{VERSION 5 0 "IBM INTEL LINUX" "5.0" } {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 Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 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 "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Author" -1 19 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 8 8 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 10 "Harj. 6 AV" }}{PARA 19 " " 0 "" {TEXT -1 8 "22.10.02" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "A lustukset" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "alias(ref=GaussianEliminatio n):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "alias(Dot=DotProduct ,Id=IdentityMatrix):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 113 "Konjugoi ntis\344\344nn\366t DotProduct:ssa ova niin mutkikkaat ja monitahoiset , ett\344 on verrattomasti helpompaa m\344\344ritell\344" }}{PARA 0 " " 0 "" {TEXT -1 103 "suosiolla oma Sis-funktio sis\344tuloksi. (Lokali sointipuutteen takia k\344ytet\344\344n \"outoja\" nimi\344, kuten i_ \+ )" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "Sis:=(u,v)->add(u[i_]* conjugate(v[i_]),i_=1..LinearAlgebra[Dimension](u));" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 2 "1." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "with(LinearAlgebra):alias(Id=IdentityMatrix):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "Sis:=(u,v)->add(u[i_]*conjugate(v[i_]),i_ =1..LinearAlgebra[Dimension](u));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "A:=<<1,1>|<-2,3>>;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "p:=Determinant(A-lambda*Id(2));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "solve(p=0,lambda);" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 11 "K\344sinlasku:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "lambda:=2+I: w:=<-2,1+I>;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "u:=map(Re,w); v:=map(Im,w);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "uv:=;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "uvI:=uv^(-1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "alpha: =Re(lambda): beta:=Im(lambda):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "C :=<|>;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "A=uv.C.uvI;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Siin\344p \344 se taas kerran!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Annetaan \+ menn\344 viel\344 Maplella kokonaan:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "(oa,ov):=Eigenvectors(A);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 30 "lambda:=oa[1]: w:=ov[1..-1,1];" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 27 "u:=map(Re,w): v:=map(Im,w);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "P:=;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "PI:=P^(-1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "Jos ollaan laiskoja, niin C voidaan ratkaista n\344in halpahintais esti:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "C:=PI.A.P;" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 256 7 "Huom 1:" }{TEXT -1 127 " C-matriis i on yksik\344sitteinen (j\344rjestyst\344 vaille). P:n sarakkeet ovat ominaisvektoreita. Ne voidaan normeerata miten hyv\344ns\344." }} {PARA 0 "" 0 "" {TEXT -1 75 "Normeeraus vaikuttaa k\344\344neismatriis iin (\"kompenspoivalla normaarauksella\")." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 257 6 "Huom 2" }{TEXT -1 43 ": Jos P:n sarakkeet otetaan j\344rj estyksess\344 " }{MPLTEXT 1 0 8 "" }{TEXT -1 108 " , niin C:n \+ sivul\344vist\344j\344n merkit vaihtavat paikkaa. T\344m\344 on se toi nen muoto, joka edllisess\344 harjoituksessa" }}{PARA 0 "" 0 "" {TEXT -1 78 "aiheutti kiusaa. Kokeile harjoituksen vuoksi johtaa kaavaa niin p\344n, niin n\344et." }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 2 "2." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "v1:=<1,1>; v2:=<1,-1>;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "lambda[1]:=3: lambda[2]:=1/3:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "x[k]=c[1]*3^k*v1+c[2]*(1/3)^ k*v2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Trajektorit \{v1,v2\}-ko ordinaatistossa toteuttavat siten ehdon" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 12 "x*y = vakio;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 113 "Jos siis merkit\344\344n koordinaatteja t\344tt\344 ov-koordinaat istossa (x,y). Saadaan siis ihan tavallinen hyperbeliparvi." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 170 "Esitet\344\344n ominaisvektorikoo rdinaatistossa, joka kierret\344\344n oikeaan asentoon k\344ytt\344m \344ll\344 3d-piirtoa. (2d-plotilla ei onnistu n\344in, silloin pit \344isi kierto kohdistaa dataan," }}{PARA 0 "" 0 "" {TEXT -1 183 "mik \344 on hiukan vaivalloisempaa). Kannattaa toki huomata, ett\344 t \344m\344 mentelm\344 soveltuu vain silloin, kun kysess\344 on koordin aatiston kierto. Jos akselien kulma ja/tai skaalat muuttuvat," }} {PARA 0 "" 0 "" {TEXT -1 77 "on ominaisvektorimatriisin antamaa kannan vaihtomuunnosta sovellettava dataan." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 169 "display(spacecurve([x,1/x,0,x=.1..10]),spacecurve([x ,-1/x,0,x=.1..10]),spacecurve([-x,1/x,0,x=.1..10]),spacecurve([-x,-1/x ,0,x=.1..10]),axes=boxed,orientation=[-135,1]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 105 "Kannattaa t\344ydent\344\344 k\344sin. Alkupisteet \+ kannattaa valita ensin ominaisvektoreilta ja piirt\344\344 suuntanuole t." }}{PARA 0 "" 0 "" {TEXT -1 151 "Sitten esim. pisteet (1,1), (-1,1) , (1,-1), (-1,-1). T\344ss\344 tapauksessa saadaan suoraan muotoa y= C/x oleva k\344yr\344parvi ominaisvektorikoordinaatistossa." }}{PARA 0 "" 0 "" {TEXT -1 44 "Sit\344 yll\344 olevassa piirroksessa on k\344y tetty." }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 2 "3." }}{PARA 0 "" 0 "" {TEXT -1 110 "A:n diagonalisointi on annettu, siit\344 saadaan itse A \+ vain kertolaskulla (ja k\344\344nteismatriisin muodostuksella)." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 80 "with(LinearAlgebra): Sis:=(u,v)->add(u[i_]*c onjugate(v[i_]),i_=1..Dimension(u));" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "A.v[1]=lambda[1]*v[1]; A .v[2]=lambda[2]*v[2];" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "A .< v[1] | v[2]> = ;" "6#/-%\".G6$%\"AG-%$< |gr>G6$&%\"vG6#\"\"\"6#&F,6#\"\"#-F)6$*&&%'lambdaG6#F.F.&F,6#F.F.6#*&& F76#F2F.&F,6#F2F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "T\344st\344 \+ saadaan A kertomalla oikealta " }{XPPEDIT 19 1 "" "6#-%$< |gr>G6$&%\"vG6#\"\"\"6#&F'6#\"\"#" }{TEXT -1 23 ":n k\344\344nteismatr iisilla." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "v1v2:=< <1,1>|<-1,1>>;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Lambda:=D iagonalMatrix([3,1/3]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 " A:=v1v2.Lambda.v1v2^(-1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "Eigenvectors(A); #Tarkistus" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 157 "Luonnollinen kysmys: Jos tunnetaan matriisin ominaisarvot ja - ve ktorit, onko matriisi 1-k\344s. m\344\344r\344tty siin\344kin tapaukse ssa, ett\344 se ei ole diagonalisoituva?" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 2 "4." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "v[1]:=<2,1 ,3>: v[2]:=<1,-2,0>; v[3]:=<6,3,-5>;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "seq(seq(Sis(v[i],v[j]),j=1..3),i=1..3);linalg[matrix ](3,3,[%]); # T\344ss\344\240kohtuullisen k\344tev\344 tapa katsoa sis \344tulokombinaatiot." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "Ma trix(3,3,(i,j)->Sis(v[i],v[j])); # T\344ss\344 on jo eleganssia, huom aa t\344m\344 matriisin muodostustapa!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "U:=; # T\344m\344 on omalla tavall aan elegantein, pit\344\344 vaan osata ajatella \"matriisiksi\"." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Transpose(U).U;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Jo se on varmaakin varmempaa!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "u:=<9,-2,4>;" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 99 "c[1]:=Sis(u,v[1])/Sis(v[1],v[1]);c[2]:=Sis(u,v [2])/Sis(v[2],v[2]);c[3]:=Sis(u,v[3])/Sis(v[3],v[3]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "add(c[j]*v[j],j=1..3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalm(%);" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 12 "Oikein meni!" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 2 "5 ." }}{EXCHG {PARA 15 "" 0 "" {TEXT -1 93 "cos ja sin integroituna koko jakson yli = 0 , joten 1 on kohtisuorassa kaikkia muita vastaan." }} {PARA 15 "> " 0 "" {XPPEDIT 19 1 "int(cos(k*x)*sin(j*x),x = -Pi .. Pi) = 0,%?;" "6$/-%$intG6$*&-%$cosG6#*&%\"kG\"\"\"%\"xGF-F--%$sinG6#*&%\" jGF-F.F-F-/F.;,$%#PiG!\"\"F7\"\"!%#%?G" }{TEXT -1 93 "koska pariton ke rtaa parillinen on pariton , ja sellaisen int. yli O:n suht. symm. v \344lin = 0." }}{PARA 15 "" 0 "" {TEXT -1 32 "J\344ljelle j\344\344v \344t siten vain n\344m\344:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 30 "cos(x)*cos(2*x); %=combine(%);" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 30 "sin(x)*sin(2*x); %=combine(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "Ja taas integroidaan cos- tai sin- funkit ota yhden tai useamman t\344yden jakson yli => 0." }}{PARA 0 "" 0 "" {TEXT -1 10 "Yleisesti:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 " cos(alpha*x)*cos(beta*x): %=combine(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "sin(alpha)*sin(beta): %=combine(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "Emme siis tarvinneet Maplen erinomaista int-kom entoa ollenkaan. No mik\344\344n ei est\344:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "F:=[1,cos(x),cos(2*x),sin(x),sin(2*x)];" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Matrix(5,5,(i,j)->int(F[i]*F [j],x=-Pi..Pi));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "T\344t\344 on syyt\344 ihastella!" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 2 "6." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "with(LinearAlgebra):alias(re f=GaussianElimination):\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "A:=<<3,1,-1,3>|<-5,1,5,-7>|<1,1,-2,8>>;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "ref(A);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Ova t LRT, joten GS soveltuu. Tehd\344\344n vaiheittain \"k\344sin laskie n\":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "v:=seq(A[1..-1,j],j =1..3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Norm(v[1],2);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "e[1]:=Normalize(v[1],2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Sis(e[1],e[1]);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "f[2]:=v[2]-Sis(v[2],e[1])*e[ 1];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "e[2]:=Normalize(f[2] ,2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "f[3]:=v[3]-Sis(v[3] ,e[1])*e[1]-Sis(v[3],e[2])*e[2];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "e[3]:=Normalize(f[3],2);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 15 "e[1],e[2],e[3];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "map(v->v*10/sqrt(5),[%]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "Tai:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "10*/sqrt(5);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Valmiill a GramSchmidt:lla:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "vvv:= seq(A[1..-1,j],j=1..3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 " GramSchmidt([vvv]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Gram Schmidt([vvv],normalized);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Q:=Matrix(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Transp ose(Q).Q;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}}}{MARK "8" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }