{VERSION 6 0 "IBM INTEL NT" "6.0" } {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 0 228 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 18 0 0 208 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 1 18 0 0 208 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 1 14 25 1 1 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 1 14 25 1 1 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 1 14 25 1 1 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 1 14 25 1 1 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 266 "" 1 18 0 0 208 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "" 1 24 0 0 0 0 0 0 0 0 0 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 "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 4 4 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Cour ier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot " -1 13 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 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 1 14 0 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 }} {SECT 0 {EXCHG {PARA 4 "" 0 "" {TEXT 269 14 "Ejercicios de " }}{PARA 4 "" 0 "" {TEXT 270 48 "(3) Ecuaciones escalares: m\351todos de RUNGE- KUTTA" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 268 15 "Ejercicio 03-12" }}{PARA 4 "" 0 "" {TEXT -1 81 "\nPara integrar una e cuaci\363n escalar no aut\363noma se dise\361an dos m\351todos num \351ricos" }}{PARA 4 "" 0 "" {TEXT -1 92 "de integraci\363n de ecuacio nes diferenciales que designaremos con los nombres de M1 y M2 . " } }{PARA 4 "" 0 "" {TEXT -1 86 "Para determinar cual es el m\341s adecua do, 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 , 25 6 , 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 53 "la soluci\363n en x=1 . Los errores que se obtienen son, " }}{PARA 4 "" 0 "" {TEXT -1 11 "para el M1 " }}{PARA 4 "" 0 "" {TEXT -1 83 " -0.8790, -1.643, -2.333, -2.977, -3.600, -4.213, -4.820, -5.424, -6.028 " }}{PARA 4 "" 0 "" {TEXT -1 13 " y para el M2" }} {PARA 4 "" 0 "" {TEXT -1 79 " -1.462, -2.543, -3.534, -4.480, \+ -5.405, -6.319, -7.228, -8.133, -9.038" }}{PARA 4 "" 0 "" {TEXT -1 91 "Con estos datos, se deben construir las siguientes 'gr\341ficas de ef iciencia' que mezclen los" }}{PARA 4 "" 0 "" {TEXT -1 26 "resultados d e M1 y M2 :" }}{PARA 4 "" 0 "" {TEXT -1 46 "a) La gr\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\351tod o parece m\341s adecuado." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "interface(labe ling=false):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Digits:=20: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "with(linalg):with(plots ):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm a nd trace have been redefined and unprotected\n" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined\n" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 259 15 "Variables lista" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "listanumpasos:=[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 "lis ta1eval:=[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 21 "La ecuaci\363n (no 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 36 "f:=(x,y)->-y+x+1:ytotalini :=[0.,1.]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "u:=x->(x*exp( x)+1)/exp(x):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 62 "M1 -> Runge-Kut ta 2 evaluaciones y orden 2 'Heun' (e. no aut.)" }{MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "for i from 1 to 9 do lista1 eval[i]:=2*listanumpasos[i]; od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 246 "Heun2:=proc(f::procedure,ytotalini::list,nit::posint ,h::numeric)\nlocal x0,y0,k,k1,k2:\nx0:=ytotalini[1]:y0:=ytotalini[2]: \nfor k from 1 to nit do\nk1:=f(x0,y0):\nk2:=f(x0+(2./3.)*h,y0+(2./3.) *h*k1):\nx0:=x0+h:y0:=y0+h*((1./4.)*k1+(3./4.)*k2):\nod:\nend:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "for i from 1 to 9 do y1:=He un2(f,ytotalini,listanumpasos[i],evalf(listapaso[i])): lista1logerro[i ]:=log[10](abs(y1-u(1.))): od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "evalf(lista1logerro,4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7+$ !%!z)!\"%$!%V;!\"$$!%LBF)$!%xHF)$!%+OF)$!%8UF)$!%?[F)$!%CaF)$!%GgF)" } }}{EXCHG {PARA 0 "" 0 "" {TEXT 261 62 "M2 -> Runge-Kutta 3 evaluacione s y orden 3 'Heun' (e. no aut.)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "for i from 1 to 9 do lista2eval[i]:=3*listanumpasos[i]; od:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 286 "Heun3:=proc(f::procedure, ytotalini::list,nit::posint,h::numeric)\nlocal x0,y0,k,k1,k2,k3:\nx0:= ytotalini[1]:y0:=ytotalini[2]:\nfor k from 1 to nit do\nk1:=f(x0,y0): \nk2:=f(x0+(1./3.)*h,y0+(1./3.)*h*k1):\nk3:=f(x0+(2./3.)*h,y0+(2./3.)* h*k2);\nx0:=x0+h:y0:=y0+h*((1./4.)*k1+(3./4.)*k3):\nod:\nend:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "for i from 1 to 9 do y1:=He un3(f,ytotalini,listanumpasos[i],evalf(listapaso[i])): lista2logerro[i ]:=log[10](abs(y1-u(1.))): od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "evalf(lista2logerro,4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7+$ !%i9!\"$$!%VDF&$!%MNF&$!%![%F&$!%0aF&$!%>jF&$!%GsF&$!%L\")F&$!%Q!*F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 54 "a) Gr\341ficas comparadas 'pas o 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(listagraa1,styl e=LINE):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "graa12:=plot(li stagraa1,style=POINT,symbol=BOX):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "graa13:=textplot([0.13, -2.34,`M1`]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "listagraa2:=[seq([listapaso[i],list a2logerro[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=CIRCLE):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "graa23:=textplot([0.1693, -3.503,`M2`]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "display( graa11,graa12,graa13,graa21,graa22,graa23);" }}{PARA 13 "" 1 "" {GLPLOT2D 496 496 496 {PLOTDATA 2 "6*-%'CURVESG6%7+7$$\"\"\"\"\"!$!5*y j'=Z!)fH!z)!#?7$$\"5+++++++++]F-$!5>G_pl)Q.Jk\"!#>7$$\"5+++++++++DF-$! 5$\\w*\\@laeKBF37$$\"5++++++++]7F-$!5(z'fly11CxHF37$$\"5++++++++]i!#@$ !50^,.ALv:+OF37$$\"5++++++++DJFA$!5VDG&R@#R_7UF37$$\"5+++++++]i:FA$!5C 3)3*>0bq>[F37$$\"5+++++++]7y!#A$!5su)G&*fF=VU&F37$$\"5+++++++D1RFQ$!5R %Qmy)RElFgF3-%'COLOURG6&%$RGBG$\"#5!\"\"$F*F*Fjn-%&STYLEG6#%%LINEG-F$6 &F&FY-F\\o6#%&POINTG-%'SYMBOLG6#%$BOXG-%%TEXTG6$7$$\"#8!\"#$!$M#F^pQ#M 16\"-F$6%7+7$F($!5'[`KJc(3gh9F37$F/$!5,DOI)>A=La#F37$F5$!5O/oIEO'*fLNF 37$F:$!5u()ocp%\\t-[%F37$F?$!5ib-]z&eI^S&F37$FE$!5;OYP+EH4>jF37$FJ$!5a W*e*['e:wA(F37$FO$!5G(H4sI%4UL\")F37$FU$!53j_*H9Ooy.*F3FYF[o-F$6&FepFY Fao-Feo6#%'CIRCLEG-Fio6$7$$\"%$p\"!\"%$!%.N!\"$Q#M2Fbp-%+AXESLABELSG6% Q!FbpFcs-%%FONTG6#%(DEFAULTG-%%VIEWG6$FgsFgs" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT 257 72 "b) Gr \341ficas comparadas 'n\372mero de evaluaciones versus log[10] del err or'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "listagrab1:=[seq([li sta1eval[i],lista1logerro[i]],i=1..9)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "grab11:=plot(listagrab1,style=LINE):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "grab12:=plot(listagrab1,style=POINT ,symbol=BOX):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "grab13:=te xtplot([171.5, -4.22,`M1`]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "listagrab2:=[seq([lista2eval[i],lista2logerro[i]],i=1..9)]:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "grab21:=plot(listagrab2,styl e=LINE):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "grab22:=plot(li stagrab2,style=POINT,symbol=CIRCLE):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "grab23:=textplot([169.2, -6.25,`M2`]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "display(grab11,grab12,grab13,grab21 ,grab22,grab23);" }}{PARA 13 "" 1 "" {GLPLOT2D 496 496 496 {PLOTDATA 2 "6*-%'CURVESG6%7+7$$\"\"#\"\"!$!5*yj'=Z!)fH!z)!#?7$$\"\"%F*$!5>G_pl) Q.Jk\"!#>7$$\"\")F*$!5$\\w*\\@laeKBF37$$\"#;F*$!5(z'fly11CxHF37$$\"#KF *$!50^,.ALv:+OF37$$\"#kF*$!5VDG&R@#R_7UF37$$\"$G\"F*$!5C3)3*>0bq>[F37$ $\"$c#F*$!5su)G&*fF=VU&F37$$\"$7&F*$!5R%Qmy)RElFgF3-%'COLOURG6&%$RGBG$ \"#5!\"\"$F*F*Fhn-%&STYLEG6#%%LINEG-F$6&F&FW-Fjn6#%&POINTG-%'SYMBOLG6# %$BOXG-%%TEXTG6$7$$\"%:A=La#F37$$\"#7F*$!5O/oIEO'*fLNF37$$\"#CF*$!5u ()ocp%\\t-[%F37$$\"#[F*$!5ib-]z&eI^S&F37$$\"#'*F*$!5;OYP+EH4>jF37$$\"$ #>F*$!5aW*e*['e:wA(F37$$\"$%QF*$!5G(H4sI%4UL\")F37$$\"$o(F*$!53j_*H9Oo y.*F3FWFin-F$6&FcpFWF_o-Fco6#%'CIRCLEG-Fgo6$7$$\"%#p\"Fgn$!$D'F^pQ#M2F `p-%+AXESLABELSG6%Q!F`pFat-%%FONTG6#%(DEFAULTG-%%VIEWG6$FetFet" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT 266 22 "c) M\351todo m\341s adecuado" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 267 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 de evaluacione s de la funci\363n." }}}}{MARK "0 3 0" 15 }{VIEWOPTS 1 1 0 3 4 1802 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }