F'*&\"#)*F')F*\"\"#F'!\"\"*&\"%Yu F'F*F'F0" }}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "# verificando...\nsubs (lambda=l[3],pc);" }}{PARA 0 "" 0 "" {XPPMATH 20 "6#$\"\"\"!\"%" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 18 "Um Sistema de ED's" }}{PARA 0 "" 0 "" {TEXT -1 50 "Vamos contar caminhos de n passos na figura abaixo" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 230 "with(plots):\none_poly := [[0,0],[0,1],[1,1],[1,0]]:\nlab:=text plot([[-0.1,-0.1,\"1\"],[-0.10,1.1,\"2\"],[1.1,1.1,\"3\"],[1.1,-0.1,\" 4\"]],color=red,font=[TIMES,ROMAN,15]):\npol:=polygonplot(one_poly,thi ckness=3,axes=none):\ndisplay(pol,lab);" }}{PARA 13 "" 1 "" {GLPLOT2D 152 152 152 {PLOTDATA 2 "6(-%)POLYGONSG6$7&7$$\"\"!F)F(7$F($\"\"\"F)7$ F+F+7$F+F(-%*THICKNESSG6#\"\"$-%%TEXTG6&7$$!\"\"F8F7Q\"16\"-%%FONTG6%% &TIMESG%&ROMANG\"#:-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-F46&7$$!#5!\"# $\"#6F8Q\"2F:F;FA-F46&7$FNFNQ\"3F:F;FA-F46&7$FNF7Q\"4F:F;FA-%*AXESSTYL EG6#%%NONEG" 1 2 0 1 10 0 2 9 1 1 2 1.000000 43.000000 45.000000 0 0 " Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" }}}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 142 "Fazendo uma an\341lise parecida com a a da aula e m que contamos caminhos em um quadrado com diagonais, vemos agora que \+ precisamos de 3 vari\341veis:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 111 "F(n), A(n), O(n), que contam n\372mero d e caminhos fechados, entre v\351rtices adjacentes e opostos, respectiv amente." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "Para obter as ED's, podemos listar os caminhos que come\347am e t erminam em um e dividi-los em duas classes:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "1, *, ..., *, " }{TEXT 256 1 "2 " }{TEXT -1 3 ", 1" }}{PARA 0 "" 0 "" {TEXT -1 14 "1, *, ..., *, " } {TEXT 257 1 "4" }{TEXT -1 3 ", 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 27 "que tem A(n) caminhos cada." }}{PARA 0 " " 0 "" {TEXT -1 12 "Isso d\341 a ED" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {XPPEDIT 18 0 "F(n)= 2*A(n-1)" "6#/-%\"FG6#%\"nG*&\" \"#\"\"\"-%\"AG6#,&F'F*F*!\"\"F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "De modo an\341logo, para \+ A(n) entre 1 e 2 temos:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "1, *, ..., *, " }{TEXT 258 1 "1" }{TEXT -1 3 ", 2" } }{PARA 0 "" 0 "" {TEXT -1 14 "1, *, ..., *, " }{TEXT 260 1 "3" }{TEXT -1 3 ", 2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "e portanto" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "A(n)=F(n-1)+O(n-1)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "De modo a n\341logo," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "O(n)=F(n-1)+O(n-1)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 26 "As condi\347\365es iniciais s\343o " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "F(0)=1, A(0)=0, O(0 )=0." }}{PARA 0 "" 0 "" {TEXT -1 62 "J\341 vimos que a forma mais r \341pida de resolver o problema \351 usar" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "unassign('A');\n# \+ tentar usar rsolve\neqs:=F(n)=2*A(n-1),A(n)=F(n-1)+O(n-1),O(n)=2*A(n-1 );\ncis:=F(0)=1,A(0)=0,O(0)=0;\nrsolve(\{eqs,cis\},\{F,A,O\});" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqsG6%/-%\"FG6#%\"nG,$*&\"\"#\"\"\" -%\"AG6#,&F*F.F.!\"\"F.F./-F0F),&-F(F1F.-%\"OGF1F./-F9F)F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$cisG6%/-%\"FG6#\"\"!\"\"\"/-%\"AGF)F*/-% \"OGF)F*" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%'rsolveG6$<(/-%\"OG6#%\" nG,$*&\"\"#\"\"\"-%\"AG6#,&F+F/F/!\"\"F/F//-%\"FGF*F,/-F1F*,&-F7F2F/-F )F2F//-F)6#\"\"!F@/-F1F?F@/-F7F?F/<%F)F1F7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "Algo parece ter dado errado, j\341 que rsolve n\343o resp ondeu." }}{PARA 0 "" 0 "" {TEXT -1 85 "Note algo mais estranho ainda. \+ Se mudarmos as condi\347\365es iniciais para as equivalentes" }}{PARA 0 "" 0 "" {TEXT -1 28 "F(1)=0, A(1)=0, O(1)=0 temos" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "cis:=F(1)=0,A(1)=1,O(1)=0;\nrsolve(\{eqs, cis\},\{F,A,O\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$cisG6%/-%\"FG6 #\"\"\"\"\"!/-%\"AGF)F*/-%\"OGF)F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# <%/-%\"FG6#%\"nG,&*&\"\"%!\"\")!\"#F(\"\"\"F/*&F+F,)\"\"#F(F/F//-%\"AG F',&*&F+F,F-F/F,*&F+F,F1F/F//-%\"OGF'F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "E agora o rsolve funciona sem problemas." }}{PARA 0 "" 0 "" {TEXT -1 108 "Vamos ver se calculando como ter\355amos feito na m \343o, descobrimos o que est\341 por tr\341s deste aparente paradoxo. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "M:=Matrix(3,3,[0,2,0,1, 0,1,0,2,0]);\n(l,S):=Eigenvectors(M);" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG-%'RTABLEG6%\"*;h!*\\\"-%'MA TRIXG6#7%7%\"\"!\"\"#F.7%\"\"\"F.F1F-%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>6$%\"lG%\"SG6$-%'RTABLEG6%\"*)o,p;-%'MATRIXG6#7%7#\"\" !7#!\"#7#\"\"#&%'VectorG6#%'columnG-F)6%\"*'\\\"y]\"-F-6#7%7%!\"\"\"\" \"FB7%F1FAFB7%FBFBFB%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "# arma a matriz diagonal\nLambda:=DiagonalMatrix(l);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'LambdaG-%'RTABLEG6%\"*K.@]\"-%'MATRIXG6#7 %7%\"\"!F.F.7%F.!\"#F.7%F.F.\"\"#%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 35 "# e a condi\347\343o inicial\nv0:=<1,0,0>;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v0G-%'RTABLEG6%\"*C=?]\"-%'MATRIXG6 #7%7#\"\"\"7#\"\"!F/&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Agora j\341 d\341 pra ver o que aconteceu. Um dos autoval ores \351 zero." }}{PARA 0 "" 0 "" {TEXT -1 50 "At\351 a\355, tudo bem . Se queremos v(n), basta calcular" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {XPPEDIT 18 0 "v(n)=A^n*v0" "6#/-%\"vG6#%\"nG*&)%\"A GF'\"\"\"%#v0GF+" }}{PARA 0 "" 0 "" {TEXT -1 26 "que tamb\351m faz sen tido via" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "A^n=S*Lambda^n*S^(-1)" "6#/)%\"AG%\"nG*(%\"SG\"\"\")%'L ambdaGF&F))F(,$F)!\"\"F)" }}{PARA 0 "" 0 "" {TEXT -1 58 "Agora o que a contece ao elevarmos a matriz diagonal a n=0?" }}{PARA 0 "" 0 "" {TEXT -1 19 "Temos um problema: " }{XPPEDIT 18 0 "0^0" "6#*$\"\"!F$" } {TEXT -1 19 " 'e indeterminado." }}{PARA 0 "" 0 "" {TEXT -1 89 "Para \+ os nossos fins, devemos defini-lo como 1, para recuperar v(0)=v(0) na \+ f\363rmula acima." }}{PARA 0 "" 0 "" {TEXT -1 79 "Mas ningu\351m conto u essa historinha para o rsolve e ele na d\372vida, n\343o faz nada." }}{PARA 0 "" 0 "" {TEXT -1 43 "Para calcular a exponencia\347\343o, po demos usar" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 31 "Lambdan:=MatrixPower(Lambda,n);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%(LambdanG-%'RTABLEG6%\"*C\">o;-%'MATRIXG6#7%7%\"\"! F.F.7%F.)!\"#%\"nGF.7%F.F.)\"\"#F2%'MatrixG" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 44 "Note que ele aqui \351 desleixado e assume que " } {XPPEDIT 18 0 "0^n=0" "6#/)\"\"!%\"nGF%" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "vn:=S.Lambdan.S^(-1).v0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#vnG-%'RTABLEG6%\"*ks9m\"-%'MATRIXG6#7%7#, &*&\"\"%!\"\")!\"#%\"nG\"\"\"F5*&F0F1)\"\"#F4F5F57#,&*&F0F1F2F5F1*&F0F 1F7F5F5F-&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "Essa \351 a solu\347\343o que o rsolve deu quando mudamos a condi\347 \343o inicial para come\347ar em n=1." }}{PARA 0 "" 0 "" {TEXT -1 33 " Note que ela n\343o funciona em n=0," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "eval(vn,n=0),v0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$- %'RTABLEG6%\"*w8!f;-%'MATRIXG6#7%7##\"\"\"\"\"#7#\"\"!F+&%'VectorG6#%' columnG-F$6%\"*C=?]\"-F(6#7%7#F-F/F/F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 " pois " }{XPPEDIT 18 0 "0^0" "6#*$\"\"!F$" }{TEXT -1 26 " \+ \351 tratado como 0 e n\343o 1." }}{PARA 0 "" 0 "" {TEXT -1 84 "Podemo s nos perguntar agora se a fun\347\343o MatrixPower n\343o podia ser u sada diretamente." }}{PARA 0 "" 0 "" {TEXT -1 15 "A resposta \351..." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "vn:=MatrixPower(M,n).v0; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#vnG-%'RTABLEG6%\"*W7]l\"-%'MATR IXG6#7%7#,$*(\"\"%!\"\")\"\"#%\"nG\"\"\",&)F1F4F5F5F5F5F57#,$*(F0F1F2F 5,&)F1,&F4F5F5F5F5F5F5F5F5F-&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 38 "Sim, poderia e daria a mesma resposta." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 19 "Um Sistema de EDO's" }}{PARA 0 " " 0 "" {TEXT -1 19 "Consideremos agora " }}{PARA 0 "" 0 "" {TEXT -1 22 "2x' - y' + x +7y =0" }}{PARA 0 "" 0 "" {TEXT -1 22 "-x' +2y +7 x - y =0" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 15 "x(0)=1, y(0)=0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "Note que esse sistema n\343o est\341 escrito da form a" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 " RTABLE(150330960,MATRIX([[diff(x(t),t)], [diff(y(t),t)]]),Vector[colum n]);" "6#-%'RTABLEG6%\"*g4L]\"-%'MATRIXG6#7$7#-%%diffG6$-%\"xG6#%\"tGF 27#-F-6$-%\"yGF1F2&%'VectorG6#%'columnG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "M.RTABLE(150454232,MATRIX([[x], [y]]),Vector[column]);" "6#-%\". G6$%\"MG-%'RTABLEG6%\"*KUX]\"-%'MATRIXG6#7$7#%\"xG7#%\"yG&%'VectorG6#% 'columnG" }}{PARA 0 "" 0 "" {TEXT -1 29 "Mas pode ser escrito da forma " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "Av'(t )+Bv(t)=0, para" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 54 "A:=Matrix(2,2,[2,-1,-1,2]);\nB:=Matrix(2,2,[1, 7,7,-1]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\"*_;?m\"-%'MATRIXG6#7$7$\"\"#!\"\"7$ F/F.%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'RTABLEG6%\"* ?\")=]\"-%'MATRIXG6#7$7$\"\"\"\"\"(7$F/!\"\"%'MatrixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Ora, portanto v'=Mv, onde " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "M:=-A^(-1).B;\nv0:=<1,0>;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG-%'RTABLEG6%\"*Oz#f;-%'MATRIXG6#7$7$! \"$#!#8\"\"$7$!\"&#F3F1%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% #v0G-%'RTABLEG6%\"*+C+n\"-%'MATRIXG6#7$7#\"\"\"7#\"\"!&%'VectorG6#%'co lumnG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Podemos novamente calcul ar autovalores e autovetores etc. Vamos l\341" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "(l,S):=evalf(Eigenvectors(M));\nLambda:=Diagonal Matrix(l);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>6$%\"lG%\"SG6$-%'RTABLE G6%\"*+ivm\"-%'MATRIXG6#7$7#$\"+$*>\"*oB!\"*7#$!+f'yb.(F3&%'VectorG6#% 'columnG-F)6%\"*'\\-u;-F-6#7$7$$!+>t:r!)!#5$\"+*R#yt5F37$$\"\"\"\"\"!F H%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'LambdaG-%'RTABLEG6%\" *)3%Qn\"-%'MATRIXG6#7$7$$\"+$*>\"*oB!\"*\"\"!7$F1$!+f'yb.(F0%'MatrixG " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "A solu\347\343o \351 " } {XPPEDIT 18 0 "exp(A*t)*v(0)=S*exp(Lambda*t)*S^(-1)*v(0)" "6#/*&-%$exp G6#*&%\"AG\"\"\"%\"tGF*F*-%\"vG6#\"\"!F***%\"SGF*-F&6#*&%'LambdaGF*F+F *F*)F1,$F*!\"\"F*-F-6#F/F*" }{TEXT -1 30 ", que podemos calcular fazen do" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "expLambdat:=MatrixExp onential(Lambda*t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+expLambdatG- %'RTABLEG6%\"*O/'o;-%'MATRIXG6#7$7$-%$expG6#,$*&$\"+$*>\"*oB!\"*\"\"\" %\"tGF6F6\"\"!7$F8-F/6#,$*&$\"+f'yb.(F5F6F7F6!\"\"%'MatrixG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "vt:=S.expLambdat.S^(-1).v0; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#vtG-%'RTABLEG6%\"*kiHn\"-%'MATR IXG6#7$7#,&*&$\"+%z=6H%!#5\"\"\"-%$expG6#,$*&$\"+$*>\"*oB!\"*F3%\"tGF3 F3F3F3*&$\"+17))3dF2F3-F56#,$*&$\"+f'yb.(F;F3F " 0 "" {MPLTEXT 1 0 37 "vt:=evalf(MatrixExponential(M*t).v0 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#vtG-%'RTABLEG6%\"*/!*Ro\"-%'M ATRIXG6#7$7#,&*&$\"+&z=6H%!#5\"\"\"-%$expG6#,$*&$\"+$*>\"*oB!\"*F3%\"t GF3F3F3F3*&$\"+07))3dF2F3-F56#,$*&$\"+f'yb.(F;F3F " 0 "" {MPLTEXT 1 0 108 "edos:=2*diff(x(t),t) - diff(y(t),t) + x(t) +7*y(t) \+ =0,\n-diff(x(t),t) +2*diff(y(t),t) +7*x(t) - y(t) =0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%edosG6$/,**&\"\"#\"\"\"-%%diffG6$-%\"xG6#%\" tGF1F*F*-F,6$-%\"yGF0F1!\"\"F.F**&\"\"(F*F4F*F*\"\"!/,*F+F6*&F)F*F2F*F **&F8F*F.F*F*F4F6F9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "ci:= x(0)=1,y(0)=0;\nevalf(dsolve([edos,ci]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ciG6$/-%\"xG6#\"\"!\"\"\"/-%\"yGF)F*" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<$/-%\"yG6#%\"tG,&*&$\"+P!4mJ&!#5\"\"\"-%$expG6#,$ *&$\"+$*>\"*oB!\"*F.F(F.F.F.!\"\"*&$\"+Q!4mJ&F-F.-F06#,$*&$\"+f'yb.(F6 F.F(F.F7F.F./-%\"xGF',&*&$\"+&z=6H%F-F.F/F.F.*&$\"+07))3dF-F.F;F.F." } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "Uma \372ltima pergunta seria o q ue ocorre com a solu\347\343o a medida que t vai a infinito?" }}{PARA 0 "" 0 "" {TEXT -1 43 "A resposta \351 que ela se alinha com o vetor \+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "<-.5316609037,.42911187 95>;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\"*S7mo\"-%'MATRIX G6#7$7#$!+P!4mJ&!#57#$\"+&z=6H%F.&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "que s\343o os coeficientes dos termos " } {XPPEDIT 18 0 "exp(lambda*t)" "6#-%$expG6#*&%'lambdaG\"\"\"%\"tGF(" } {TEXT -1 14 ", com o maior " }{XPPEDIT 18 0 "lambda" "6#%'lambdaG" } {TEXT -1 1 "." }}}}}{MARK "5" 0 }{VIEWOPTS 1 1 0 3 2 1804 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }{RTABLE_HANDLES 149815380 165074028 149906116 166901688 150781496 150210332 150201824 166819124 166147264 165901376 165501244 166201652 150188120 165927936 167002400 166756200 167402496 167384088 166860436 167296264 168399004 168661240 }{RTABLE M7R0 I6RTABLE_SAVE/149815380X,%)anythingG6"6"[gl!"%!!!#*"$"$"#F"")"#p"#**"#H"#W"##*! 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