Newtonin menetelm\303\244 yht\303\244l\303\266systeemeille ja optimointiin. 28.3.2001 ..L/newtonsys.mws restart:with(LinearAlgebra):with(linalg):with(plottools): with(plots):setoptions3d(axes=boxed,orientation=[-30,50]): #read("c:\134\134opetus\134\134maple\134\134v201.mpl"): #read("/home/apiola/opetus/peruskurssi/v2-3/maple/ohjelmat.mpl"): #read("/home/apiola/opetus/peruskurssi/v2-3/201/maple/v201.mpl"): read("/p/edu/mat-1.414/maple/ohjelmat.mpl"): read("/p/edu/mat-1.414/maple/v201.mpl"):

Warning, the name changecoords has been redefined Warning, the previous binding of the name GramSchmidt has been removed and it now has an assigned value Warning, the protected names norm and trace have been redefined and unprotected

Ep\303\244lineaarinen yht\303\244l\303\266systeemi: Esim: Ratkaistava: {f1=0, f2=0} miss\303\244 f1:=y*exp(x)-2;f2:=x^2+y-4; F:=[f1,f2]; J:=Matrix(jacobian(F,[x,y])); subs(x=a,y=b,J): #Kokeillaan, t\303\244m\303\244 on loistavaa (verrattuna vanhaan linalgiin). Nyt onkin mit\303\244 nautinnollisinta alkaa leikitell\303\244 Newtonilla ja muilla vastaavilla. Huomaa, ett\303\244 implicitplot:lle t\303\244ytyy yht\303\244l\303\266t antaa joukkona eik\303\244 listana. Pit\303\244nee ajatella, ett\303\244 kyseess\303\244 on yht\303\244l\303\266systeemi. kuvaajat:=implicitplot({f1=0,f2=0},x=-1..3,y=-3..4):kuvaajat; Olkoon xa alkupiste, xc nykypiste ("current"), X-matriisin sarakkeiksi ker\303\244t\303\244\303\244n reittidata. xa:=<1.5,0.5>:xc:=xa: X:=<<xa>>: # Alkupiste, ker\303\244t\303\244\303\244n pisteet matriisin X sarakkeiksi. Ensin yksi Newtonaskel komento kerrallaan: Jc:=subs(x=xc[1],y=xc[2],J); # Lasketaan Jakobiaani nykypisteess\303\244 xc. Fc:=Vector(subs(x=xc[1],y=xc[2],F)); # Lasketaan vektorifunktion F arvo nykypisteess\303\244 xc. h:=LinearSolve(Jc,-Fc); # Ratkaistaan yht\303\244l\303\266systeemi Jc.h = - Fc xc:=xc+h; # P\303\244ivitet\303\244\303\244n nykypiste. X:=<X | xc>;# Ker\303\244t\303\244\303\244n pisteet X-matriisiin ja palataan alkuun riville Jc:=subs(...); Yll\303\244 olevaa voi iteroida nuolin\303\244pp\303\244imell\303\244 tai hiirell\303\244. Seuraava vaihe on k\303\244ytt\303\244\303\244 for-silmukkaa tms. xa:=<1.5,0.5>:xc:=xa: X:=<<xa>>: for i from 0 to 2 do Jc:=subs(x=xc[1],y=xc[2],J); Fc:=Vector(subs(x=xc[1],y=xc[2],F)); h:=LinearSolve(Jc,-Fc); xc:=xc+h; X:=<X | xc>; od: X;Transpose(X);reitti:=convert(%,listlist); Reittipisteet voidaan antaa suoraan plot:lle. reitinkuva:=plot(reitti,style=line,symbol=circle,color=brown):reittinumerot:=seq([op(reitti[i]),` `||`i`],i=1..nops(reitti)); rnumkuva:=textplot([reittinumerot],align=RIGHT): display([kuvaajat,reitinkuva,rnumkuva]);display([kuvaajat,reitinkuva,rnumkuva],view=[1..2.2,0..1]); T\303\244ss\303\244 n\303\244ytettiin perusty\303\266kaluideoita, ei valmista viimeistely\303\244. Reittien piirtely on hyvin interaktiivista, v\303\244lill\303\244 kannattaa ottaa yhdistysjanat, v\303\244lil\303\244 taas ei. Numerointi antaa ainakin j\303\244rjestyksen, jossa reitti\303\244 kuljettiin. Seuraava vaihe on kirjoittaa edell\303\244 oleva proc:ksi. Sen pit\303\244isi nyt sujua k\303\244den k\303\244\303\244nteess\303\244. Hiukan riesaa koitui siit\303\244, ett\303\244 annoin ensin Fun:n lausekkeena, silloin meniv\303\244t globaalit x ja y solmuun vastaavien lokaalien kanssa. Homma on luontevammin hallittavissa m\303\244\303\244rittelem\303\244ll\303\244 funktioksi. print(Newton2); F:=(x,y)->[y*exp(x)-2,x^2+y-4]; X:=Newton2(F,<1,1>,4); Reitinpiirto voidaan toteuttaa edell\303\244 olevilla komennoilla. T\303\244ss\303\244 on omat hyv\303\244t puolensa. Usein reittimatriisista kannattaa ottaa vain osa kuvaan. Kootaan nyt reitinpiirto omaksi proc:ksi (t\303\244m\303\244 vaihe sujui ilman mint\303\244\303\244n ongelmia). Tuossa on hiukan tarpeettomasti lokaaleja, mutta vaiva minimoitui. print(plotpath); F:=(x,y)->[y*exp(x)-2,x^2+y-4]:xa:=<1.,1.>:X:=Newton2(F,xa,5); plotpath(F,X,xa[1]-1..xa[1]+1,xa[2]-1..xa[2]+1);alkupiste=xa;display(%%,view=[1.9..2.1,0.2..0.6]);

T\303\244ss\303\244 on Harrin tyylin\303\244yte. Jokainen voi kokeilla/kehitell\303\244 mieltymystens\303\244 mukaan. VisualNewton := proc(f1, f2, x0, xrange, yrange, steps) local J, Jf, xk, i, path, x, y, ipic, ppic; with(linalg): with(LinearAlgebra): with(plots): Jf := proc(xp, yp, J) local setval; setval := (item) -> subs(x = xp, y = yp, item); RETURN(map(setval, J)); end: J := jacobian([f1(x,y), f2(x,y)], [x,y]); xk := x0; path := [convert(xk, list)]; for i to steps do xk := xk - LinearSolve(Matrix(Jf(xk[1],xk[2], J)), <f1(xk[1], xk[2]), f2(xk[1], xk[2])>): path := [op(path), convert(xk, list)]; od: ipic := implicitplot({f1(x,y), f2(x,y)}, x=xrange, y=yrange, grid=[40, 40]): ppic := PLOT(CURVES(path), SCALING(CONSTRAINED), AXESSTYLE(NONE)): xk, [ppic, ipic]; end: f1:=(x,y)->y*exp(x)-2;f2:=(x,y)->x^2+y-4; xk, pic := VisualNewton(f1, f2, <5.0, -5.0>, -5..5, -5..5, 10): display(pic); xk, pic := VisualNewton(f1, f2, <5.0, 5.0>, -5..5, -5..5, 10): display(pic);

T\303\244ss\303\244 on viimevuosta toistaiseksi, pit\303\244\303\244 ensit\303\266iksi muuttaa LinearAlgebra-tyyliin. f:=(x-2)^4+(x-2*y)^2; g:=grad(f,[x,y]); H:=hessian(f,[x,y]); contourplot(f,x=-1..4,y=-5..5); p0:=[0.8,4]; H0:=evm(H,[x,y],p0); g0:=evm(g,[x,y],p0); h:=-linsolve(H0,g0); p1:=evalm(p0+h); Nminstep:=proc(f,p0) # Yksi askel Newtonin menetelm\303\244ss\303\244 optimointiteht\303\244v\303\244\303\244n sovellettuna. # Argumentit: # f funktio, muuttujien lkm = nops(p) # p lista local vars, _x,H,H0,h,g,g0,i; vars:=[seq(_x[i],i=1..nops(p0))]; H:=linalg[hessian](f(op(vars)),vars); g:=linalg[grad](f(op(vars)),vars); H0:=evm(H,vars,evalf(p0)); g0:=evm(g,vars,evalf(p0)); convert(evalm(p0-linsolve(H0,g0)),list); end: Esim: Edell\303\244 oleva f-lauseke on nyt m\303\244\303\244ritelt\303\244v\303\244 oikeaksi funktioksi. F:=unapply(f,x,y); p0;Nminstep(F,p0); i:='i': for i from 1 to 6 do P[i]:=Nminstep(F,P[i-1]) od: pisteet:=seq(P[i],i=0..6); pfg:=seq([i,op(P[i]),F(op(P[i])),norm(evm(grad(F(x,y),[x,y]),[x,y],P[i]))],i=0..6); matrix([pfg]);

T\303\244ss\303\244 nyt sopiva tyyli (samanlainen kuin Newtonissa yll\303\244: x0:=1,1: x0v:=<x0>: xc:=x0: xcv:=<x0>: X:=<<xcv>>:F:=f(x0):f:=(x,y)->4*x^2-4*x*y+2*y^2: g:=[D[1](f),D[2](f)]: ## T\303\244h\303\244n takaisin, t\303\244st\303\244 alasp\303\244in iteroidaan u:=-Normalize(Vector(g(xc))): jana:=v2l(evalm(xcv+t*u)): phi:=unapply(simplify(f(op(jana))),t): Lmax:=1:phi(0)-phi(Lmax): while (phi(Lmax) > phi(0)) do Lmax:=Lmax/2 od: Lmax;

NiMiIiI=

plot(phi(t),t=0..Lmax): td:=[0,Lmax/2,Lmax]; fd:=map(phi,td);

NiM+JSN0ZEc3JSIiISMiIiIiIiMiIiI= NiM+JSNmZEc3JSIiIyIiIiIiIw==

ip:=interp(td,fd,t);

NiM+JSNpcEcsKComIiIlIiIiKSUidEciIiMiIiIiIiIqJiIiJSIiIiUidEciIiIhIiIiIiMiIiI=

tmin:=solve(diff(ip,t)=0,t); xcv:=xcv+tmin*u; xc:=op(v2l(xcv));

NiM+JSV0bWluRyMiIiIiIiM= 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 NiM+JSN4Y0c2JCMiIiIiIiMiIiI=

X:=< X | xcv>; F:=F,f(xc);

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 NiM+JSJGRzYkIiIjIiIi

display(plotonlypath(X,0..2,0..2),contourplot(f(x,y),x=0..1.6,y=0..1.6,contours=[F])); ## Takaisin

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op(f);F:=unapply([f(x,y),f(x,y)],x,y); op(plotpath): Kokeillaan, meneek\303\266 hommat helpommin, jos otamme usean muuttujan funktion argumentiksi listan jonon sijasta. T\303\244st\303\244 on sekin etu, ett\303\244 m\303\244\303\244rittely toimii yht\303\244 hyvin vektorille. f:=(x,y)->4*x^2-4*x*y+2*y^2;x0:=[2,3];g:=grad(f(x,y),[x,y]);#gf:=unapply(g,x,y); g:=Vector(g); u:=subs(x=x0[1],y=x0[2],g); G:=Vector(grad(f(x,y),[x,y]));subs(x=x0[1],y=x0[2],G); Vector(%); N\303\244iss\303\244 oli jotain ihmeellist\303\244, on v\303\244h\303\244n viel\303\244kin. [1,1];Vector(%); RTABLE(136885404,VECTOR([4, 4])); T\303\244mm\303\266isen otuksen kanssa en halua tapella. Tehd\303\244\303\244n "k\303\244sin", siis ilman grad-funktiota. x0:=1,1: x0v:=<x0>;

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

f:=(x,y)->4*x^2-4*x*y+2*y^2; g:=[D[1](f),D[2](f)];

NiM+JSJmR2YqNiQlInhHJSJ5RzYiNiQlKW9wZXJhdG9yRyUmYXJyb3dHNiIsKComIiIlIiIiKSUieEciIiMiIiIiIiIqKCIiJSIiIiUieUciIiIlInhHIiIiISIiKiYiIiMiIiIpJSJ5RyIiIyIiIiIiIjYiNiI2Ig== NiM+JSJnRzckZio2JCUieEclInlHNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0c2IiwmKiYiIikiIiIlInhHIiIiIiIiKiYiIiUiIiIlInlHIiIiISIiNiI2IjYiZio2JCUieEclInlHNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0c2IiwmKiYiIiUiIiIlInhHIiIiISIiKiYiIiUiIiIlInlHIiIiIiIiNiI2IjYi

Vector(g(x0));

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWFjdGlvbkdGJDYlLUkobWZlbmNlZEdGJDYoLUYjNiUtSSdtdGFibGVHRiQ2Ni1JJG10ckdGJDYmLUkkbXRkR0YkNigtSSNtbkdGJDYkUSI0RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUpcm93YWxpZ25HUSFGJy8lLGNvbHVtbmFsaWduR0ZFLyUrZ3JvdXBhbGlnbkdGRS8lKHJvd3NwYW5HUSIxRicvJStjb2x1bW5zcGFuR0ZMRkNGRkZILUY3NiYtRjo2KC1GPTYkUSIwRidGQEZDRkZGSEZKRk1GQ0ZGRkgvJSZhbGlnbkdRJWF4aXNGJy9GRFEpYmFzZWxpbmVGJy9GR1EmcmlnaHRGJy9GSVEnfGZybGVmdHxockYnLyUvYWxpZ25tZW50c2NvcGVHUSV0cnVlRicvJSxjb2x1bW53aWR0aEdRJWF1dG9GJy8lJndpZHRoR0Zeby8lK3Jvd3NwYWNpbmdHUSYxLjBleEYnLyUuY29sdW1uc3BhY2luZ0dRJjAuOGVtRicvJSlyb3dsaW5lc0dRJW5vbmVGJy8lLGNvbHVtbmxpbmVzR0Zpby8lJmZyYW1lR0Zpby8lLWZyYW1lc3BhY2luZ0dRLDAuNGVtfjAuNWV4RicvJSplcXVhbHJvd3NHUSZmYWxzZUYnLyUtZXF1YWxjb2x1bW5zR0ZjcC8lLWRpc3BsYXlzdHlsZUdGY3AvJSVzaWRlR0Zmbi8lMG1pbmxhYmVsc3BhY2luZ0dGZm8vJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnRkBGQC9JK21zZW1hbnRpY3NHRiRRKkNvbFZlY3RvckYnLyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnRl9xLyUrYWN0aW9udHlwZUdRLnJ0YWJsZWFkZHJlc3NGJy8lKXJ0YWJsZWlkR1EqMTM2NzkyMDA0RidGXHFGQA==

u:=-Vector(g(x0));

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

jana:=v2l(evalm(x0v+t*u));

NiM+JSVqYW5hRzckLCYqJiIiJSIiIiUidEciIiIhIiIiIiIiIiIiIiI=

phi:=simplify(f(op(jana)));

NiM+JSRwaGlHLCgqJiIjayIiIiklInRHIiIjIiIiIiIiKiYiIzsiIiIlInRHIiIiISIiIiIjIiIi

plot(phi,t=0..1);

LSUlUExPVEc2Ji0lJ0NVUlZFU0c2JDdTNyQkIiIhISIiJCIjPyEiIjckJCIybW1tbVQ6KHpAISM9JCIxQCRRRyZIbCJvIiEjOjckJCIyTkxMZTl1aTIlISM9JCIxKVIubGdRVFgiISM6NyQkIjFubW0iel8iNGkhIzwkIjIqNDUuYiV5S0QiISM7NyQkIjFtbW1UJnBoTikhIzwkIjIxVl52XicqKTQ2ISM7NyQkIjJLTCRlKj0pSFw1ISM8JCIyeCozcHkqemQtIiEjOzckJCIybW07ei8zdUMiISM8JCIyYSl6bCpILysrIiEjOzckJCIyKSoqKlw3TFJEWCIhIzwkIjJpL11jPmFpLSIhIzs3JCQiMmxtO3pSJ29rOyEjPCQiMlIlXF54dTA1NiEjOzckJCIyKSoqKlxpNWBoKD0hIzwkIjF2JnlmTEI0RCIhIzo3JCQiMk5MTCQzRW4kNCMhIzwkIjI0RUwzVVRiWCIhIzs3JCQiMm5tbVQhUkUmRyMhIzwkIjI9MThbbUxmbyIhIzs3JCQiK0QuJjRdIyEjNSQiMnRFJip6NEA6KyMhIzs3JCQiK3ZCXzxGISM1JCIyLlQ/JkghPSR5QiEjOzckJCIrdidIaSNIISM1JCIydVFXNFJQIyl6IyEjOzckJCIybW07eipldjpKISM8JCIyJ0g0Plgpb3lBJCEjOzckJCIxTExMMzQ3VEwhIzskIjJ4LyM0ZU1lKXokISM7NyQkIjFMTExMWS5LTiEjOyQiMSMqKltMXztITCUhIzo3JCQiLUQibzdUdiQhIzckIjFAVDAwOTw4XSEjOjckJCIxTExMJFEqb11SISM7JCIxLCdmSCJHKXptJiEjOjckJCItRCI9bGo7JSEjNyQiMS0yMWQqPUxXJyEjOjckJCItdlYmUjxQJSEjNyQiMXR6Y3lcJ3BCKCEjOjckJCIyTkxMZTlFZ2UlISM8JCIxMXZCJjMwRTcpISM6NyQkIjJMTCRlUiIzR3klISM8JCIxLCEzWk1wdykqKSEjOjckJCIybW1tVDVrXSpcISM8JCIxOExPI0hCaigqKiEjOjckJCIxbW0ielJRYkAmISM7JCIyWlxpaW1KazUiISM6NyQkIi12PT5ZMmEhIzckIjI4Ilt4TXQ/MTchIzo3JCQiMW1tO3pYdTljISM7JCIyck1aYXBqIz44ISM6NyQkIiomeSkpR2UhIiokIjJ4O1dCcFA9VyIhIzo3JCQiMSoqKipcaV9RUWchIzskIjInUmgsUkRWbjohIzo3JCQiMSoqKlw3eSUzVGkhIzskIjJELSN6WSQqSCVwIiEjOjckJCIxKioqKlxQIVtoWSchIzskIjFtTE1BW0tUPSEjOTckJCIxTExMJFF4JG9tISM7JCIxamYnPjFrKnk+ISM5NyQkIit2LkklKW8hIzUkIjInM2shZSE9cUpAISM6NyQkIjFtbSJ6cGUqenEhIzskIjIjM0QlKXkoZV9GIyEjOjckJCIrRFwnUUgoISM1JCIxJ0hEb1M2eVYjISM5NyQkIjFLTGU5UzgmXCghIzskIjJsd1lxczNoZiMhIzo3JCQiMSoqKlxpPz1icSghIzskIjJLVD0qeng2bkYhIzo3JCQiMUxMTDNzPzZ6ISM7JCIxRDBCOXd5UkghIzk3JCQiLURKWGFFIikhIzckIjJOInA8UyRmajckISM6NyQkIjFubW1tKlJSTCkhIzskIjImKmUmKSlbNm02TCEjOjckJCIxbW07YTwuWSYpISM7JCIxTCgqKil5SSZvXSQhIzk3JCQiMUxMZTl0T2MoKSEjOyQiMjFbS25AOmhxJCEjOjckJCIqJlFrXCopISIqJCIwJXpuJykqM1UqUSEjODckJCIxTEwkM2RnNjwqISM7JCIxeTZcRWltOlQhIzk3JCQiMW1tbW14R3AkKiEjOyQiMV4kR2NPaCE+ViEjOTckJCItRCJvSzBlKiEjNyQiMmsmZkAlZWQ5YSUhIzo3JCQiLXY9NXMjeSohIzckIjJXTms0KipvJ2ZaISM6NyQkIiM1ISIiJCIkKyYhIiItJSZDT0xPUkc2JiUkUkdCRyQiIzUhIiIkIiIhISIiJCIiISEiIi0lJVZJRVdHNiQ7JCIiISEiIiQiIzUhIiIlKERFRkFVTFRHLSUrQVhFU0xBQkVMU0c2JFEidDYiUSE2Ii0lJVJPT1RHNictJSlCT1VORFNfWEc2IyQiJCFSISIiLSUpQk9VTkRTX1lHNiMkIiQ/IiEiIi0lLUJPVU5EU19XSURUSEc2IyQiJUlOISIiLSUuQk9VTkRTX0hFSUdIVEc2IyQiJVNDISIiLSUpQ0hJTERSRU5HNiI=

f:=(x,y)->4*x^2-4*x*y+2*y^2; g:=[D[1](f),D[2](f)]; x0:=1,1: x0v:=<x0>; u:=-Vector(g(x0)); jana:=v2l(evalm(x0v+t*u)); phi:=simplify(f(op(jana))); plot(phi,t=0..1);

