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{SECT 0 {EXCHG {PARA 4 "" 0 "" {TEXT 289 14 "Ejercicios de " }}{PARA 
4 "" 0 "" {TEXT 290 48 "(3) Ecuaciones escalares: m\351todos de RUNGE-
KUTTA" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 288 15 
"Ejercicio 03-13" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" 
{TEXT -1 92 "El problema escalar   y' = - y + x + 1 ,  y(0) = 1 , del \+
que se conoce la verdadera soluci\363n" }}{PARA 4 "" 0 "" {TEXT -1 15 
"       y(x) =  " }{XPPEDIT 18 0 "(x*exp(x)+1)/exp(x);" "6#*&,&*&%\"xG
\"\"\"-%$expG6#F&F'F'F'F'F'-F)6#F&!\"\"" }}{PARA 4 "" 0 "" {TEXT 257 
121 "se integra num\351ricamente en el intervalo  [ 0 , 1 ]  con los m
\351todos  RK2 ,  RK3  y  RK4 , que son los siguientes m\351todos" }}
{PARA 4 "" 0 "" {TEXT 258 72 "de Runge-Kutta de 2 ,  3   y  4  evaluac
iones, y \363rdenes  2 ,  3  y  4 :" }}{PARA 4 "" 0 "" {TEXT 268 52 "-
 el m\351todo de Heun de dos evaluaciones, con tablero" }}{PARA 4 "" 
0 "" {TEXT 269 12 "       0  | " }}{PARA 4 "" 0 "" {TEXT 270 15 "    2
/3  |  2/3" }}{PARA 4 "" 0 "" {TEXT 271 21 "    -----------------" }}
{PARA 4 "" 0 "" {TEXT 272 22 "          |  1/4   3/4" }}{PARA 4 "" 0 "
" {TEXT 273 53 "- el m\351todo de Heun de tres evaluaciones, con table
ro" }}{PARA 4 "" 0 "" {TEXT 275 12 "       0  | " }}{PARA 4 "" 0 "" 
{TEXT 276 15 "    1/3  |  1/3" }}{PARA 4 "" 0 "" {TEXT 279 22 "    2/3
  |     0   2/3" }}{PARA 4 "" 0 "" {TEXT -1 12 "   ---------" }{TEXT 
277 16 "----------------" }}{PARA 4 "" 0 "" {TEXT 278 31 "           |
  1/4      0    3/4" }}{PARA 4 "" 0 "" {TEXT 274 70 "- el m\351todo cl
\341sico de Runge-Kutta de cuatro evaluaciones, con tablero" }}{PARA 
4 "" 0 "" {TEXT 280 12 "       0  | " }}{PARA 4 "" 0 "" {TEXT 281 15 "
    1/2  |  1/2" }}{PARA 4 "" 0 "" {TEXT 284 22 "    1/2  |     0   1/
2" }}{PARA 4 "" 0 "" {TEXT 285 29 "       1  |     0      0    1" }}
{PARA 4 "" 0 "" {TEXT -1 12 "   ---------" }{TEXT 282 16 "------------
----" }}{PARA 4 "" 0 "" {TEXT 283 35 "          |  1/6    1/3   1/3   \+
1/6" }}{PARA 4 "" 0 "" {TEXT 259 69 "Las listas de 8 datos que se cons
ideran corresponden a los ocho pasos" }}{PARA 4 "" 0 "" {TEXT 260 4 " \+
   " }{XPPEDIT 18 0 "1/(2^4);" "6#*&\"\"\"F$*$\"\"#\"\"%!\"\"" }{TEXT 
-1 11 " = 0.625 , " }{XPPEDIT 18 0 "1/(2^5);" "6#*&\"\"\"F$*$\"\"#\"\"
&!\"\"" }{TEXT -1 13 " = 0.03125 , " }{XPPEDIT 18 0 "1/(2^6);" "6#*&\"
\"\"F$*$\"\"#\"\"'!\"\"" }{TEXT -1 13 " = 0.01563 , " }{XPPEDIT 18 0 "
1/(2^7);" "6#*&\"\"\"F$*$\"\"#\"\"(!\"\"" }{TEXT -1 14 " = 0.007813 , \+
" }{XPPEDIT 18 0 "1/(2^8);" "6#*&\"\"\"F$*$\"\"#\"\")!\"\"" }{TEXT -1 
14 " = 0.003906 , " }{XPPEDIT 18 0 "1/(2^9);" "6#*&\"\"\"F$*$\"\"#\"\"
*!\"\"" }{TEXT -1 14 " = 0.001953 , " }{XPPEDIT 18 0 "1/(2^10);" "6#*&
\"\"\"F$*$\"\"#\"#5!\"\"" }{TEXT -1 15 " = 0.0009766 , " }{XPPEDIT 18 
0 "1/(2^11);" "6#*&\"\"\"F$*$\"\"#\"#6!\"\"" }{TEXT -1 14 " = 0.000488
3 ." }}{PARA 4 "" 0 "" {TEXT -1 46 "Los n\372meros de pasos son, en to
dos los casos, " }}{PARA 4 "" 0 "" {TEXT -1 48 "    16 , 32 , 64 , 128
 , 256 , 512 , 1024 , 2048" }}{PARA 4 "" 0 "" {TEXT -1 44 "Los n\372me
ros 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 para el RK4 " }}{PARA 4 "" 0 "" {TEXT -1 52 "    64 , 128 , 256 , 5
12 , 1024 , 2048 , 4096 , 8192" }}{PARA 4 "" 0 "" {TEXT -1 59 "Para ca
da m\351todo se calcula el  log[10] del error cometido." }}{PARA 4 "" 
0 "" {TEXT -1 84 "Como podemos ver, para nuestros m\351todos, vamos ob
teniendo los siguientes logaritmos " }}{PARA 4 "" 0 "" {TEXT -1 28 "de
 los errores; para el RK2 " }}{PARA 4 "" 0 "" {TEXT -1 66 "    -3.600,
 -4.213, -4.820, -5.424, -6.028, -6.630, -7.233, -7.835" }}{PARA 4 "" 
0 "" {TEXT -1 12 "para el RK3 " }}{PARA 4 "" 0 "" {TEXT -1 66 "    -5.
405, -6.319, -7.228, -8.133, -9.038, -9.942, -10.85, -11.75" }}{PARA 
4 "" 0 "" {TEXT -1 13 "y para el RK4" }}{PARA 4 "" 0 "" {TEXT -1 66 " \+
   -7.307, -8.523, -9.733, -10.94, -12.15, -13.35, -14.55, -15.76" }}
{PARA 4 "" 0 "" {TEXT -1 91 "Con estos datos, se deben construir las s
iguientes 'gr\341ficas de eficiencia' 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 'paso versus log[10] del error'." }}
{PARA 4 "" 0 "" {TEXT -1 64 "b) La gr\341fica 'n\372mero de evaluacion
es 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 pu
ede comprobar la" }}{PARA 4 "" 0 "" {TEXT -1 87 "tendencia a dar recta
s con pendiente igual en m\363dulo al orden, usando bien la recta de \+
" }}{PARA 4 "" 0 "" {TEXT -1 84 "regresi\363n de los datos, o mejor la
 pendiente de alguno de los \372ltimos 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\372mero de evaluaciones versus log[10] d
el error'. Convendr\341 " }}{PARA 4 "" 0 "" {TEXT -1 67 "establecer la
s mismas comprobaciones que en el apartado precedente." }}}{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 37 "with(stats):with(linalg):with(plots):" }}{PARA 7 "" 
1 "" {TEXT -1 80 "Warning, the protected names norm and trace have bee
n redefined and unprotected\n" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning,
 the name changecoords has been redefined\n" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT 286 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 "listalogpaso:=[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 "lista2l
ogeval:=[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 "lista3logerro:=[0,0,0,0,0,0,0,0]:" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "lista4eval:=[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 "lista4lo
gerro:=[0,0,0,0,0,0,0,0]:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 321 21 "La \+
ecuaci\363n (no aut.)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "li
stanumpasos:=[16,32,64,128,256,512,1024,2048]:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 59 "for i from 1 to 8 do listapaso[i]:=1./listanum
pasos[i]; od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "for i from
 1 to 8 do listalogpaso[i]:=evalf(log[10](listapaso[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 322 63 "RK2 -> Runge-Kutt
a 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 8 do lista2e
val[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](lista2eva
l[i])); od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 246 "Heun2:=proc
(f::procedure,ytotalini::list,nit::posint,h::numeric)\nlocal x0,y0,k,k
1,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 8 do y1:=Heun2(f,ytotalini,listanump
asos[i],evalf(listapaso[i])): lista2logerro[i]:=log[10](abs(y1-u(1.)))
: od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "evalf(lista2logerr
o,4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7*$!%+O!\"$$!%8UF&$!%?[F&$!%C
aF&$!%GgF&$!%ImF&$!%LsF&$!%NyF&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 323 
63 "RK3 -> Runge-Kutta 3 evaluaciones y orden 3 'Heun' (e. no aut.)" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "for i from 1 to 8 do lista
3eval[i]:=3*listanumpasos[i]; od:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 73 "for i from 1 to 8 do lista3logeval[i]:=evalf(log[10](
lista3eval[i])); od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 286 "He
un3:=proc(f::procedure,ytotalini::list,nit::posint,h::numeric)\nlocal \+
x0,y0,k,k1,k2,k3:\nx0:=ytotalini[1]:y0:=ytotalini[2]:\nfor k from 1 to
 nit do\nk1:=f(x0,y0):\nk2:=f(x0+(1./3.)*h,y0+(1./3.)*h*k1):\nk3:=f(x0
+(2./3.)*h,y0+(2./3.)*h*k2);\nx0:=x0+h:y0:=y0+h*((1./4.)*k1+(3./4.)*k3
):\nod:\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "for i fro
m 1 to 8 do y1:=Heun3(f,ytotalini,listanumpasos[i],evalf(listapaso[i])
): lista3logerro[i]:=log[10](abs(y1-u(1.))): od:" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 23 "evalf(lista3logerro,4);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7*$!%0a!\"$$!%>jF&$!%GsF&$!%L\")F&$!%Q!*F&$!%U**F&$!%&3
\"!\"#$!%v6F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 287 61 "RK4 -> Runge-Ku
tta 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 list
a4eval[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 0 332 "Rk
ut4:=proc(f::procedure,ytotalini::list,nit::posint,h::numeric)\nlocal \+
x0,y0,k,k1,k2,k3,k4:\nx0:=ytotalini[1]:y0:=ytotalini[2]:\nfor k from 1
 to nit do\nk1:=f(x0,y0):\nk2:=f(x0+(1./2.)*h,y0+(1./2.)*h*k1):\nk3:=f
(x0+(1./2.)*h,y0+(1./2.)*h*k2);\nk4:=f(x0+h,y0+h*k3);\nx0:=x0+h:y0:=y0
+h*((1./6.)*k1+(1./3.)*k2+(1./3.)*k3+(1./6.)*k4):\nod:\nend:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "for i from 1 to 8 do y1:=Rk
ut4(f,ytotalini,listanumpasos[i],evalf(listapaso[i])): lista4logerro[i
]:=log[10](abs(y1-u(1.))): od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 23 "evalf(lista4logerro,4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7*$
!%2t!\"$$!%B&)F&$!%L(*F&$!%%4\"!\"#$!%:7F-$!%N8F-$!%b9F-$!%w:F-" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 256 54 "a) Gr\341ficas comparadas 'paso v
ersus log[10] del error'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 
"listagraa2:=[seq([listapaso[i],lista2logerro[i]],i=1..8)]:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "graa21:=plot(listagraa2,paso
,log10_error=0..-16,style=LINE):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 48 "graa22:=plot(listagraa2,style=POINT,symbol=BOX):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "graa23:=textplot([0.05, -3.1
7,`RK2`]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "listagraa3:=[
seq([listapaso[i],lista3logerro[i]],i=1..8)]:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 36 "graa31:=plot(listagraa3,style=LINE):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "graa32:=plot(listagraa3,style=POINT
,symbol=CIRCLE):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "graa33:
=textplot([0.05, -5.11,`RK3`]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 58 "listagraa4:=[seq([listapaso[i],lista4logerro[i]],i=1..8)]:" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "graa41:=plot(listagraa4,sty
le=LINE):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "graa42:=plot(l
istagraa4,style=POINT,symbol=CROSS):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 38 "graa43:=textplot([0.05, -7.01,`RK4`]):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "display(graa21,graa22,graa23,graa31
,graa32,graa33,graa41,graa42,graa43);" }}{PARA 13 "" 1 "" {GLPLOT2D 
496 496 496 {PLOTDATA 2 "6--%'CURVESG6%7*7$$\"5++++++++]i!#@$!50^,.ALv
:+O!#>7$$\"5++++++++DJF*$!5VDG&R@#R_7UF-7$$\"5+++++++]i:F*$!5C3)3*>0bq
>[F-7$$\"5+++++++]7y!#A$!5su)G&*fF=VU&F-7$$\"5+++++++D1RF;$!5R%Qmy)REl
FgF-7$$\"5++++++]7`>F;$!54[85j6$\\.j'F-7$$\"5++++++]il(*!#B$!5l[;3_;vs
KsF-7$$\"5++++++D\"G)[FK$!5_`)=AHeY\\$yF--%'COLOURG6&%$RGBG$\"#5!\"\"$
\"\"!FenFZ-%&STYLEG6#%%LINEG-F$6&F&FS-Fgn6#%&POINTG-%'SYMBOLG6#%$BOXG-
%%TEXTG6$7$$\"\"&!\"#$!$<$FioQ$RK26\"-F$6%7*7$F($!5ib-]z&eI^S&F-7$F/$!
5;OYP+EH4>jF-7$F4$!5aW*e*['e:wA(F-7$F9$!5G(H4sI%4UL\")F-7$F?$!53j_*H9O
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*7$F($!5;()ecNb$>tI(F-7$F/$!5s!>v$>(ohF_)F-7$F4$!5A+\\k<'pRDt*F-7$F9$!
5hcv\"o.z[R4\"Ffq7$F?$!5`(y9*43A]97Ffq7$FD$!5!HEk];G&)\\L\"Ffq7$FI$!5^
4Bj+$pUaX\"Ffq7$FO$!5<b7d=EcCw:FfqFSFfn-F$6&FgrFSF\\o-F`o6#%&CROSSG-Fd
o6$7$Fgo$!$,(FioQ$RK4F]p-%+AXESLABELSG6%Q%pasoF]pQ,log10_errorF]p-%%FO
NTG6#%(DEFAULTG-%%VIEWG6$Fcu;$!#;FenFZ" 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 261 72 "b) Gr\341ficas comparadas 'n\372mero de evalua
ciones versus log[10] del error'" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 59 "listagrab2:=[seq([lista2eval[i],lista2logerro[i]],i=1
..8)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "grab21:=plot(list
agrab2,num_eval,log10_error=0..-16,style=LINE):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 48 "grab22:=plot(listagrab2,style=POINT,symbol=BOX
):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "grab23:=textplot([400
0, -7.11,`RK2`]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "listag
rab3:=[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(listagrab
3,style=POINT,symbol=CIRCLE):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 39 "grab33:=textplot([4000, -10.65,`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
(listagrab4,style=LINE):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 
"grab42:=plot(listagrab4,style=POINT,symbol=CROSS):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 39 "grab43:=textplot([4000, -13.72,`RK4`]):" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "display(grab21,grab22,gra
b23,grab31,grab32,grab33,grab41,grab42,grab43);" }}{PARA 13 "" 1 "" 
{GLPLOT2D 496 496 496 {PLOTDATA 2 "6--%'CURVESG6%7*7$$\"#K\"\"!$!50^,.
ALv:+O!#>7$$\"#kF*$!5VDG&R@#R_7UF-7$$\"$G\"F*$!5C3)3*>0bq>[F-7$$\"$c#F
*$!5su)G&*fF=VU&F-7$$\"$7&F*$!5R%Qmy)RElFgF-7$$\"%C5F*$!54[85j6$\\.j'F
-7$$\"%[?F*$!5l[;3_;vsKsF-7$$\"%'4%F*$!5_`)=AHeY\\$yF--%'COLOURG6&%$RG
BG$\"#5!\"\"$F*F*FX-%&STYLEG6#%%LINEG-F$6&F&FQ-FZ6#%&POINTG-%'SYMBOLG6
#%$BOXG-%%TEXTG6$7$$\"%+SF*$!$6(!\"#Q$RK26\"-F$6%7*7$$\"#[F*$!5ib-]z&e
I^S&F-7$$\"#'*F*$!5;OYP+EH4>jF-7$$\"$#>F*$!5aW*e*['e:wA(F-7$$\"$%QF*$!
5G(H4sI%4UL\")F-7$$\"$o(F*$!53j_*H9Ooy.*F-7$$\"%O:F*$!5n%**[]()*pjT**F
-7$$\"%sIF*$!5WB?!=yi1X3\"!#=7$$\"%WhF*$!5`+qC*RgK[<\"FarFQFY-F$6&F]pF
QFin-F]o6#%'CIRCLEG-Fao6$7$Fdo$!%l5FhoQ$RK3Fjo-F$6%7*7$F/$!5;()ecNb$>t
I(F-7$F4$!5s!>v$>(ohF_)F-7$F9$!5A+\\k<'pRDt*F-7$F>$!5hcv\"o.z[R4\"Far7
$FC$!5`(y9*43A]97Far7$FH$!5!HEk];G&)\\L\"Far7$FM$!5^4Bj+$pUaX\"Far7$$
\"%#>)F*$!5<b7d=EcCw:FarFQFY-F$6&FdsFQFin-F]o6#%&CROSSG-Fao6$7$Fdo$!%s
8FhoQ$RK4Fjo-%+AXESLABELSG6%Q)num_evalFjoQ,log10_errorFjo-%%FONTG6#%(D
EFAULTG-%%VIEWG6$Fbv;$!#;F*FX" 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 262 66 "c) Gr\341ficas comparadas 'log[10] del paso versus log
[10] del error'" }}}{EXCHG {PARA 4 "" 0 "" {TEXT 317 92 "Hay un tipo d
e gr\341ficas que sirve para comprobar el orden 'efectivo' de los m
\351todos, o sea, " }}{PARA 4 "" 0 "" {TEXT 291 90 "el orden con que, \+
en la pr\341ctica, parecen comportarse. En estas gr\341ficas se debe p
resentar" }}{PARA 4 "" 0 "" {TEXT 292 94 "en el eje de abscisas el log
[10] del paso empleado y en el de ordenadas el log[10] del error. " }}
{PARA 4 "" 0 "" {TEXT 293 90 "Entonces el aspecto de la gr\341fica se \+
aproxima a una recta (o se puede aproximar utilizando" }}{PARA 4 "" 0 
"" {TEXT 294 91 "la regresi\363n lineal). La pendiente de esa recta es
 el orden con que se comporta el m\351todo. " }}{PARA 4 "" 0 "" {TEXT 
295 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 296 37 ") . Considera
ndo que, aproximadamente" }}{PARA 4 "" 0 "" {TEXT 297 20 "se tiene   E
(h) = K " }{XPPEDIT 18 0 "h^p" "6#)%\"hG%\"pG" }{TEXT 298 78 "  , resu
lta que el cociente incremental (la pendiente aproximada de la recta) \+
" }}{PARA 4 "" 0 "" {TEXT 299 43 "entre dos puntos correspondientes a \+
pasos  " }{XPPEDIT 18 0 "h[1]" "6#&%\"hG6#\"\"\"" }{TEXT 300 3 " < " }
{XPPEDIT 18 0 "h[2]" "6#&%\"hG6#\"\"#" }{TEXT 301 5 "  es " }}{PARA 4 
"" 0 "" {TEXT 302 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 303 89 "Esta pendiente se puede estudiar \+
cuando se utilizan los segmentos correspondientes a los " }}{PARA 4 "
" 0 "" {TEXT 307 69 "menores valores de  h , los mas representativos d
el efecto del orden." }}{PARA 4 "" 0 "" {TEXT 304 101 "Algo semejante \+
ocurre cuando  se presenta en el eje de abscisas el log[10] del numero
 de evaluaciones" }}{PARA 4 "" 0 "" {TEXT 308 97 "y en el de ordenadas
 el log[10] del error. Entonces tambi\351n el aspecto de la gr\341fica
 se aproxima " }}{PARA 4 "" 0 "" {TEXT 318 98 "a una recta (o se puede
 aproximar utilizando la regresi\363n lineal). Pero la pendiente de es
a recta " }}{PARA 4 "" 0 "" {TEXT 319 91 "es ahora igual a menos el or
den con que se comporta el m\351todo, debido a que en el cociente " }}
{PARA 4 "" 0 "" {TEXT 320 39 "antes expuesto el denominador es ahora \+
" }}{PARA 4 "" 0 "" {TEXT 311 5 "     " }{XPPEDIT 18 0 "log[10] (eval(
h[2]))-log[10] (eval(h[1]))" "6#,&-&%$logG6#\"#56#-%%evalG6#&%\"hG6#\"
\"#\"\"\"-&F&6#F(6#-F+6#&F.6#F1!\"\"" }{TEXT 309 3 " = " }{XPPEDIT 18 
0 "log[10] (m/h[2])-log[10] (m/h[1])" "6#,&-&%$logG6#\"#56#*&%\"mG\"\"
\"&%\"hG6#\"\"#!\"\"F,-&F&6#F(6#*&F+F,&F.6#F,F1F1" }{TEXT 310 4 "  = \+
" }}{PARA 4 "" 0 "" {TEXT 312 12 "         =  " }{XPPEDIT 18 0 "log[10
](m)-log[10] (h[2])-log[10](m)+log[10] (h[1])" "6#,*-&%$logG6#\"#56#%
\"mG\"\"\"-&F&6#F(6#&%\"hG6#\"\"#!\"\"-&F&6#F(6#F*F4-&F&6#F(6#&F16#F+F
+" }{TEXT 313 4 "  = " }{XPPEDIT 18 0 "log[10] (h[1])-log[10] (h[2])" 
"6#,&-&%$logG6#\"#56#&%\"hG6#\"\"\"F--&F&6#F(6#&F+6#\"\"#!\"\"" }
{TEXT 314 21 "                     " }}{PARA 4 "" 0 "" {TEXT 305 87 "e
n el supuesto de que  m  es el n\372mero de evaluaciones por paso del \+
m\351todo en cuesti\363n." }}{PARA 4 "" 0 "" {TEXT 306 101 "Esta idea \+
se emplea especialmente cuando el m\351todo es de paso variable y se d
esea averiguar su orden " }}{PARA 4 "" 0 "" {TEXT 315 104 "efectivo, y
a que, entonces, no tiene sentido hablar del paso del m\351todo, pero \+
s\355 que lo tiene hablar de " }}{PARA 4 "" 0 "" {TEXT 316 42 "las eva
luaciones realizadas por el mismo. " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 61 "listagrac2:=[seq([listalogpaso[i],lista2logerro[i]],i
=1..8)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "grac21:=plot(li
stagrac2,log10_paso,log10_error=0..-16,style=LINE):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 48 "grac22:=plot(listagrac2,style=POINT,symbo
l=BOX):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "grac23:=textplot
([-1.57, -3.44,`RK2`]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "
listagrac3:=[seq([listalogpaso[i],lista3logerro[i]],i=1..8)]:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "grac31:=plot(listagrac3,styl
e=LINE):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "grac32:=plot(li
stagrac3,style=POINT,symbol=CIRCLE):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 39 "grac33:=textplot([-1.57, -5.63,`RK3`]):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "listagrac4:=[seq([listalogpaso[i],l
ista4logerro[i]],i=1..8)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
36 "grac41:=plot(listagrac4,style=LINE):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 50 "grac42:=plot(listagrac4,style=POINT,symbol=CROSS):" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "grac43:=textplot([-1.57, -
7.83,`RK4`]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "display(gr
ac21,grac22,grac23,grac31,grac32,grac33,grac41,grac42,grac43);" }}
{PARA 13 "" 1 "" {GLPLOT2D 496 496 496 {PLOTDATA 2 "6--%'CURVESG6%7*7$
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F*7$$!58<()Q)R(*zh!=F*$!5C3)3*>0bq>[F*7$$!5lOoykp*4s5#F*$!5su)G&*fF=VU
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F$6&FdrFPFin-F]o6#%&CROSSG-Fao6$7$Fdo$!$$yFfoQ$RK4Fjo-%+AXESLABELSG6%Q
+log10_pasoFjoQ,log10_errorFjo-%%FONTG6#%(DEFAULTG-%%VIEWG6$F`u;$!#;FX
FW" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" 
"Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" 
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'CURVESG6%7*7$$!54yCfl#)*>T?\"!#>$!50^,.ALv:+OF*7$$!5h(f!*>$y*\\^]\"F*
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*$!54[85j6$\\.j'F*7$$!5@&>\")Rm&**H5IF*$!5l[;3_;vsKsF*7$$!5u9$z.B&*H8J
$F*$!5_`)=AHeY\\$yF*-%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!FXFW-%&STYLEG6#%
%LINEG-F$6&F&FP-FZ6#%&POINTG-%'SYMBOLG6#%$BOXG-%%TEXTG6$7$$!$d\"!\"#$!
$W$FfoQ$RK26\"-F$6%7*7$F($!5ib-]z&eI^S&F*7$F.$!5;OYP+EH4>jF*7$F3$!5aW*
e*['e:wA(F*7$F8$!5G(H4sI%4UL\")F*7$F=$!53j_*H9Ooy.*F*7$FB$!5n%**[]()*p
jT**F*7$FG$!5WB?!=yi1X3\"!#=7$FL$!5`+qC*RgK[<\"FcqFPFY-F$6&F]pFPFin-F]
o6#%'CIRCLEG-Fao6$7$Fdo$!$j&FfoQ$RK3Fjo-F$6%7*7$F($!5;()ecNb$>tI(F*7$F
.$!5s!>v$>(ohF_)F*7$F3$!5A+\\k<'pRDt*F*7$F8$!5hcv\"o.z[R4\"Fcq7$F=$!5`
(y9*43A]97Fcq7$FB$!5!HEk];G&)\\L\"Fcq7$FG$!5^4Bj+$pUaX\"Fcq7$FL$!5<b7d
=EcCw:FcqFPFY-F$6&FdrFPFin-F]o6#%&CROSSG-Fao6$7$Fdo$!$$yFfoQ$RK4Fjo-%+
AXESLABELSG6%Q+log10_pasoFjoQ,log10_errorFjo-%%FONTG6#%(DEFAULTG-%(SCA
LINGG6#%,CONSTRAINEDG-%%VIEWG6$F`u;$!#;FXFW" 1 2 0 1 10 0 2 9 1 4 1 
1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+
4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" }}}}{EXCHG {PARA 
0 "" 0 "" {TEXT 264 19 "Rectas de regresi\363n" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 54 "fit[leastsquare[[x,y]]]([listalogpaso,lista2lo
gerro]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"yG,&$!5SDDjX7j)))=\"!#
>\"\"\"*&$\"5X&G[L%)*p<3?F(F)%\"xGF)F)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 54 "fit[leastsquare[[x,y]]]([listalogpaso,lista3logerro])
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"yG,&$!5.()yaE$[C&*y\"!#>\"\"
\"*&$\"5hck=aj\"o'3IF(F)%\"xGF)F)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 54 "fit[leastsquare[[x,y]]]([listalogpaso,lista4logerro])
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"yG,&$!5)=c/d)R\\]&[#!#>\"\"\"
*&$\"5,Txmx)H6,,%F(F)%\"xGF)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 265 
29 "Pendiente del \372ltimo segmento" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 70 "(lista2logerro[8]-lista2logerro[7])/(listalogpaso[8]-
listalogpaso[7]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5G\"ebbfVG0+#!
#>" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "(lista3logerro[8]-lis
ta3logerro[7])/(listalogpaso[8]-listalogpaso[7]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"5:47._zRc+I!#>" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 70 "(lista4logerro[8]-lista4logerro[7])/(listalogpaso[8]-
listalogpaso[7]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5&>b9Mnd')H,%!
#>" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 263 84 "d) Gr\341ficas comparadas \+
'log[10] del n\372mero de evaluaciones versus log[10] del error'" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "listagrad2:=[seq([lista2loge
val[i],lista2logerro[i]],i=1..8)]:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 66 "grad21:=plot(listagrad2,log10_eval,log10_error=0..-16
,style=LINE):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "grad22:=pl
ot(listagrad2,style=POINT,symbol=BOX):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 36 "grad23:=textplot([3., -5.32,`RK2`]):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "listagrad3:=[seq([lista3logeval[i],
lista3logerro[i]],i=1..8)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
36 "grad31:=plot(listagrad3,style=LINE):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 51 "grad32:=plot(listagrad3,style=POINT,symbol=CIRCLE):" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "grad33:=textplot([3., -8.
27,`RK3`]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "listagrad4:=
[seq([lista4logeval[i],lista4logerro[i]],i=1..8)]:" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 36 "grad41:=plot(listagrad4,style=LINE):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "grad42:=plot(listagrad4,styl
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