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}}{PARA 4 "" 0 " " {TEXT -1 93 "El problema de dos cuerpos est\341 finalmente gobernado por el siguiente sistema de primer orden" }}{PARA 4 "" 0 "" {TEXT -1 15 " y'_1 = y_3" }}{PARA 4 "" 0 "" {TEXT -1 15 " y'_2 = y_4" } }{PARA 4 "" 0 "" {TEXT -1 14 " y'_3 = - " }{XPPEDIT 18 0 "y_1/(sqr t((y_1)^2+(y_2)^2))^3" "6#*&%$y_1G\"\"\"*$-%%sqrtG6#,&*$F$\"\"#F%*$%$y _2GF,F%\"\"$!\"\"" }}{PARA 4 "" 0 "" {TEXT -1 14 " y'_4 = - " } {XPPEDIT 18 0 "y_2/(sqrt((y_1)^2+(y_2)^2))^3" "6#*&%$y_2G\"\"\"*$-%%sq rtG6#,&*$%$y_1G\"\"#F%*$F$F-F%\"\"$!\"\"" }}{PARA 4 "" 0 "" {TEXT -1 109 "donde, como es habitual, ( y_1, y_2 ) representa la posici\363 n de la part\355cula y ( y_3, y_4 ) representa " }}{PARA 4 "" 0 "" {TEXT -1 44 "su velocidad. Para las condiciones iniciales" }}{PARA 4 " " 0 "" {TEXT -1 56 " y_1(0) = 1 , y_2(0) = 0 , y_3(0) = 0 , y_4( 0) = 1" }}{PARA 4 "" 0 "" {TEXT -1 93 "la soluci\363n exacta, que es l a que se busca num\351ricamente, es una \363rbita circular de per\355o do 2" }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT 256 3 " , " }}{PARA 4 "" 0 "" {TEXT 258 23 "exactamente la \363rbita " }{TEXT -1 38 "( cos x , \+ sen x , - sen x , cos x )" }{TEXT 259 36 " . Se integra num\351ric amente en el " }}{PARA 4 "" 0 "" {TEXT 260 18 "intervalo [ 0 , 4" } {XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT 261 86 " ] (dos vueltas) con los \+ m\351todos RK2 , RK3 y RK4 , que son los siguientes m\351todos" }} {PARA 4 "" 0 "" {TEXT 262 72 "de Runge-Kutta de 2 , 3 y 4 evaluac iones, y \363rdenes 2 , 3 y 4 :" }}{PARA 4 "" 0 "" {TEXT 276 52 "- el m\351todo de Heun de dos evaluaciones, con tablero" }}{PARA 4 "" 0 "" {TEXT 277 12 " 0 | " }}{PARA 4 "" 0 "" {TEXT 278 15 " 2 /3 | 2/3" }}{PARA 4 "" 0 "" {TEXT 279 21 " -----------------" }} {PARA 4 "" 0 "" {TEXT 280 22 " | 1/4 3/4" }}{PARA 4 "" 0 " " {TEXT 281 53 "- el m\351todo de Heun de tres evaluaciones, con table ro" }}{PARA 4 "" 0 "" {TEXT 283 12 " 0 | " }}{PARA 4 "" 0 "" {TEXT 284 15 " 1/3 | 1/3" }}{PARA 4 "" 0 "" {TEXT 287 22 " 2/3 | 0 2/3" }}{PARA 4 "" 0 "" {TEXT -1 12 " ---------" }{TEXT 285 16 "----------------" }}{PARA 4 "" 0 "" {TEXT 286 31 " | 1/4 0 3/4" }}{PARA 4 "" 0 "" {TEXT 282 70 "- el m\351todo cl \341sico de Runge-Kutta de cuatro evaluaciones, con tablero" }}{PARA 4 "" 0 "" {TEXT 288 12 " 0 | " }}{PARA 4 "" 0 "" {TEXT 289 15 " 1/2 | 1/2" }}{PARA 4 "" 0 "" {TEXT 292 22 " 1/2 | 0 1/ 2" }}{PARA 4 "" 0 "" {TEXT 293 29 " 1 | 0 0 1" }} {PARA 4 "" 0 "" {TEXT -1 12 " ---------" }{TEXT 290 16 "------------ ----" }}{PARA 4 "" 0 "" {TEXT 291 35 " | 1/6 1/3 1/3 \+ 1/6" }}{PARA 4 "" 0 "" {TEXT 263 69 "Las listas de 8 datos que se cons ideran corresponden a los ocho pasos" }}{PARA 4 "" 0 "" {TEXT 264 4 " \+ " }{XPPEDIT 18 0 "Pi/2^2" "6#*&%#PiG\"\"\"*$\"\"#F'!\"\"" }{TEXT -1 12 " = 0.7854 , " }{XPPEDIT 18 0 "Pi/2^3" "6#*&%#PiG\"\"\"*$\"\"#\" \"$!\"\"" }{TEXT -1 12 " = 0.3827 , " }{XPPEDIT 18 0 "Pi/2^4" "6#*&%#P iG\"\"\"*$\"\"#\"\"%!\"\"" }{TEXT -1 12 " = 0.1963 , " }{XPPEDIT 18 0 "Pi/2^5" "6#*&%#PiG\"\"\"*$\"\"#\"\"&!\"\"" }{TEXT -1 11 " = 0.0982 , " }{XPPEDIT 18 0 "Pi/2^6" "6#*&%#PiG\"\"\"*$\"\"#\"\"'!\"\"" }{TEXT -1 12 " = 0.0491 , " }{XPPEDIT 18 0 "Pi/2^7" "6#*&%#PiG\"\"\"*$\"\"#\" \"(!\"\"" }{TEXT -1 12 " = 0.0245 , " }{XPPEDIT 18 0 "Pi/2^8" "6#*&%#P iG\"\"\"*$\"\"#\"\")!\"\"" }{TEXT -1 12 " = 0.0123 , " }{XPPEDIT 18 0 "Pi/2^9" "6#*&%#PiG\"\"\"*$\"\"#\"\"*!\"\"" }{TEXT -1 9 " = 0.0061" }} {PARA 4 "" 0 "" {TEXT -1 46 "Los n\372meros de pasos son, en todos los casos, " }}{PARA 4 "" 0 "" {TEXT -1 48 " 16 , 32 , 64 , 128 , 256 \+ , 512 , 1024 , 2048" }}{PARA 4 "" 0 "" {TEXT -1 44 "Los n\372meros de \+ evaluaciones son para el RK2 " }}{PARA 4 "" 0 "" {TEXT -1 50 " 32 , 64 , 128 , 256 , 512 , 1024 , 2048 , 4096" }}{PARA 4 "" 0 "" {TEXT -1 12 "para el RK3 " }}{PARA 4 "" 0 "" {TEXT -1 50 " 48 , 96 , 192 \+ , 384 , 768 , 1536 , 3072 , 6144" }}{PARA 4 "" 0 "" {TEXT -1 14 "y par a el RK4 " }}{PARA 4 "" 0 "" {TEXT -1 52 " 64 , 128 , 256 , 512 , 1 024 , 2048 , 4096 , 8192" }}{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 12 " ( y_1( " }{TEXT 265 1 "4 " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 10 " ) , y_2( " }{TEXT 266 1 "4" }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 10 " ) , y_3( " }{TEXT 267 1 "4" }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 10 " ) , y_4( " } {TEXT 268 1 "4" }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 9 " ) ) " }}{PARA 4 "" 0 "" {TEXT -1 70 "error medido en norma2. Como podemos ve r, para nuestros m\351todos, vamos" }}{PARA 4 "" 0 "" {TEXT -1 65 "obt eniendo los siguientes logaritmos de los errores; para el RK2 " }} {PARA 4 "" 0 "" {TEXT -1 85 " 0.4343 , 0.3456 , -0.1695 , -0.7862 , - 1.412 , - 2.030 , - 2.641 , - 3.248" }}{PARA 4 "" 0 "" {TEXT -1 12 "para el RK3 " }}{PARA 4 "" 0 "" {TEXT -1 92 " -0.008691 , \+ - 1.459 , - 3.037 , - 4.912 , - 6.062 , - 6.770 , - 6.782 , - 7. 7859" }}{PARA 4 "" 0 "" {TEXT -1 14 "y para el RK4 " }}{PARA 4 "" 0 " " {TEXT -1 87 " 0.3978 , - 1.267 , - 2.707 , - 4.096 , - 5.445 \+ , - 6.973 , - 7.007 , - 8.084" }}{PARA 4 "" 0 "" {TEXT -1 91 "Con e stos datos, se deben construir las siguientes 'gr\341ficas de eficienc ia' que mezcles los" }}{PARA 4 "" 0 "" {TEXT -1 34 "resultados de RK2 , RK3 y RK4 :" }}{PARA 4 "" 0 "" {TEXT -1 46 "a) La gr\341fica 'pas o 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 88 "c) La gr\341fica 'log[10] del paso versus log[10] del error'. En \351sta se puede comprobar la" }}{PARA 4 "" 0 "" {TEXT -1 87 "tendencia a dar rectas con pendiente igual en m\363dul o al orden, usando bien la recta de " }}{PARA 4 "" 0 "" {TEXT -1 84 "r egresi\363n de los datos, o mejor la pendiente de alguno de los \372lt imos segmentos que " }}{PARA 4 "" 0 "" {TEXT -1 19 "componen la recta. " }}{PARA 4 "" 0 "" {TEXT -1 87 "d) La gr\341fica 'log[10] del n\372m ero de evaluaciones versus log[10] del error'. Convendr\341 " }}{PARA 4 "" 0 "" {TEXT -1 67 "establecer las mismas comprobaciones que en el \+ apartado precedente." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "rest art:" }}}{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 37 "with(stats):with(linalg):wit h(plots):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and 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 295 15 "Variables lista" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "listanumpasos:=[0,0, 0,0,0,0,0,0]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "listpaso:= [0,0,0,0,0,0,0,0]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "lista logpaso:=[0,0,0,0,0,0,0,0]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "lista2eval:=[0,0,0,0,0,0,0,0]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "lista2logeval:=[0,0,0,0,0,0,0,0]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "lista2logerro:=[0,0,0,0,0,0,0,0]:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "lista3eval:=[0,0,0,0,0,0,0,0 ]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "lista3logeval:=[0,0,0 ,0,0,0,0,0]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "lista3loger ro:=[0,0,0,0,0,0,0,0]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "l ista4eval:=[0,0,0,0,0,0,0,0]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "lista4logeval:=[0,0,0,0,0,0,0,0]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "lista4logerro:=[0,0,0,0,0,0,0,0]:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 294 17 "El sistema (aut.)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "listanumpasos:=[16,32,64,128,256,512,1024,2048]:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "for i from 1 to 8 do listapa so[i]:=4*Pi/listanumpasos[i]; od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "for i from 1 to 8 do listalogpaso[i]:=evalf(log[10](l istapaso[i])); od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "f:=[ x->x[3],x->x[4],x->-(x[1]/sqrt(x[1]^2+x[2]^2)^3),x->-(x[2]/sqrt(x[1]^2 +x[2]^2)^3)]:yini:=[1.,0.,0.,1.]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "u:=x->[cos(x),sin(x),-sin(x),cos(x)]:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 296 60 "RK2 -> Runge-Kutta 2 evaluaciones y orde n 2 'Heun' (s. aut.)" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "for i from 1 to 8 do lista2eval[i]:=2*listanumpasos[i ]; od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "for i from 1 to 8 do lista2logeval[i]:=evalf(log[10](lista2eval[i])); od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 191 "Heun2:=proc(f::list,yini::list,nit ::posint,h::numeric)\nlocal k,k1,k2:\nglobal y0:\ny0:=yini:\nfor k fro m 1 to nit do\nk1:=f(y0):\nk2:=f(y0+(2./3.)*h*k1):\ny0:=y0+h*((1./4.)* k1+(3./4.)*k2):\nod:\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "for i from 1 to 8 do y1:=Heun2(f,yini,listanumpasos[i],evalf(list apaso[i])): lista2logerro[i]:=log[10](linalg[norm](y1-u(4*Pi),2)): od: " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 297 60 "RK3 -> Runge-Kutta 3 evaluac iones y orden 3 'Heun' (s. aut.)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "for i from 1 to 8 do lista3eval[i]:=3*listanumpasos[i ]; od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "for i from 1 to 8 do lista3logeval[i]:=evalf(log[10](lista3eval[i])); od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 218 "Heun3:=proc(f::list,yini::list,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(y0+(2./3. )*h*k2);\ny0:=y0+h*((1./4.)*k1+(3./4.)*k3):\nod:\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "for i from 1 to 8 do y1:=Heun3(f,y ini,listanumpasos[i],evalf(listapaso[i])): lista3logerro[i]:=log[10](l inalg[norm](y1-u(4*Pi),2)): od:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 298 61 "RK4 -> Runge-Kutta 4 evaluaciones y orden 4 cl\341sico (s. aut.)" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "for i \+ from 1 to 8 do lista4eval[i]:=4*listanumpasos[i]; od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "for i from 1 to 8 do lista4logeval[ i]:=evalf(log[10](lista4eval[i])); od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 259 "Rkut4:=proc(f::list,yini::list,nit::posint,h::numeri c)\nlocal k,k1,k2,k3,k4:\nglobal y0:\ny0:=yini:\nfor k from 1 to nit d o\nk1:=f(y0):\nk2:=f(y0+(1./2.)*h*k1):\nk3:=f(y0+(1./2.)*h*k2);\nk4:=f (y0+h*k3);\ny0:=y0+h*((1./6.)*k1+(1./3.)*k2+(1./3.)*k3+(1./6.)*k4):\no d:\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "for i from 1 t o 8 do y1:=Rkut4(f,yini,listanumpasos[i],evalf(listapaso[i])): lista4l ogerro[i]:=log[10](linalg[norm](y1-u(4*Pi),2)): od:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 257 54 "a) Gr\341ficas comparadas 'paso versus log[10] del error'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "listagraa2:= [seq([listapaso[i],lista2logerro[i]],i=1..8)]:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 36 "graa21:=plot(listagraa2,style=LINE):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "graa22:=plot(listagraa2,styl e=POINT,symbol=BOX):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "gra a23:=textplot([0.1567, -0.7317,`RK2`]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "listagraa3:=[seq([listapaso[i],lista3logerro[i]],i=1. .8)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "graa31:=plot(lista graa3,style=LINE):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "graa3 2:=plot(listagraa3,style=POINT,symbol=CIRCLE):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 41 "graa33:=textplot([0.2036, -3.904,`RK3`]):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "listagraa4:=[seq([listapaso[ i],lista4logerro[i]],i=1..8)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "graa41:=plot(listagraa4,style=LINE):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "graa42:=plot(listagraa4,style=POINT,symbol=CROSS ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "graa43:=textplot([0.1 892, -2.086,`RK4`]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "dis play(graa21,graa22,graa23,graa31,graa32,graa33,graa41,graa42,graa43); " }}{PARA 13 "" 1 "" {GLPLOT2D 496 496 496 {PLOTDATA 2 "6--%'CURVESG6% 7*7$$\"5i4$[uRj\")R&y!#?$\"5rC\\Kf[:(QM%F*7$$\"5\"[:C()p\"3*p#RF*$\"5Z <%y-H4\"ycMF*7$$\"5Tx?O\\3a\\j>F*$!5ceU^p\"F*7$$\"5.(Q5oC/xu\")*! #@$!5;\"Q7!p\\P2iyF*7$$\"5_$>0M7_Q(3\\F:$!5K.;C\"4UiCT\"!#>7$$\"5w'f-< 1EpVX#F:$!54rv;8LVuI?FB7$$\"5Q)H^3.j%=F7F:$!5'eE!z\\SC!>k#FB7$$\"5*=\\ cU::Bf8'!#A$!5$Q$*RVO7^([KFB-%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!FenFZ-%& STYLEG6#%%LINEG-F$6&F&FS-Fgn6#%&POINTG-%'SYMBOLG6#%$BOXG-%%TEXTG6$7$$ \"%n:!\"%$!%$!5LP_$H.Ge16'FB7$FD$ !5_Ua$[m)Hy?qFB7$FI$!5lG&)o7eSwF\")FB7$FN$!5Fe#[_;li,H*FBFSFfn-F$6&F`p FSF\\o-F`o6#%'CIRCLEG-Fdo6$7$$\"%O?Fio$!%/R!\"$Q$RK3F]p-F$6%7*7$F($\"5 sQx$o$!5xx2mL2G:LaFB7$FD$!5KT5r " 0 "" {MPLTEXT 1 0 59 "listagrab2:=[seq([lista2eval [i],lista2logerro[i]],i=1..8)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "grab21:=plot(listagrab2,style=LINE):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "grab22:=plot(listagrab2,style=POINT,symbol=BOX): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "grab23:=textplot([1892, -2.086,`RK2`]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "listagr ab3:=[seq([lista3eval[i],lista3logerro[i]],i=1..8)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "grab31:=plot(listagrab3,style=LINE):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "grab32:=plot(listagrab3,styl e=POINT,symbol=CIRCLE):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 " grab33:=textplot([1687, -6.77,`RK3`]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "listagrab4:=[seq([lista4eval[i],lista4logerro[i]],i=1 ..8)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "grab41:=plot(list agrab4,style=LINE):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "grab 42:=plot(listagrab4,style=POINT,symbol=CROSS):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 38 "grab43:=textplot([1890, -5.66,`RK4`]):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "display(grab21,grab22,grab23 ,grab31,grab32,grab33,grab41,grab42,grab43);" }}{PARA 13 "" 1 "" {GLPLOT2D 496 496 496 {PLOTDATA 2 "6--%'CURVESG6%7*7$$\"#K\"\"!$\"5rC \\Kf[:(QM%!#?7$$\"#kF*$\"5Z<%y-H4\"ycMF-7$$\"$G\"F*$!5ceU^p\"F-7$ $\"$c#F*$!5;\"Q7!p\\P2iyF-7$$\"$7&F*$!5K.;C\"4UiCT\"!#>7$$\"%C5F*$!54r v;8LVuI?FB7$$\"%[?F*$!5'eE!z\\SC!>k#FB7$$\"%'4%F*$!5$Q$*RVO7^([KFB-%'C OLOURG6&%$RGBG$\"#5!\"\"$F*F*FY-%&STYLEG6#%%LINEG-F$6&F&FR-Fen6#%&POIN TG-%'SYMBOLG6#%$BOXG-%%TEXTG6$7$$\"%#*=F*$!%'3#!\"$Q$RK26\"-F$6%7*7$$ \"#[F*$!5'3k0hV#pS\"p)!#A7$$\"#'*F*$!5P3l/+J)[#f9FB7$$\"$#>F*$!5gRm,qm b^PIFB7$$\"$%QF*$!5`]$z:VJZ0\"\\FB7$$\"$o(F*$!5LP_$H.Ge16'FB7$$\"%O:F* $!5_Ua$[m)Hy?qFB7$$\"%sIF*$!5lG&)o7eSwF\")FB7$$\"%WhF*$!5Fe#[_;li,H*FB FRFZ-F$6&F^pFRFjn-F^o6#%'CIRCLEG-Fbo6$7$$\"%(o\"F*$!$x'!\"#Q$RK3F[p-F$ 6%7*7$F/$\"5sQx$o$!54M\\:)*fb7'4%FB7$FD$!5xx2mL2G:LaFB7$FI$!5KT5r)F*$!5t=t8Kxh=4#*FBFRFZ-F$6&FhsFRFjn-F^o6#%&CROSSG-Fbo 6$7$$\"%!*=F*$!$m&FdsQ$RK4F[p-%+AXESLABELSG6%Q!F[pFcv-%%FONTG6#%(DEFAU LTG-%%VIEWG6$FgvFgv" 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" "Curve 7" "Curve 8" "Curve 9" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT 270 66 "c) Gr\341ficas comparadas 'log[10] del paso versus log[10] del error'" }}}{EXCHG {PARA 4 "" 0 "" {TEXT 302 48 "Recordemos lo que ya \+ sabemos de otros problemas:" }}{PARA 4 "" 0 "" {TEXT 330 92 "Hay un ti po de gr\341ficas que sirve para comprobar el orden 'efectivo' de los \+ m\351todos, o sea, " }}{PARA 4 "" 0 "" {TEXT 304 90 "el orden con que, en la pr\341ctica, parecen comportarse. En estas gr\341ficas se debe \+ presentar" }}{PARA 4 "" 0 "" {TEXT 305 94 "en el eje de abscisas el lo g[10] del paso empleado y en el de ordenadas el log[10] del error. " } }{PARA 4 "" 0 "" {TEXT 306 90 "Entonces el aspecto de la gr\341fica se aproxima a una recta (o se puede aproximar utilizando" }}{PARA 4 "" 0 "" {TEXT 307 91 "la regresi\363n lineal). La pendiente de esa recta \+ es el orden con que se comporta el m\351todo. " }}{PARA 4 "" 0 "" {TEXT 308 59 "En efecto, si el m\351todo es de orden p , el error ser \341 O(" }{XPPEDIT 18 0 "h^p" "6#)%\"hG%\"pG" }{TEXT 309 37 ") . Con siderando que, aproximadamente" }}{PARA 4 "" 0 "" {TEXT 310 20 "se tie ne E(h) = K " }{XPPEDIT 18 0 "h^p" "6#)%\"hG%\"pG" }{TEXT 311 78 " \+ , resulta que el cociente incremental (la pendiente aproximada de la r ecta) " }}{PARA 4 "" 0 "" {TEXT 312 43 "entre dos puntos correspondien tes a pasos " }{XPPEDIT 18 0 "h[1]" "6#&%\"hG6#\"\"\"" }{TEXT 313 3 " < " }{XPPEDIT 18 0 "h[2]" "6#&%\"hG6#\"\"#" }{TEXT 314 5 " es " }} {PARA 4 "" 0 "" {TEXT 315 5 " " }{XPPEDIT 18 0 "(log[10] (E(h[2))- log[10] (E(h[1])))/(log[10] (h[2])-log[10] (h[1]))" "6#*&,&-&%$logG6# \"#56#-%\"EG6#&%\"hG6#\"\"#\"\"\"-&F'6#F)6#-F,6#&F/6#F2!\"\"F2,&-&F'6# F)6#&F/6#F1F2-&F'6#F)6#&F/6#F2F;F;" }{TEXT -1 4 " = " }{XPPEDIT 18 0 "(log[10] (K)+p*log[10](h[2])-log[10] (K)-p*log[10](h[1]))/(log[10] (h [2])-log[10] (h[1]))" "6#*&,*-&%$logG6#\"#56#%\"KG\"\"\"*&%\"pGF,-&F'6 #F)6#&%\"hG6#\"\"#F,F,-&F'6#F)6#F+!\"\"*&F.F,-&F'6#F)6#&F46#F,F,F;F,,& -&F'6#F)6#&F46#F6F,-&F'6#F)6#&F46#F,F;F;" }{TEXT -1 24 " = p \+ " }}{PARA 4 "" 0 "" {TEXT 316 89 "Esta pendiente se puede es tudiar cuando se utilizan los segmentos correspondientes a los " }} {PARA 4 "" 0 "" {TEXT 320 69 "menores valores de h , los mas represen tativos del efecto del orden." }}{PARA 4 "" 0 "" {TEXT 317 101 "Algo s emejante ocurre cuando se presenta en el eje de abscisas el log[10] d el numero de evaluaciones" }}{PARA 4 "" 0 "" {TEXT 321 97 "y en el de \+ ordenadas el log[10] del error. Entonces tambi\351n el aspecto de la g r\341fica se aproxima " }}{PARA 4 "" 0 "" {TEXT 335 98 "a una recta (o se puede aproximar utilizando la regresi\363n lineal). Pero la pendie nte de esa recta " }}{PARA 4 "" 0 "" {TEXT 336 91 "es ahora igual a me nos el orden con que se comporta el m\351todo, debido a que en el coci ente " }}{PARA 4 "" 0 "" {TEXT 337 39 "antes expuesto el denominador e s ahora " }}{PARA 4 "" 0 "" {TEXT 324 5 " " }{XPPEDIT 18 0 "log[10 ] (eval(h[2]))-log[10] (eval(h[1]))" "6#,&-&%$logG6#\"#56#-%%evalG6#&% \"hG6#\"\"#\"\"\"-&F&6#F(6#-F+6#&F.6#F1!\"\"" }{TEXT 322 3 " = " } {XPPEDIT 18 0 "log[10] (m/h[2])-log[10] (m/h[1])" "6#,&-&%$logG6#\"#56 #*&%\"mG\"\"\"&%\"hG6#\"\"#!\"\"F,-&F&6#F(6#*&F+F,&F.6#F,F1F1" }{TEXT 323 4 " = " }}{PARA 4 "" 0 "" {TEXT 325 12 " = " }{XPPEDIT 18 0 "log[10](m)-log[10] (h[2])-log[10](m)+log[10] (h[1])" "6#,*-&%$lo gG6#\"#56#%\"mG\"\"\"-&F&6#F(6#&%\"hG6#\"\"#!\"\"-&F&6#F(6#F*F4-&F&6#F (6#&F16#F+F+" }{TEXT 326 4 " = " }{XPPEDIT 18 0 "log[10] (h[1])-log[1 0] (h[2])" "6#,&-&%$logG6#\"#56#&%\"hG6#\"\"\"F--&F&6#F(6#&F+6#\"\"#! \"\"" }{TEXT 327 21 " " }}{PARA 4 "" 0 "" {TEXT 318 87 "en el supuesto de que m es el n\372mero de evaluaciones por \+ paso del m\351todo en cuesti\363n." }}{PARA 4 "" 0 "" {TEXT 319 101 "E sta idea se emplea especialmente cuando el m\351todo es de paso variab le y se desea averiguar su orden " }}{PARA 4 "" 0 "" {TEXT 328 104 "ef ectivo, ya que, entonces, no tiene sentido hablar del paso del m\351to do, pero s\355 que lo tiene hablar de " }}{PARA 4 "" 0 "" {TEXT 329 42 "las evaluaciones realizadas por el mismo. " }}{PARA 4 "" 0 "" {TEXT 331 95 "Y esto es justamente lo que sucede en este caso en el qu e hay que comparar con m\351todos de paso " }}{PARA 4 "" 0 "" {TEXT 332 97 "variable; por ello no emplearemos las gr\341ficas 'log[10] del paso versus log[10] del error' , que " }}{PARA 4 "" 0 "" {TEXT 333 106 "carecen ahora de sentido, sino las gr\341ficas 'log[10] del n\372 mero de evaluaciones versus log[10] del error'," }}{PARA 4 "" 0 "" {TEXT 334 70 "que son las que ahora pueden servir para detectar el 'or den efectivo'." }{TEXT -1 0 "" }{TEXT 303 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "listagrac2:=[seq([listalogpaso[i],lista2logerro[ i]],i=1..8)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "grac21:=pl ot(listagrac2,style=LINE):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "grac22:=plot(listagrac2,style=POINT,symbol=BOX):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "grac23:=textplot([-1.048, -0.6154,`RK2`]) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "listagrac3:=[seq([list alogpaso[i],lista3logerro[i]],i=1..8)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "grac31:=plot(listagrac3,style=LINE):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "grac32:=plot(listagrac3,style=POINT ,symbol=CIRCLE):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "grac33: =textplot([-0.8809, -4.416,`RK3`]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "listagrac4:=[seq([listalogpaso[i],lista4logerro[i]],i =1..8)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "grac41:=plot(li stagrac4,style=LINE):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "gr ac42:=plot(listagrac4,style=POINT,symbol=CROSS):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "grac43:=textplot([-0.8283, -2.6,`RK4`]):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "display(grac21,grac22,grac23 ,grac31,grac32,grac33,grac41,grac42,grac43);" }}{PARA 13 "" 1 "" {GLPLOT2D 496 496 496 {PLOTDATA 2 "6--%'CURVESG6%7*7$$!53O&GQj=,\"\\5! 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