{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 }{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 "Headi ng 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 "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{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 LV" }}{PARA 19 " " 0 "" {TEXT -1 12 "25.10.02 HA" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 20 "Alustukset, ohjelmat" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 " read(\"/home/apiola/opetus/peruskurssi/v2-3/302/L/v302.mpl\");" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "# read(\"/p/edu/mat-1.414/ma ple/v302.mpl\"); # Poista kommentti t\344st\344 ja lis\344\344 edelle !" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "#read(\"c:\\\\usr\\\\h eikki\\\\s02\\\\v302.mpl\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "with(LinearAlgebra): with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "op(trajenuoli);" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 1 "1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "print(SisF); # S is\344tulo funktioavaruudessa, emme kuitenkaan t\344ss\344 \"maanl\344 heisess\344\" tavassa tarvitse sit\344." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Kaikkein opettavaisinta on laskea aivan k\344sinlaskua si muloiden:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "f1:=x:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "a0:=int(f1*1,x=-Pi..Pi)/int( 1*1,x=-Pi..Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "a1:=int(f1*cos(x),x=-Pi..Pi) /int(cos(x)^2,x=-Pi..Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "a2:=int(f1*cos(2*x),x=-Pi..Pi)/int(cos(2*x)^2,x=-Pi..Pi);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "N\344m\344 oltaisiin voitu p\344 \344tell\344 integroitavien parittomuuden perusteella suoraan." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "b1:=int(f1*sin(x),x=-Pi..Pi) /int(sin(x)^2,x=-Pi..Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "b2:=int(f1*sin(2*x),x=-Pi..Pi)/int(sin(2*x)^2,x=-Pi..Pi);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "projf1:=b1*sin(x)+b2*sin(2*x );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "plot([f1,projf1],x=-P i..Pi,title=\"Suoran y=x paras approx trig. 2. ast. polynomilla\");" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "f2:=x^2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "a0:=int(f2*1,x=-Pi..Pi)/int(1*1,x=-Pi..Pi );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "a1:=int(f2*cos(x),x=-Pi..Pi)/int(cos(x)^2,x=- Pi..Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "a2:=int(f2*cos( 2*x),x=-Pi..Pi)/int(cos(2*x)^2,x=-Pi..Pi);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 53 "b1:=int(f2*sin(x),x=-Pi..Pi)/int(sin(x)^2,x=-Pi..Pi );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "b2:=int(f2*sin(2*x),x =-Pi..Pi)/int(sin(2*x)^2,x=-Pi..Pi);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "b-kertoimet ovat heti n\344ht\344viss\344 0:ksi, koska parillin en (f2) kertaa pariton (sin) on pariton." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "projf2:=a0+a1*cos(x)+a2*cos(2*x)+b1*sin(x)+b2*sin(2*x );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "plot([f2,projf2],x=-P i..Pi,title=\"Paraabelin y=x^2 paras approx trig. 2. ast. polynomilla \");" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "Parittoman/parillisen fu nktion PNS-approksimaatiossa on vain parittomia/parillisia trig. terme j\344 (sinej\344/cosineja)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 75 "Vaihtoehtoinen \"elegantti\" tapa, tusk in kannattaa lukea ainakaan kokeisiin." }}{PARA 0 "" 0 "" {TEXT -1 79 "T\344ss\344 ns. \"elegantti\" tapa, jota harjoituksissa l\344hdin tav oittelemaan (suotta)." }}{PARA 0 "" 0 "" {TEXT -1 165 "Toisin kuin har joituksissa neuvoin, tein nyt funktiotyylill\344, enk\344 lauseketyyli ll\344. Yleiseen koodikokoelmaan ei kannata laittaa aivan huteria ker tak\344ytt\366funktioita." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "F:=[x->1,x->cos(x),x->cos(2*x),x->s in(x),x->sin(2*x)];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "Pro j:=f->SisF(f,F[1],-Pi,Pi)/SisF(F[1],F[1],-Pi,Pi)*F[1]+add(SisF(f,F[j], -Pi,Pi)/SisF(F[j],F[j],-Pi,Pi)*F[j],j=2..nops(F));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Pt:=Proj(t->t);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 6 "Pt(t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Pt2:=Proj(t->t^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "Pt2(x );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "Pt2laus:=subs(Pi(x)=P i,Pt2(x)); # T\344m\344 on tyhmyytt\344 Maplen taholta!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Pt2:=unapply(Pt2laus,x);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "Pt2(x);" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 61 "T\344m\344n temppuilun j\344lkeen vasta tuo vakio \+ k\344ytt\344ytyy kunnolla." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "ff:=(x->a*x)+(x->b);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " ff(t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "plot([t,Pt(t)],t= -Pi..Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "plot([t^2,Pt2( t)],t=-Pi..Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 2 "2." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "alias(TR=LinearAlgebra[Transpose]):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "A:=<<2,3-4*I>|<3+4*I,2>>;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "TR(A);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "map(conjugat e,%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "A on hermiittinen. Siit \344 seuraa: ominaisarvot reaaliset (spektri makaa reaaliakselilla)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Eigenvalues(A);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Niinp\344 n\344kyy makaavan." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "(b)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "B:=<<1/2,(1/2)*I*sqrt(3)>|<(1/2)*I*sqrt(3),1/2>>;" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 119 "Symmetrinen on, mutta ei hermiit tinen. Symmetrisyys ei ole erityisen hy\366dyllinen ominaisuus komplek siselle matriisille." }}{PARA 0 "" 0 "" {TEXT -1 25 "Katsotaan unitaar isuutta:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "HermitianTransp ose(B).B;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "On unitaarinen. Omin aisarvojen pit\344isi sijaita kompleksitason yksikk\366ympyr\344n keh \344ll\344," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Eigenvalues( B);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "map(abs,%);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Niinp\344 vaan!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "c)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "C:=<<0,1-I,0>|<1+I,0,1-I>|<0,1+I,0>>;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "HermitianTranspose(C);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Eigenvalues(C);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "T\344ss\344 nyt oli toinen esimerkki hermiittisest\344. " }}}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 2 "3." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "A:=<<.5,-.3>|<.6,1.4>>;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "trajenuoli(A,<1,1>,10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "(oa,ov):=Eigenvectors(A);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 33 "v1:=ov[1..-1,1]: v2:=ov[1..-1,2]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 94 "kuva1:=display(arrow(v1,shape=arrow),arrow(v2,shape =arrow),traje(A,v1,10),view=[-2..2,-2..2]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "op(traje);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "arrow([3*v1,3*v2,-3*v1,-3*v2],shape=arrow);[oa,ov];" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "Suora y=x vastaa ominaisarvoa 1.09 9 > 1, sill\344 virtaus ulosp\344in. Toisella taas sis\344\344np\344in ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "ov2:=display(trajenuoli(A,0.1*v2,40),trajenuoli(A,-0 .1*v2,40)):ov1:=display(trajenuoli(A,4*v1,40),trajenuoli(A,-4*v1,40)): \ndisplay(ov1,ov2);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "ov 2:=display(traje(A,0.1*v2,40),traje(A,-0.1*v2,40)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "ov1:=display(traje(A,4*v1,40),traje(A,-4* v1,40)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "ovt:=display(ov 1,ov2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "kuva1:=display(o vt,traje(A,<3,1.8>,20),traje(A,<3.8,2>,20)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "display(kuva1,traje(A,<0,1>,10),traje(A,<-4,0>, 10),traje(A,<-3.6,-1.46>,40),traje(A,<-3.7,-0.85>,10),view=[-5..5,-5.. 5]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "ovt:=display(ov1,ov2):" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 65 "kuva1:=display(ovt,trajenuoli(A,<3,1.8>,20), traje(A,<3.8,2>,20)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 168 "d isplay(kuva1,trajenuoli(A,<0,1>,10),trajenuoli(A,<-4,0>,10),traje(A,<- 3.6,-1.46>,40),trajenuoli(A,<-3.7,-0.85>,10),arrow([4*v1,4*v2],color=y ellow),view=[-5..5,-5..5]);" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 2 "4 ." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "read(\"/home/apiola/ope tus/peruskurssi/v2-3/302/L/v302.mpl\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "A:=<<0.8,-4>|<0.3,1.5>>;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 25 "(oa,ov):=Eigenvectors(A);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "w:=ov[1..-1,1]; lambda:=oa[1]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "u:=map(Re,w); v:=map(Im,w);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "a:=Re(lambda): b:=Im(lambda):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "uv:=;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "C:=<|>;" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 15 "A,uv.C.uv^(-1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "r:=abs(lambda);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Ckierto:=<|>;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 17 "arrow([3*u,3*v]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "display(trajenuoli(A,0.1*u,10),traje(A,0.1*u,12)); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "display(trajenuoli(C,0. 1*u,10),traje(C,0.1*u,12));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "display(trajenuoli(Ckierto,u,10),traje(Ckierto,0.3*u,50));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 " " {TEXT -1 2 "5." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "A:=<<0,. 18,0>|<0,0,.71>|<.33,0,.94>>;A,<\"juveniles\",\"subadults\",\"adults\" >;\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "x[k]=<'j[k]','s[k]' ,'a[k]'>;\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "(oa,ov):=Eig envectors(A);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "lambda[1 ]:=oa[1];\nlambda[3]:=oa[3];\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "Im(%);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "lambda[3 ]:=Re(lambda[3]);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "abs( lambda[1]);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "op(trajenu oli);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "iterjono:=proc (A , x0, n) \nlocal x, k; \nx[0] := x0; for k to n do x[k] := A.x[k-1] en d do; \n[seq(x[k],k=0..n)];\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "A;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "displ ay(trajenuoli(A,<0,15,10>,10),axes=boxed);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 3 "ov;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "w: =ov[1..-1,1]; rv:=map(Re,ov[1..-1,3]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "u:=map(Re,w); v:=map(Im,w);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "display(arrow([10*u,10*v,10*rv]),trajenuoli(A,<0 ,15,10>,40),axes=boxed);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "P \366ll\366jen m\344\344r\344 tulee hiljalleen alas reaalista ominaisar voa vastaavan ominaisvektorin suunnassa. Katsotaan numeroina:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "iterjono(A,<0,15,10>,40);" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "Otetaan viel\344 40 iteraatiota \+ siit\344, mihin viimeksi tultiin (suunnnilleen)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "iterjono(A,<3.3,0.6,9.8>,40);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Huonostihan t\344ss\344 p\366ll\366ille k \344y." }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 2 "6." }}{PARA 0 "" 0 "" {TEXT -1 100 "T\344ss\344 olisi se vastaava hyv\344 tarina, mutta anta apa sen olla. Itse suoritus on aivan vastaavanlainen." }}}}{MARK "8" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }