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{SECT 0 {EXCHG {PARA 4 "" 0 "" {TEXT 268 14 "Ejercicios de " }}{PARA 
4 "" 0 "" {TEXT 269 46 "(5) Sistemas aut\363nomos: m\351todos de RUNGE
-KUTTA" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 267 
15 "Ejercicio 05-06" }}{PARA 4 "" 0 "" {TEXT -1 93 "\nPara un problema
 (en este caso de Inteligencia Artificial) se dise\361an dos  m\351tod
os num\351ricos" }}{PARA 4 "" 0 "" {TEXT -1 92 "de integraci\363n de e
cuaciones diferenciales que designaremos con los nombres de  M1  y  M2
 . " }}{PARA 4 "" 0 "" {TEXT -1 86 "Para determinar cual es el m\341s \+
adecuado, se realiza la integraci\363n de un problema test" }}{PARA 4 
"" 0 "" {TEXT -1 38 "entre  0  y  1  y se emplean los pasos" }}{PARA 
4 "" 0 "" {TEXT 262 2 "  " }{TEXT -1 4 "1. ," }{TEXT 264 2 "  " }
{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 8 " = 0.5 ," }
{TEXT 263 2 "  " }{XPPEDIT 18 0 "1/2^2" "6#*&\"\"\"F$*$\"\"#F&!\"\"" }
{TEXT -1 10 " = 0.25 , " }{XPPEDIT 18 0 "1/2^3" "6#*&\"\"\"F$*$\"\"#\"
\"$!\"\"" }{TEXT -1 11 " = 0.125 , " }{XPPEDIT 18 0 "1/2^4" "6#*&\"\"
\"F$*$\"\"#\"\"%!\"\"" }{TEXT -1 12 " = 0.0625 , " }{XPPEDIT 18 0 "1/2
^5" "6#*&\"\"\"F$*$\"\"#\"\"&!\"\"" }{TEXT -1 12 " = 0.0313 , " }
{XPPEDIT 18 0 "1/2^6" "6#*&\"\"\"F$*$\"\"#\"\"'!\"\"" }{TEXT -1 12 " =
 0.0156 , " }{XPPEDIT 18 0 "1/2^7" "6#*&\"\"\"F$*$\"\"#\"\"(!\"\"" }
{TEXT -1 11 " = 0.0078 ," }{TEXT 265 2 "  " }{XPPEDIT 18 0 "1/2^8" "6#
*&\"\"\"F$*$\"\"#\"\")!\"\"" }{TEXT -1 10 " = 0.0039 " }}{PARA 4 "" 0 
"" {TEXT -1 46 "Los n\372meros de pasos son, en todos los casos, " }}
{PARA 4 "" 0 "" {TEXT -1 44 "    1 , 2 , 4 , 8 , 16 , 32 , 64 , 128 , \+
256" }}{PARA 4 "" 0 "" {TEXT -1 42 "Los n\372meros de evaluaciones son
 para el M1" }}{PARA 4 "" 0 "" {TEXT -1 46 "    2 , 4 , 8 , 16 , 32 , \+
64 , 128 , 256 , 512" }}{PARA 4 "" 0 "" {TEXT -1 12 "y para el M2" }}
{PARA 4 "" 0 "" {TEXT -1 47 "    3 , 6 , 12 , 24 , 48 , 96 , 192 , 384
 , 768" }}{PARA 4 "" 0 "" {TEXT -1 80 "Para cada m\351todo se calcula \+
el  log[10] del error cometido en la aproximaci\363n de" }}{PARA 4 "" 
0 "" {TEXT -1 77 "la soluci\363n en x=1 , error medido en norma2. Los \+
errores que se obtienen son," }}{PARA 4 "" 0 "" {TEXT -1 11 "para el M
1 " }}{PARA 4 "" 0 "" {TEXT -1 71 "    1.716, 1.912, .8447, -2.149, -3
.849, -4.701, -5.381, -6.014, -6.630" }}{PARA 4 "" 0 "" {TEXT -1 13 " \+
y para el M2" }}{PARA 4 "" 0 "" {TEXT -1 73 "    2.180, 2.202, .04186,
 -4.047, -4.878, -5.861, -6.814, -7.743, -8.659 " }}{PARA 4 "" 0 "" 
{TEXT -1 91 "Con estos datos, se deben construir las siguientes 'gr
\341ficas de eficiencia' que mezclen los" }}{PARA 4 "" 0 "" {TEXT -1 
26 "resultados de  M1  y  M2 :" }}{PARA 4 "" 0 "" {TEXT -1 46 "a) La g
r\341fica 'paso versus log[10] del error'." }}{PARA 4 "" 0 "" {TEXT 
-1 64 "b) La gr\341fica 'n\372mero de evaluaciones versus log[10] del \+
error'." }}{PARA 4 "" 0 "" {TEXT -1 54 "c) Finalmente, se dir\341 qu
\351 m\351todo parece m\341s adecuado." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 
"interface(labeling=false):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
11 "Digits:=20:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "with(lin
alg):with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protect
ed names norm and trace have been redefined and unprotected\n" }}
{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been r
edefined\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 259 15 "Variables lista" }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "listanu
mpasos:=[0,0,0,0,0,0,0,0,0]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 30 "listpaso:=[0,0,0,0,0,0,0,0,0]:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "lista1eval:=[0,0,0,0,0,0,0,0,0]:" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 35 "lista1logerro:=[0,0,0,0,0,0,0,0,0]:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "lista2eval:=[0,0,0,0,0,0,0,0
,0]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "lista2logerro:=[0,0
,0,0,0,0,0,0,0]:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 258 17 "El sistema (
aut.)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "listanumpasos:=[1 \+
, 2 , 4 , 8 , 16 , 32 , 64 , 128 , 256]:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 59 "for i from 1 to 9 do listapaso[i]:=1./listanumpasos[i
]; od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "f:=[x->-10*x[1]+x
[2],x->-10*x[2]]:yini:=[1.,1.]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 43 "u:=x->[exp(-10*x)+x*exp(-10*x),exp(-10*x)]:" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT 260 59 "M1 -> Runge-Kutta 2 evaluaciones y orden 2 'He
un' (s. aut.)" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 59 "for i from 1 to 9 do lista1eval[i]:=2*listanumpasos[i
]; od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 191 "Heun2:=proc(f::l
ist,yini::list,nit::posint,h::numeric)\nlocal k,k1,k2:\nglobal y0:\ny0
:=yini:\nfor k from 1 to nit do\nk1:=f(y0):\nk2:=f(y0+(2./3.)*h*k1):\n
y0:=y0+h*((1./4.)*k1+(3./4.)*k2):\nod:\nend:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 133 "for i from 1 to 9 do y1:=Heun2(f,yini,listanump
asos[i],evalf(listapaso[i])): lista1logerro[i]:=log[10](linalg[norm](y
1-u(1.),2)): od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "evalf(l
ista1logerro,4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7+$\"%;<!\"$$\"%7>
F&$\"%Z%)!\"%$!%\\@F&$!%\\QF&$!%,ZF&$!%\"Q&F&$!%9gF&$!%ImF&" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 261 59 "M2 -> Runge-Kutta 3 evaluaciones \+
y orden 3 'Heun' (s. aut.)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
59 "for i from 1 to 9 do lista2eval[i]:=3*listanumpasos[i]; od:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 218 "Heun3:=proc(f::list,yini::l
ist,nit::posint,h::numeric)\nlocal k,k1,k2,k3:\nglobal y0:\ny0:=yini:
\nfor k from 1 to nit do\nk1:=f(y0):\nk2:=f(y0+(1./3.)*h*k1):\nk3:=f(y
0+(2./3.)*h*k2);\ny0:=y0+h*((1./4.)*k1+(3./4.)*k3):\nod:\nend:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "for i from 1 to 9 do y1:=He
un3(f,yini,listanumpasos[i],evalf(listapaso[i])): lista2logerro[i]:=lo
g[10](linalg[norm](y1-u(1.),2)): od:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "evalf(lista2logerro,4);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#7+$\"%!=#!\"$$\"%-AF&$\"%'=%!\"&$!%ZSF&$!%y[F&$!%heF&$!%9oF&$!%V
xF&$!%f')F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 54 "a) Gr\341ficas co
mparadas 'paso versus log[10] del error'" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 58 "listagraa1:=[seq([listapaso[i],lista1logerro[i]],i=1.
.9)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "graa11:=plot(lista
graa1,style=LINE):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "graa1
2:=plot(listagraa1,style=POINT,symbol=BOX):" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 40 "graa13:=textplot([0.1186, -1.335,`M1`]):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "listagraa2:=[seq([listapaso[
i],lista2logerro[i]],i=1..9)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 36 "graa21:=plot(listagraa2,style=LINE):" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 51 "graa22:=plot(listagraa2,style=POINT,symbol=CIRCL
E):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "graa23:=textplot([0.
1693, -3.503,`M2`]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "dis
play(graa11,graa12,graa13,graa21,graa22,graa23);" }}{PARA 13 "" 1 "" 
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9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3
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72 "b) Gr\341ficas comparadas 'n\372mero de evaluaciones versus log[10
] del error'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "listagrab1:
=[seq([lista1eval[i],lista1logerro[i]],i=1..9)]:" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 36 "grab11:=plot(listagrab1,style=LINE):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "grab12:=plot(listagrab1,styl
e=POINT,symbol=BOX):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "gra
b13:=textplot([176.5, -5.081,`M1`]):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 59 "listagrab2:=[seq([lista2eval[i],lista2logerro[i]],i=1
..9)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "grab21:=plot(list
agrab2,style=LINE):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "grab
22:=plot(listagrab2,style=POINT,symbol=CIRCLE):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 37 "grab23:=textplot([182, -7.132,`M2`]):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "display(grab11,grab12,grab13
,grab21,grab22,grab23);" }}{PARA 13 "" 1 "" {GLPLOT2D 496 496 496 
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{PARA 0 "" 0 "" {TEXT 266 22 "c) M\351todo m\341s adecuado" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 270 83 "El m\351todo M2 \+
funciona mejor en cuanto el paso se hace adecuadamente peque\361o, lo \+
que" }}{PARA 256 "" 0 "" {TEXT -1 58 "corresponde al mayor n\372mero d
e evaluaciones de la funci\363n." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}
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