{VERSION 3 0 "IBM INTEL LINUX" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 256 "helvetica" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 257 "helvetica" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "helvetica" 0 14 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "helvetica " 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "helvetica" 1 18 0 0 0 0 1 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "helvetica" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "helvetica " 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "helvetica" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "helvetica" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 22 "Numerische Optimierung" }}{PARA 257 "" 0 "" {TEXT -1 72 "Lightversion f\374r Computer, die mit der Vollversion Schwierigkeiten haben" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 20 "a) Gradientenmethode" }{MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart; with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "f := (x, y) -> 1-sin(x)*c os(y)*(1+y^2):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "fx := unapply(dif f(f(x, y), x), x, y):\nfy := unapply(diff(f(x, y), y), x, y):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "graph := plot3d(f(x, y), x=-2..2, y =-1.8..1.8, axes=boxed, color=COLOR(RGB, .5, .75, .5)):" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 137 "x[0] := 1.2: y[0] := 1/2: z[0] := evalf(f(x[0 ], y[0]))+0.01:\nh := 0.2:\nphi := Pi/5: a := evalf(cos(Pi/2-phi)): b \+ := evalf(sin(Pi/2-phi)):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 316 "for i \+ from 1 to 25 do\ndx := evalf(fx(x[i-1], y[i-1])*h);\ndy := evalf(fy(x[ i-1], y[i-1])*h);\nx[i] := evalf(x[i-1] + dx):\ny[i] := evalf(y[i-1] + dy):\nz[i] := evalf(f(x[i], y[i])) + 0.01:\ndz := z[i] - z[i-1];\nPfe il[i] := PLOT3D(CURVES([[x[i-1], y[i-1], z[i-1]], [x[i], y[i], z[i]]], COLOR(RGB,1,0,0),THICKNESS(3))):\nod:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "display3d([seq(\n display3d([graph, seq(Pfeil[i], \+ i=1..j)]),\n j=0..25)], insequence=true, orientation=[-30,65]);" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 206 "graph := plot3d(f(x, y), x =-2..2, y=-2..2.3, color=COLOR(RGB, 0.5,0.75,0.5)):\ndisplay3d([seq(\n display3d([graph, seq(Pfeil[i], i=1..j)]),\n j=0..25)], insequen ce=true, orientation=[-30,65], axes=boxed);" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 25 "b) Die Idee der Tunnelung" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "fsolve(\{fx(u,v), fy(u,v)\}, \{u,v\}, v=0.01..1, u=-2..0):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "a ssign(%):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 170 "display3d([plot3d(f(u , v)-0.015, x=-2..2, y=-2..3, color=yellow),\nplot3d(f(x, y), x=-2..2, y=-2..2.3, color=COLOR(RGB, 0.5,0.75,0.5))], orientation=[-30,65], ax es=boxed);" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 108 "c) Boltzmann-Ver teilung f\374r verschiedene Temperaturen und ihre Anwendung auf zw\366 lf Fortschreitungsrichtungen" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 252 "display([seq(display([plot(exp(-E/kT), E=0..1 0, color=red, thickness=3),\n textplot([9, 0.8, sprintf(\"kT = %6.2f \", kT)], font=[TIMES, ROMAN, 16])]), kT = [1000,500,250,100,80,50,25, 10,8,5,2.5,1,0.8,0.5,0.25,0.1,0.08,0.05,0.02,0.01])],\ninsequence=true );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 194 "for i from 0 to 11 d o\nrx[i] := evalf(cos(i*Pi/6), 5): ry[i] := evalf(sin(i*Pi/6), 5):\nod :\ndisplay([\n seq(plot([[0,0], [rx[i], ry[i]]], color=red), i=0..11) ],\n axes=boxed, scaling=constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 378 "T0 := 10:\nx0 := x[10]: y0 := y[10]: f0 := f(x0, y0) :\nfor k from 1 to 50 do\nE0 := 0:\nfor i from 0 to 11 do\nE[i] := eva lf(exp((f(x0 + rx[i], y0 + rx[i]) - f0)*k^1.5/T0), 5):\nE0 := E0 + E[i ]\nod:\nBild[k] := display([seq(\n plot([[0,0], [rx[i]*E[i]/E0, ry[i] *E[i]/E0]], color=red),\n i=0..11)]):\nod:\ndisplay([seq(Bild[k], k=1 ..50)], insequence=true, axes=boxed, scaling=constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "seq(evalf(E[i]), i=0..11);" }{TEXT -1 0 "" }}{PARA 12 "" 1 "" {XPPMATH 20 "6.$\"&29&!#=$\"&x;)F%$\"&?:)!# 8$\"&N+\"!\"%$\"&?'e\"\"#$\"&d*=\"\"($\"&%y9\"\")F1F.F+F(F&" }}} {EXCHG {PARA 261 "" 0 "" {TEXT -1 29 "d) Der Metropolis-Algorithmus" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "Zufalls zahl := rand(1..100000);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%,Zufalls zahlGR6\"6#%\"tGF&F&C%>%&_seedG-%%iremG6$,$F+\"-\"3p'>uU\"-*)********* *>8$F+,&-F-6$F3\"'++5\"\"\"F8F8F&6#F+F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "isprime(999999999989);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "numtheory[ order](427419669081, 999999999989);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #\"-))**********" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "readlib (randomize): randomize(57206210305): _seed;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\",0.@1s&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 867 "h := 0.45:\nT0 := 10:\nxmin := -2: xmax := 2:\nymin := -1.8: ymax := 3:\nfor k from 1 to 25 do\nE0 := 0: \nfor i from 0 to 11 do\nxk[i] := evalf(x[k-1] + rx[i]*h):\nyk[i] := evalf(y[k-1] + ry[i]*h):\nzk[i] := evalf(f(xk[i], yk[i])):\ndz[i] := zk[i] - z[k-1]:\nE[i] := evalf(e xp(dz[i]*ln(4*k)/T0), 5): E0 := E0 + E[i]\nod:\nG[0] := 0:\nfor i from 0 to 11 do\nG[i+1] := G[i] + evalf(E[i]/E0*100000, 5):\nod:\nW := Zuf allszahl():\nfor i from 0 to 11 do\nif ( (G[i] < W) and (W <= G[i+1]) \+ ) then j := i: break: fi:\nod:\nx[k] := evalf(x[k-1] + rx[j]*h):\nif ( x[k] > xmax) then xmax := x[k] elif\n (x[k] < xmin) then xmin := x[k ] fi:\ny[k] := evalf(y[k-1] + ry[j]*h):\nif (y[k] > ymax) then ymax := y[k] elif\n (y[k] < ymin) then ymin := y[k] fi:\nz[k] := evalf(f(x[ k], y[k])) + 0.3:\nPfeil[k] := PLOT3D(CURVES([[x[k-1], y[k-1], z[k-1]] , [x[k], y[k], z[k]]], COLOR(RGB,1,0,0), THICKNESS(3))):\nod:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 239 "graph := plot3d(f(x, y), x=xmin-0. 1..xmax+0.1, y=ymin-0.1..ymax+0.1, axes=boxed, color=COLOR(RGB, .5, .7 5, .5)):h := 0.2:\ndisplay3d([seq(\n display3d([graph, seq(Pfeil[k], k=1..j)]),\n j=0..25)], insequence=true, orientation=[-30,65]);\n " }{TEXT 256 0 "" }{TEXT 257 0 "" }{TEXT 259 212 "Wenn Sie diese Befeh lsgruppe mehrfach ausf\374hren, bekommen Sie Pfade zu neuen Zufallszah len. Die Initialisierung des Zufallsgenerators in der Befehlsgruppe vo rher darf dabei aber nat\374rlich nicht wiederholt werden!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "2 0 0" 8 }{VIEWOPTS 1 1 0 1 1 1803 }