NiM+JSJmR2YqNiQlInhHJSJ5RzYiNiQlKW9wZXJhdG9yRyUmYXJyb3dHNiIsKComIiIlIiIiKSUieEciIiMiIiIiIiIqKCIiJSIiIiUieUciIiIlInhHIiIiISIiKiYiIiMiIiIpJSJ5RyIiIyIiIiIiIjYiNiI2Ig== NiM+JSJnRzckZio2JCUieEclInlHNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0c2IiwmKiYiIikiIiIlInhHIiIiIiIiKiYiIiUiIiIlInlHIiIiISIiNiI2IjYiZio2JCUieEclInlHNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0c2IiwmKiYiIiUiIiIlInhHIiIiISIiKiYiIiUiIiIlInlHIiIiIiIiNiI2IjYi 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUYjNictSSNtaUdGJDYlUSJ1RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEjOj1GJy9GNVEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj8vJSlzdHJldGNoeUdGPy8lKnN5bW1ldHJpY0dGPy8lKGxhcmdlb3BHRj8vJS5tb3ZhYmxlbGltaXRzR0Y/LyUnYWNjZW50R0Y/LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JKG1hY3Rpb25HRiQ2JS1JKG1mZW5jZWRHRiQ2KC1GIzYlLUknbXRhYmxlR0YkNjYtSSRtdHJHRiQ2Ji1JJG10ZEdGJDYoLUYjNiUtRjg2LVEqJnVtaW51czA7RidGO0Y9RkBGQkZERkZGSEZKL0ZNUSwwLjIyMjIyMjJlbUYnL0ZQRmJvLUkjbW5HRiQ2JFEiNEYnRjtGOy8lKXJvd2FsaWduR1EhRicvJSxjb2x1bW5hbGlnbkdGam8vJStncm91cGFsaWduR0Zqby8lKHJvd3NwYW5HUSIxRicvJStjb2x1bW5zcGFuR0ZhcEZob0ZbcEZdcC1GZ242Ji1Gam42KC1GZW82JFEiMEYnRjtGaG9GW3BGXXBGX3BGYnBGaG9GW3BGXXAvJSZhbGlnbkdRJWF4aXNGJy9GaW9RKWJhc2VsaW5lRicvRlxwUSZyaWdodEYnL0ZecFEnfGZybGVmdHxockYnLyUvYWxpZ25tZW50c2NvcGVHRjMvJSxjb2x1bW53aWR0aEdRJWF1dG9GJy8lJndpZHRoR0ZocS8lK3Jvd3NwYWNpbmdHUSYxLjBleEYnLyUuY29sdW1uc3BhY2luZ0dRJjAuOGVtRicvJSlyb3dsaW5lc0dRJW5vbmVGJy8lLGNvbHVtbmxpbmVzR0Zjci8lJmZyYW1lR0Zjci8lLWZyYW1lc3BhY2luZ0dRLDAuNGVtfjAuNWV4RicvJSplcXVhbHJvd3NHRj8vJS1lcXVhbGNvbHVtbnNHRj8vJS1kaXNwbGF5c3R5bGVHRj8vJSVzaWRlR0ZhcS8lMG1pbmxhYmVsc3BhY2luZ0dGYHIvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnRjtGOy9JK21zZW1hbnRpY3NHRiRRKkNvbFZlY3RvckYnLyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnRmhzLyUrYWN0aW9udHlwZUdRLnJ0YWJsZWFkZHJlc3NGJy8lKXJ0YWJsZWlkR1EqMTM2NTI3NzAwRidGZXNGO0Zlc0Y7 NiM+JSVqYW5hRzckLCYqJiIiJSIiIiUidEciIiIhIiIiIiIiIiIiIiI= NiM+JSRwaGlHLCgqJiIjayIiIiklInRHIiIjIiIiIiIiKiYiIzsiIiIlInRHIiIiISIiIiIjIiIi LSUlUExPVEc2Ji0lJ0NVUlZFU0c2JDdTNyQkIiIhISIiJCIjPyEiIjckJCIybW1tbVQ6KHpAISM9JCIxQCRRRyZIbCJvIiEjOjckJCIyTkxMZTl1aTIlISM9JCIxKVIubGdRVFgiISM6NyQkIjFubW0iel8iNGkhIzwkIjIqNDUuYiV5S0QiISM7NyQkIjFtbW1UJnBoTikhIzwkIjIxVl52XicqKTQ2ISM7NyQkIjJLTCRlKj0pSFw1ISM8JCIyeCozcHkqemQtIiEjOzckJCIybW07ei8zdUMiISM8JCIyYSl6bCpILysrIiEjOzckJCIyKSoqKlw3TFJEWCIhIzwkIjJpL11jPmFpLSIhIzs3JCQiMmxtO3pSJ29rOyEjPCQiMlIlXF54dTA1NiEjOzckJCIyKSoqKlxpNWBoKD0hIzwkIjF2JnlmTEI0RCIhIzo3JCQiMk5MTCQzRW4kNCMhIzwkIjI0RUwzVVRiWCIhIzs3JCQiMm5tbVQhUkUmRyMhIzwkIjI9MThbbUxmbyIhIzs3JCQiK0QuJjRdIyEjNSQiMnRFJip6NEA6KyMhIzs3JCQiK3ZCXzxGISM1JCIyLlQ/JkghPSR5QiEjOzckJCIrdidIaSNIISM1JCIydVFXNFJQIyl6IyEjOzckJCIybW07eipldjpKISM8JCIyJ0g0Plgpb3lBJCEjOzckJCIxTExMMzQ3VEwhIzskIjJ4LyM0ZU1lKXokISM7NyQkIjFMTExMWS5LTiEjOyQiMSMqKltMXztITCUhIzo3JCQiLUQibzdUdiQhIzckIjFAVDAwOTw4XSEjOjckJCIxTExMJFEqb11SISM7JCIxLCdmSCJHKXptJiEjOjckJCItRCI9bGo7JSEjNyQiMS0yMWQqPUxXJyEjOjckJCItdlYmUjxQJSEjNyQiMXR6Y3lcJ3BCKCEjOjckJCIyTkxMZTlFZ2UlISM8JCIxMXZCJjMwRTcpISM6NyQkIjJMTCRlUiIzR3klISM8JCIxLCEzWk1wdykqKSEjOjckJCIybW1tVDVrXSpcISM8JCIxOExPI0hCaigqKiEjOjckJCIxbW0ielJRYkAmISM7JCIyWlxpaW1KazUiISM6NyQkIi12PT5ZMmEhIzckIjI4Ilt4TXQ/MTchIzo3JCQiMW1tO3pYdTljISM7JCIyck1aYXBqIz44ISM6NyQkIiomeSkpR2UhIiokIjJ4O1dCcFA9VyIhIzo3JCQiMSoqKipcaV9RUWchIzskIjInUmgsUkRWbjohIzo3JCQiMSoqKlw3eSUzVGkhIzskIjJELSN6WSQqSCVwIiEjOjckJCIxKioqKlxQIVtoWSchIzskIjFtTE1BW0tUPSEjOTckJCIxTExMJFF4JG9tISM7JCIxamYnPjFrKnk+ISM5NyQkIit2LkklKW8hIzUkIjInM2shZSE9cUpAISM6NyQkIjFtbSJ6cGUqenEhIzskIjIjM0QlKXkoZV9GIyEjOjckJCIrRFwnUUgoISM1JCIxJ0hEb1M2eVYjISM5NyQkIjFLTGU5UzgmXCghIzskIjJsd1lxczNoZiMhIzo3JCQiMSoqKlxpPz1icSghIzskIjJLVD0qeng2bkYhIzo3JCQiMUxMTDNzPzZ6ISM7JCIxRDBCOXd5UkghIzk3JCQiLURKWGFFIikhIzckIjJOInA8UyRmajckISM6NyQkIjFubW1tKlJSTCkhIzskIjImKmUmKSlbNm02TCEjOjckJCIxbW07YTwuWSYpISM7JCIxTCgqKil5SSZvXSQhIzk3JCQiMUxMZTl0T2MoKSEjOyQiMjFbS25AOmhxJCEjOjckJCIqJlFrXCopISIqJCIwJXpuJykqM1UqUSEjODckJCIxTEwkM2RnNjwqISM7JCIxeTZcRWltOlQhIzk3JCQiMW1tbW14R3AkKiEjOyQiMV4kR2NPaCE+ViEjOTckJCItRCJvSzBlKiEjNyQiMmsmZkAlZWQ5YSUhIzo3JCQiLXY9NXMjeSohIzckIjJXTms0KipvJ2ZaISM6NyQkIiM1ISIiJCIkKyYhIiItJSZDT0xPUkc2JiUkUkdCRyQiIzUhIiIkIiIhISIiJCIiISEiIi0lJVZJRVdHNiQ7JCIiISEiIiQiIzUhIiIlKERFRkFVTFRHLSUrQVhFU0xBQkVMU0c2JFEidDYiUSE2Ii0lJVJPT1RHNictJSlCT1VORFNfWEc2IyQiJCFSISIiLSUpQk9VTkRTX1lHNiMkIiQ/IiEiIi0lLUJPVU5EU19XSURUSEc2IyQiJUlOISIiLSUuQk9VTkRTX0hFSUdIVEc2IyQiJVNDISIiLSUpQ0hJTERSRU5HNiI=

t1:=solve(diff(phi,t)=0,t);

NiM+JSN0MUcjIiIiIiIp

x1v:=x0v+t1*u;x1:=op(v2l(x1v));f(x1);subs(t=t1,phi);

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 NiM+JSN4MUc2JCMiIiIiIiMiIiI= NiMiIiI= NiMiIiI=

f(op(%));simplify(%);plot(%,t=0..1);

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No nyth\303\244n menee n\303\244tisti, t\303\244st\303\244 on tosi kiva jatkaa!! T\303\244ss\303\244 on koodi viime vuodelta, lis\303\244\303\244 ohjelmat.mpl:ss\303\244 op(SDstep); f:=x->(x[1]-2)^4+(x[1]-2*x[2])^2; p0:=[1,1]; Lmax:=1.; op(F); op(f); g:=grad(f([x,y]),[x,y]);g:=Vector(op(g)); s0:=-Normalize(subs(x=p0[1],y=p0[2],g)); fs:=unapply(simplify(f(evalm(p0+lambda*s0))),lambda); plot(fs,0..1); SDstep:=proc(f,p0,Lmax) local vars,_x,g,s,fs,lambda,i,t,p,dp; vars:=[seq(_x[i],i=1..nops(p0))]; g:=linalg[grad](f(vars),vars); s:=-normalize(evm(g,vars,p0)); fs:=unapply(f(evalm(p0+lambda*s0)),lambda); lambda:=Lmax; while(fs(lambda)>fs(0)) do lambda:=lambda/2. od; p:=interp([0,lambda/2,lambda],map(fs,[0,lambda/2,lambda]),t); dp:=diff(p,t); lambda:=solve(dp=0,t); [v2l(evalm(p0+lambda*s0)),fs(lambda)]; end: SDstep(f,p0,100); P:='P': P[0]:=p0; for i to 6 do pp:=SDstep(f,P[i-1],10):P[i]:=pp[1]:od: pp:=SDstep(f,P[0],10); pp[1]; op(%); seq(P[i],i=0..6); op(f);

Oheinen rutiini suorittaa gradienttimenetelm\303\244\303\244 annetun m\303\244\303\244r\303\244n askeleita. Rutiini palauttaa minimipisteen (tai siis ainakin approksimaation sille) ja kuvan, johon on piirretty menetelm\303\244n kulkema polku ja vastaavat tasa-arvok\303\244yr\303\244t. GradientMethod := proc(fun, xrange, yrange, init, steps) local x, y, xk, path, levels, gr, gf1, gf2, g, gder, i, tval, cpic, ppic, dir, detail; with(plots): with(linalg): gr := grad(fun(x,y), [x,y]); gf1 := unapply(gr[1], (x,y)); gf2 := unapply(gr[2], (x,y)); g := unapply(fun(x - t*gf1(x,y), y-t*gf2(x,y)), t, x, y); gder := unapply(diff(g(t,x,y), t), t, x, y); xk := init; path := [xk]; levels := [fun(op(xk))]; for i to steps do dir := [gf1(op(xk)), gf2(op(xk))]; if dir[1] = 0 and dir[2] = 0 then break; fi; tval := solve(gder(t, op(xk)) = 0, t); xk := evalf(xk - tval * dir); path := [op(path), xk]; levels := [op(levels), fun(op(xk))]; od; if steps > 10 then detail := [50, 50]; else detail := [30, 30]; fi; cpic := contourplot(fun(x,y), x=xrange, y=yrange, contours=levels, axes=none, grid=detail): ppic := PLOT(CURVES(path), SCALING(CONSTRAINED), AXESSTYLE(NONE)): xk, [ppic, cpic]; end: