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considera el par encaja do RK4(5) de FEHLBERG dado por " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 4 "" 0 "" {TEXT 256 14 " " }{XPPEDIT 18 0 "0" "6#\" \"!" }{TEXT 339 5 " | " }}{PARA 4 "" 0 "" {TEXT 257 13 " \+ " }{XPPEDIT 18 0 "1/4" "6#*&\"\"\"F$\"\"%!\"\"" }{TEXT 260 14 " | \+ " }{XPPEDIT 18 0 "1/4" "6#*&\"\"\"F$\"\"%!\"\"" }{TEXT 261 2 " " }}{PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 283 12 " " } {XPPEDIT 18 0 "3/8" "6#*&\"\"$\"\"\"\"\")!\"\"" }{TEXT 284 13 " | \+ " }{XPPEDIT 18 0 "3/32" "6#*&\"\"$\"\"\"\"#K!\"\"" }{TEXT 285 2 " " }{TEXT -1 1 " " }{TEXT 302 8 " " }{XPPEDIT 18 0 "9/32" "6# *&\"\"*\"\"\"\"#K!\"\"" }{TEXT 303 2 " " }}{PARA 4 "" 0 "" {TEXT 286 12 " " }{XPPEDIT 18 0 "12/13" "6#*&\"#7\"\"\"\"#8!\"\"" } {TEXT 287 9 " | " }{XPPEDIT 18 0 "1932/2197" "6#*&\"%K>\"\"\"\"% (>#!\"\"" }{TEXT 288 2 " " }{TEXT -1 1 " " }{TEXT 294 2 " " } {XPPEDIT 18 0 "-7200/2197" "6#,$*&\"%+s\"\"\"\"%(>#!\"\"F(" }{TEXT 295 10 " " }{XPPEDIT 18 0 "7296/2197" "6#*&\"%'H(\"\"\"\"%(># !\"\"" }{TEXT 304 2 " " }}{PARA 4 "" 0 "" {TEXT 289 15 " \+ " }{XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT 334 10 " | " } {XPPEDIT 18 0 "439/216" "6#*&\"$R%\"\"\"\"$;#!\"\"" }{TEXT 290 2 " " }{TEXT -1 1 " " }{TEXT 297 9 " " }{XPPEDIT 18 0 "-8" "6#,$\"\" )!\"\"" }{TEXT 335 11 " " }{XPPEDIT 18 0 "3680/513" "6#*&\"% !o$\"\"\"\"$8&!\"\"" }{TEXT 296 8 " " }{XPPEDIT 18 0 "-845/4104 " "6#,$*&\"$X)\"\"\"\"%/T!\"\"F(" }{TEXT 305 2 " " }}{PARA 4 "" 0 "" {TEXT 291 14 " " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#! \"\"" }{TEXT 292 9 " | " }{XPPEDIT 18 0 "-8/27" "6#,$*&\"\")\"\" \"\"#F!\"\"F(" }{TEXT 293 2 " " }{TEXT -1 1 " " }{TEXT 299 11 " \+ " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 300 7 " " }{XPPEDIT 18 0 "-3544/2565" "6#,$*&\"%WN\"\"\"\"%lD!\"\"F(" }{TEXT 301 11 " \+ " }{XPPEDIT 18 0 "1859/4104" "6#*&\"%f=\"\"\"\"%/T!\"\"" }{TEXT 298 5 " " }{XPPEDIT 18 0 "-11/40" "6#,$*&\"#6\"\"\"\"#S!\"\"F(" } {TEXT 306 2 " " }}{PARA 4 "" 0 "" {TEXT -1 47 " ------------------- -------------------------" }{TEXT 258 42 "---------------------------- --------------" }}{PARA 4 "" 0 "" {TEXT 259 22 " orden 4 | \+ " }{XPPEDIT 18 0 "25/216" "6#*&\"#D\"\"\"\"$;#!\"\"" }{TEXT 307 2 " \+ " }{TEXT -1 1 " " }{TEXT 309 12 " " }{XPPEDIT 18 0 "0" "6# \"\"!" }{TEXT 310 11 " " }{XPPEDIT 18 0 "1408/2565" "6#*&\"% 39\"\"\"\"%lD!\"\"" }{TEXT 311 10 " " }{XPPEDIT 18 0 "2197/41 04" "6#*&\"%(>#\"\"\"\"%/T!\"\"" }{TEXT 308 7 " " }{XPPEDIT 18 0 "-1/5" "6#,$*&\"\"\"F%\"\"&!\"\"F'" }{TEXT 312 2 " " }{TEXT -1 6 " \+ " }{TEXT 336 1 " " }{XPPEDIT 18 0 "0" "6#\"\"!" }}{PARA 4 "" 0 " " {TEXT 313 1 " " }{TEXT -1 46 " ------------------------------------ --------" }{TEXT 314 43 "-------------------------------------------" }}{PARA 4 "" 0 "" {TEXT 315 23 " orden 5 | " }{XPPEDIT 18 0 "16/135" "6#*&\"#;\"\"\"\"$N\"!\"\"" }{TEXT 316 2 " " }{TEXT -1 1 " " }{TEXT 318 11 " " }{XPPEDIT 18 0 "0" "6#\"\"!" }{TEXT 319 9 " " }{XPPEDIT 18 0 "6656/12825" "6#*&\"%cm\"\"\"\"&DG\"! \"\"" }{TEXT 320 9 " " }{XPPEDIT 18 0 "28561/56430" "6#*&\"&h& G\"\"\"\"&Ik&!\"\"" }{TEXT 317 5 " " }{XPPEDIT 18 0 "-9/50" "6#,$* &\"\"*\"\"\"\"#]!\"\"F(" }{TEXT 321 2 " " }{TEXT -1 4 " " }{TEXT 337 1 " " }{XPPEDIT 18 0 "2/55" "6#*&\"\"#\"\"\"\"#b!\"\"" }}{PARA 0 " " 0 "" {TEXT 322 1 " " }}{PARA 4 "" 0 "" {TEXT 323 1 " " }{TEXT -1 78 " ------------------------------------------------------------------- ---------" }{TEXT 324 11 "-----------" }}{PARA 4 "" 0 "" {TEXT 325 26 " est | " }{XPPEDIT 18 0 "1/360" "6#*&\"\"\"F$\"$g$ !\"\"" }{TEXT 326 2 " " }{TEXT -1 1 " " }{TEXT 328 12 " " }{XPPEDIT 18 0 "0" "6#\"\"!" }{TEXT 329 6 " " }{XPPEDIT 18 0 "-12 8/4275" "6#,$*&\"$G\"\"\"\"\"%vU!\"\"F(" }{TEXT 330 6 " " } {XPPEDIT 18 0 "-2197/75240" "6#,$*&\"%(>#\"\"\"\"&S_(!\"\"F(" }{TEXT 327 9 " " }{XPPEDIT 18 0 "1/50" "6#*&\"\"\"F$\"#]!\"\"" } {TEXT 331 2 " " }{TEXT -1 4 " " }{TEXT 338 1 " " }{XPPEDIT 18 0 "2 /55" "6#*&\"\"#\"\"\"\"#b!\"\"" }}{PARA 0 "" 0 "" {TEXT 332 1 " " }} {PARA 4 "" 0 "" {TEXT 333 2 " " }}{PARA 4 "" 0 "" {TEXT -1 73 "a) Se \+ comprobar\341 mediante las gr\341ficas de eficiencia correspondientes \+ que" }}{PARA 4 "" 0 "" {TEXT -1 79 "se trata efectivamente de m\351tod os de orden 4 y 5 . Para ello se utilizar\341 en" }}{PARA 4 "" 0 " " {TEXT -1 20 "[0,0.8] el problema" }}{PARA 4 "" 0 "" {TEXT -1 9 " \+ y' = " }{XPPEDIT 18 0 "y^2" "6#*$%\"yG\"\"#" }{TEXT -1 14 " , y(0) = 1 ," }}{PARA 4 "" 0 "" {TEXT 265 18 "de soluci\363n exacta" }}{PARA 4 "" 0 "" {TEXT 266 11 " y(x) = " }{XPPEDIT 18 0 "1/(1-x)" "6#*&\" \"\"F$,&F$F$%\"xG!\"\"F'" }{TEXT 267 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 268 62 "b) Tambi\351n se emplear\341 e l par para integrar el mismo problema" }}{PARA 4 "" 0 "" {TEXT 271 76 "con una tolerancia de E = 0.0001 en el intervalo [0,0.8] . Se \+ calcular\341n" }}{PARA 4 "" 0 "" {TEXT 272 67 "las evaluaciones realiz adas y (ya que se conoce la soluci\363n exacta)" }}{PARA 4 "" 0 "" {TEXT 273 67 "se comparar\341 el error E admitido con el verdadero e rror cometido." }}{PARA 4 "" 0 "" {TEXT 281 65 "Se estimar\341 adem \341s el orden efectivo que muestra el par encajado." }}}{EXCHG {PARA 4 "" 0 "" {TEXT 269 2 "a)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Digits:=30:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "interface(labeling=false): " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 274 62 "RK4(5) -> RUNGE-KUTTA-FEHLBE RG \363rdenes 4 y 5 cl\341sico (e. aut.)" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 231 "a[2,1]:=1/4:a[3,1]:=3/32:a[ 3,2]:=9/32:a[4,1]:=1932/2197:a[4,2]:=-7200/2197:a[4,3]:=7296/2197:a[5, 1]:=439/216:a[5,2]:=-8:a[5,3]:=3680/513:a[5,4]:=-845/4104:a[6,1]:=-8/2 7:a[6,2]:=2:a[6,3]:=-3544/2565:a[6,4]:=1859/4104:a[6,5]:=-11/40:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "b[1]:=25/216:b[3]:=1408/2565 :b[4]:=2197/4104:b[5]:=-1/5:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "bt[1]:=16/135:bt[3]:=6656/12825:bt[4]:=28561/56430:bt[5]:=-9/50: bt[6]:=2/55:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "d[1]:=1/360 :d[3]:=-128/4275:d[4]:=-2197/75240:d[5]:=1/50:d[6]:=2/55:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 330 "a[2,1]:=evalf(a[2,1]):a[3,1]:=eval f(a[3,1]):a[3,2]:=evalf(a[3,2]):a[4,1]:=evalf(a[4,1]):a[4,2]:=evalf(a[ 4,2]):a[4,3]:=evalf(a[4,3]):a[5,1]:=evalf(a[5,1]):a[5,2]:=evalf(a[5,2] ):a[5,3]:=evalf(a[5,3]):a[5,4]:=evalf(a[5,4]):a[6,1]:=evalf(a[6,1]):a[ 6,2]:=evalf(a[6,2]):a[6,3]:=evalf(a[6,3]):a[6,4]:=evalf(a[6,4]):a[6,5] :=evalf(a[6,5]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "b[1]:=e valf(b[1]):b[3]:=evalf(b[3]):b[4]:=evalf(b[4]):b[5]:=evalf(b[5]):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "bt[1]:=evalf(bt[1]):bt[3]:= evalf(bt[3]):bt[4]:=evalf(bt[4]):bt[5]:=evalf(bt[5]):bt[6]:=evalf(bt[6 ]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "d[1]:=evalf(d[1]):d[ 3]:=evalf(d[3]):d[4]:=evalf(d[4]):d[5]:=evalf(d[5]):d[6]:=evalf(d[6]): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 321 "RKF45etapas:=proc(f::p rocedure,y0::numeric,h::numeric)\nglobal a,k1,k2,k3,k4,k5,k6:\nk1:=f(y 0):\nk2:=f(y0+h*a[2,1]*k1):\nk3:=f(y0+h*(a[3,1]*k1+a[3,2]*k2)):\nk4:=f (y0+h*(a[4,1]*k1+a[4,2]*k2+a[4,3]*k3)):\nk5:=f(y0+h*(a[5,1]*k1+a[5,2]* k2+a[5,3]*k3+a[5,4]*k4)):\nk6:=f(y0+h*(a[6,1]*k1+a[6,2]*k2+a[6,3]*k3+a [6,4]*k4+a[6,5]*k5)):\nend:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 270 45 "L a ecuaci\363n (e. aut.) y la verdadera soluci\363n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "f:=y->y^2;yv:=x->1/(1-x);xini:=0;xend:=0.8; yini:=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"yG6\"6$%)oper atorG%&arrowGF(*$)9$\"\"#\"\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#yvGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&\"\"\"F-,&F-F-9$!\"\"F0F (F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%xiniG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%xendG$\"\")!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%yiniG\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 275 43 "Las gr \341ficas de eficiencia, para el orden 4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 199 "listanumpasos:=[1024,2048]:listapaso:=[(xend-xini)/e valf(listanumpasos[1]),0.8/evalf(listanumpasos[2])]:listalogpaso:=[eva lf(log[10](listapaso[1])),evalf(log[10](listapaso[2]))]:listalogerror: =[0,0]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 195 "x0:=xini:y0:=yi ni:h:=listapaso[1]:for k from 1 to listanumpasos[1] do RKF45etapas(f,y 0,h);y1:=y0+h*(b[1]*k1+b[3]*k3+b[4]*k4+b[5]*k5);x0:=x0+h;y0:=y1; od:li stalogerror[1]:=log[10](abs(yv(x0)-y0)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 195 "x0:=xini:y0:=yini:h:=listapaso[2]:for k from 1 to li stanumpasos[2] do RKF45etapas(f,y0,h);y1:=y0+h*(b[1]*k1+b[3]*k3+b[4]*k 4+b[5]*k5);x0:=x0+h;y0:=y1; od:listalogerror[2]:=log[10](abs(yv(x0)-y0 )):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "(listalogerror[2]-li stalogerror[1])/(listalogpaso[2]-listalogpaso[1]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"?)yoN8ZZKv6n4&*[*R!#H" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 276 43 "Las gr\341ficas de eficiencia, para el orden 5" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 199 "listanumpasos:=[1024,2048]: listapaso:=[(xend-xini)/evalf(listanumpasos[1]),0.8/evalf(listanumpaso s[2])]:listalogpaso:=[evalf(log[10](listapaso[1])),evalf(log[10](lista paso[2]))]:listalogerror:=[0,0]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 208 "x0:=xini:y0:=yini:h:=listapaso[1]:for k from 1 to li stanumpasos[1] do RKF45etapas(f,y0,h);y1:=y0+h*(bt[1]*k1+bt[3]*k3+bt[4 ]*k4+bt[5]*k5+bt[6]*k6);x0:=x0+h;y0:=y1; od:listalogerror[1]:=log[10]( abs(yv(x0)-y0)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 208 "x0:=xi ni:y0:=yini:h:=listapaso[2]:for k from 1 to listanumpasos[2] do RKF45e tapas(f,y0,h);y1:=y0+h*(bt[1]*k1+bt[3]*k3+bt[4]*k4+bt[5]*k5+bt[6]*k6); x0:=x0+h;y0:=y1; od:listalogerror[2]:=log[10](abs(yv(x0)-y0)):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "(listalogerror[2]-listaloger ror[1])/(listalogpaso[2]-listalogpaso[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?B')>@Vv^XsS " 0 "" {MPLTEXT 1 0 11 "Digits:=30:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 279 85 "Tolerancia 0.0001 y tolerancia unidad; \+ paso inicial y contador de evaluaciones a 0 ." }}{PARA 0 "" 0 "" {TEXT 343 32 "Luego se arranca el par encajado" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 537 "E:=0.0001;TU:=E/(xend-xini);cont:=0:conr:=0:x 0:=xini:y0:=yini:hmax:=0.1:h:=hmax:h:=min(h,hmax):\nwhile (x0=1. then x0:=x0+h;x0:=min(x0,xend); y0:=y1;h:=alpha*h;h1:=min(h,xend-x0);if h1%\"EG$\"\"\"!\"%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#TUG$\"?+++++++++++++]7!#L" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 179 "print(`el numero de evaluac iones para E=`,E,` es `,cont);print(`y ademas hay`,conr,`evaluaciones \+ fallidas`);print(`el error verdadero cometido para E=`,E,` es `,abs(yv (xend)-y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%Bel~numero~de~evaluac iones~para~E=G$\"\"\"!\"%%%~es~G\"#[" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6%%-y~ademas~hayG\"#O%6evaluaciones~fallidasG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%Del~error~verdadero~cometido~para~E=G$\"\"\"!\"%%%~es~ G$\";Uth3o%4wcAQJl&!#H" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 280 75 "Orden \+ efectivo del par encajado. Ahora hay que buscar otra tolerancia para \+ " }}{PARA 0 "" 0 "" {TEXT 340 75 "la que el n\372mero de evaluaciones \+ (que se almacena en cont) sea superior al " }}{PARA 0 "" 0 "" {TEXT 341 79 "precedente; sin ese requisito tendr\355amos la misma abscisa p ara los dos extremos" }}{PARA 0 "" 0 "" {TEXT 342 76 "del segmento, lo que invalidar\355a la gr\341fica. E=0.00001 cumple este requisito" } {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "listaloger ror:=[0,0]:listalogerror[1]:=log[10](abs(yv(xend)-y0)):listalogeval:=[ 0,0]:listalogeval[1]:=log[10](evalf(cont)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 622 "E:=0.00001:TU:=E/(xend-xini):cont:=0:conr:=0:x0:= xini:y0:=yini:hmax:=0.1:h:=hmax:h:=min(h,hmax):\nwhile (x0=1. then x0:=x0+h;x0:=min(x0,xend);y0: =y1;h:=alpha*h;h1:=min(h,xend-x0);if h1 " 0 "" {MPLTEXT 1 0 179 " print(`el numero de evaluaciones para E=`,E,` es `,cont);print(`y adem as hay`,conr,`evaluaciones fallidas`);print(`el error verdadero cometi do para E=`,E,` es `,abs(yv(xend)-y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%Bel~numero~de~evaluaciones~para~E=G$\"\"\"!\"&%%~es~G\"#%)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%%-y~ademas~hayG\"#s%6evaluaciones~fall idasG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%Del~error~verdadero~cometido ~para~E=G$\"\"\"!\"&%%~es~G$\":![KDWZ(=5y$fjk!#H" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "(listalogerror[2]-listalogerror[1])/(listalog eval[2]-listalogeval[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!?Q\"*y> 9ZZ@#QeQx^(Q!#H" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 282 84 "y el par se m uestra con un orden efectivo cercano a 4 ; no hemos empleado factores " }}{PARA 0 "" 0 "" {TEXT 344 78 "de correcci\363n del error estimado , y tampoco hemos puesto en discusi\363n el paso " }}{PARA 0 "" 0 "" {TEXT 345 56 "recomendado (que son cosas que en la pr\341ctica se hace n) " }}}}{MARK "0 3 0" 15 }{VIEWOPTS 1 1 0 3 4 1802 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }