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0 0 0 0 0 0 0 1 }{CSTYLE "" -1 305 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 306 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 307 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 308 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 309 "" 1 18 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 310 "" 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 "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Ti mes" 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 }} {SECT 0 {EXCHG {PARA 4 "" 0 "" {TEXT 268 14 "Ejercicios de " }}{PARA 4 "" 0 "" {TEXT 310 55 "(7) M\351todos de RUNGE-KUTTA impl\355citos y \+ semi-impl\355citos" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 286 18 "Ejercicio 07-07 " }{TEXT 309 39 " (del EXAMEN EXTRAORD INARIO de 04JUL06)" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 83 "a) B\372squense los m\351todos con el mayor orden posible para la familia de RUNGE-KUTTA " }}{PARA 4 "" 0 "" {TEXT -1 32 "impl \355citos del siguiente tablero" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 4 "" 0 "" {TEXT 256 21 " 0 | 0" }}{PARA 4 "" 0 " " {TEXT 257 5 " " }{XPPEDIT 18 0 "2*alpha;" "6#*&\"\"#\"\"\"%&alph aGF%" }{TEXT 264 10 " | " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT 272 9 " " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }}{PARA 4 "" 0 "" {TEXT -1 12 " ---------" }{TEXT 258 24 "------------------ ------" }}{PARA 4 "" 0 "" {TEXT 259 20 " | " } {XPPEDIT 18 0 "beta;" "6#%%betaG" }{TEXT 265 10 " " } {XPPEDIT 18 0 "beta;" "6#%%betaG" }{TEXT 266 1 " " }}{PARA 4 "" 0 "" {TEXT -1 8 " donde " }{XPPEDIT 18 0 "0 < alpha;" "6#2\"\"!%&alphaG" } {TEXT -1 6 " y " }{XPPEDIT 18 0 "0 < beta;" "6#2\"\"!%%betaG" } {TEXT -1 21 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 4 "" 0 "" {TEXT -1 40 "(desde aqu\355 es Ejercicio del cap\355tu lo 8)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 80 " b) Calc\372lese la funci\363n de estabilidad para los m\351todos de la familia propuesta." }}{PARA 4 "" 0 "" {TEXT -1 54 "Tambi\351n para lo s m\351todos del orden m\341ximo alcanzable " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 81 "c) Compru\351bese que dic hos m\351todos de orden m\341ximi son A-estables. B\372squese alg\372n " }}{PARA 4 "" 0 "" {TEXT -1 88 "otro m\351todo de la familia que tam bi\351n sea A-estable, describiendo en este caso la regi\363n" }} {PARA 4 "" 0 "" {TEXT -1 14 "de estabilidad" }}}{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 14 "y1:=f(x,y(x)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "y 2:=subs(diff(y(x),x)=y1,diff(y1,x)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "y3:=subs(diff(y(x),x)=y1,diff(y2,x)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Y1:=subs(\{y(x)=y0,x=x0\},y1):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Y2:=subs(\{y(x)=y0,x=x0\},y2 ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Y3:=subs(\{y(x)=y0,x= x0\},y3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "k1_:=unapply(f (x0,y0),h):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "k2_:=unapply (f(x0+2*alpha*h,y0+alpha*h*kp1_(h)+alpha*h*kp2_(h)),h):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "k1_0:=k1_(0):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "k2_0:=k2_(0):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "k11:=D(k1_):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "k21:=D(k2_):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "k110: =subs(\{kp1_(0)=k1_0,kp2_(0)=k2_0\},k11(0)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "k210:=subs(\{kp1_(0)=k1_0,kp2_(0)=k2_0\},k21(0)) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "k12:=D(k11):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "k22:=D(k21):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "k120:=subs(\{D(kp1_)(0)=k110,D(kp2_ )(0)=k210,kp1_(0)=k1_0,kp2_(0)=k2_0\},k12(0)):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 79 "k220:=subs(\{D(kp1_)(0)=k110,D(kp2_)(0)=k210,k p1_(0)=k1_0,kp2_(0)=k2_0\},k22(0)):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 263 2 "a)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 27 "condiciones para el orden 1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Fi_0:=expand(be ta*k1_0+beta*k2_0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Fi_0G,$*(\" \"#\"\"\"%%betaGF(-%\"fG6$%#x0G%#y0GF(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "cond1a:=coeff(Y1,f(x0,y0))=coeff(Fi_0,f(x0,y0));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond1aG/\"\"\",$*&\"\"#F&%%betaGF&F &" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "solve(cond1a,\{beta\}) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#/%%betaG#\"\"\"\"\"#" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 261 27 "condiciones para el orden 2" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Fi10:=expand(beta*k110+beta* k210);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Fi10G,&**\"\"#\"\"\"%%bet aGF(--&%\"DG6#F(6#%\"fG6$%#x0G%#y0GF(%&alphaGF(F(*,F'F(F)F(--&F-6#F'F/ F1F(F4F(-F0F1F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "cond2a :=(1/2)*coeff(Y2,D[1](f)(x0,y0))=coeff(Fi10,D[1](f)(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond2aG/#\"\"\"\"\"#,$*(F(F'%%betaGF'%&al phaGF'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "cond2b:=(1/2)*c oeff(coeff(Y2,D[2](f)(x0,y0)),f(x0,y0))=coeff(coeff(Fi10,D[2](f)(x0,y0 )),f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond2bG/#\"\"\"\" \"#,$*(F(F'%%betaGF'%&alphaGF'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "solucion:=solve(\{cond1a,cond2a,cond2b\},\{alpha,beta \});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)solucionG<$/%%betaG#\"\"\" \"\"#/%&alphaGF(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 262 28 "condiciones \+ para el orden 3 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Fi20:=e xpand(beta*k120+beta*k220);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Fi20 G,,**\"\"%\"\"\"%%betaGF(--&%\"DG6$F(F(6#%\"fG6$%#x0G%#y0GF()%&alphaG \"\"#F(F(*,\"\")F(F)F(--&F-6$F(F6F/F1F(F4F(-F0F1F(F(*,F'F(F)F(F4F()F=F 6F(--&F-6$F6F6F/F1F(F(*,F'F(F)F(--&F-6#F6F/F1F(F4F(--&F-6#F(F/F1F(F(*, F'F(F)F()FEF6F(F4F(F=F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "cond3a:=(1/6)*coeff(Y3,D[1,1](f)(x0,y0))=(1/2)*coeff(Fi20,D[1,1](f )(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3aG/#\"\"\"\"\"', $*(\"\"#F'%%betaGF')%&alphaGF+F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "cond3b:=(1/6)*coeff(coeff(Y3,D[1,2](f)(x0,y0)),f(x0, y0))=(1/2)*coeff(coeff(Fi20,D[1,2](f)(x0,y0)),f(x0,y0));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%'cond3bG/#\"\"\"\"\"$,$*(\"\"%F'%%betaGF')%&al phaG\"\"#F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "cond3c:=( 1/6)*coeff(coeff(Y3,D[2,2](f)(x0,y0)),f(x0,y0)^2)=(1/2)*coeff(coeff(Fi 20,D[2,2](f)(x0,y0)),f(x0,y0)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %'cond3cG/#\"\"\"\"\"',$*(\"\"#F'%%betaGF')%&alphaGF+F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "cond3d:=(1/6)*coeff(coeff(Y3,D[2]( f)(x0,y0)),D[1](f)(x0,y0))=(1/2)*coeff(coeff(Fi20,D[2](f)(x0,y0)),D[1] (f)(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3dG/#\"\"\"\"\" ',$*(\"\"#F'%%betaGF')%&alphaGF+F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "cond3e:=(1/6)*coeff(coeff(Y3,D[2](f)(x0,y0)^2),f(x0, y0))=(1/2)*coeff(coeff(Fi20,D[2](f)(x0,y0)^2),f(x0,y0));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%'cond3eG/#\"\"\"\"\"',$*(\"\"#F'%%betaGF')%&al phaGF+F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "solve(\{cond1 a,cond2a,cond2b,cond3a,cond3b,cond3c,cond3d,cond3e\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 269 95 "Es decir, no se alcanza orden 3 . El or den m\341ximo alcanzable es 2 y se alcanza con el m\351todo" }} {PARA 0 "" 0 "" {TEXT 273 19 "dado por el tablero" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 274 18 " 0 | 0" }} {PARA 4 "" 0 "" {TEXT 275 16 " 1 | " }{XPPEDIT 18 0 "1/2;" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT 280 9 " " }{XPPEDIT 18 0 "1/2; " "6#*&\"\"\"F$\"\"#!\"\"" }}{PARA 4 "" 0 "" {TEXT -1 12 " --------- " }{TEXT 276 24 "------------------------" }}{PARA 4 "" 0 "" {TEXT 277 17 " | " }{XPPEDIT 18 0 "1/2;" "6#*&\"\"\"F$\"\"#!\" \"" }{TEXT 278 9 " " }{XPPEDIT 18 0 "1/2;" "6#*&\"\"\"F$\"\"#! \"\"" }{TEXT 279 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 270 2 "b)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 267 23 "funci\363n de estabilidad " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f:=(x,y)->lambda*y;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf *6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF)*&%'lambdaG\"\"\"9%F/F)F)F)" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "ka1:=unapply(f(x0,y0),h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ka1Gf*6#%\"hG6\"6$%)operatorG%&ar rowGF(*&%'lambdaG\"\"\"%#y0GF.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "ka2:=unapply(f(x0+2*alpha*h,y0+alpha*h*kp1(h)+alpha*h *kp2(h)),h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ka2Gf*6#%\"hG6\"6$% )operatorG%&arrowGF(*&%'lambdaG\"\"\",(%#y0GF.*(%&alphaGF.9$F.-%$kp1G6 #F3F.F.*(F2F.F3F.-%$kp2GF6F.F.F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "test:=solve(\{k1=subs(\{kp1(h)=k1,kp2(h)=k2\},ka1(h)) ,k2=subs(\{kp1(h)=k1,kp2(h)=k2\},ka2(h))\},\{k1,k2\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%testG<$/%#k1G*&%'lambdaG\"\"\"%#y0GF*/%#k2G,$** F)F*F+F*,&F*F**(F)F*%&alphaGF*%\"hGF*F*F*,&F*!\"\"F1F*F5F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "k1:=subs(test,k1);k2:=subs(test,k2) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k1G*&%'lambdaG\"\"\"%#y0GF'" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k2G,$**%'lambdaG\"\"\"%#y0GF(,&F(F (*(F'F(%&alphaGF(%\"hGF(F(F(,&F(!\"\"F+F(F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "y1:=expand(y0+h*(beta*k1+beta*k2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G,*%#y0G\"\"\"**%\"hGF'%%betaGF'%'lambdaGF 'F&F'F'*,F)F'F*F'F+F'F&F',&F'!\"\"*(F+F'%&alphaGF'F)F'F'F.F.*.F*F'F+\" \"#F&F'F-F.F0F'F)F2F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "y1 :=collect(subs(lambda=z/h,y1),y0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#y1G*&,*\"\"\"F'*&%%betaGF'%\"zGF'F'*(F)F'F*F',&F'!\"\"*&F*F'%&alpha GF'F'F-F-**F)F'F*\"\"#F,F-F/F'F-F'%#y0GF'" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 41 "r:=unapply(collect(simplify(y1/y0),z),z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rGf*6#%\"zG6\"6$%)operatorG%&arrowGF(*&, &\"\"\"!\"\"*&,&%&alphaGF.*&\"\"#F.%%betaGF.F/F.9$F.F.F.,&F.F/*&F2F.F6 F.F.F/F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 271 65 "que es la funci \363n de estabilidad de la familia general propuesta." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 293 8 "Cuando " }{XPPEDIT 18 0 "alpha;" "6#%&al phaG" }{TEXT 294 6 " y " }{XPPEDIT 18 0 "beta;" "6#%%betaG" }{TEXT 295 69 " valen 1/2 , o sea, en el \372nico caso de orden m\341ximo 2 , la funci\363n" }}{PARA 0 "" 0 "" {TEXT 296 20 "de estabil\361idad v ale" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "r1:=unapply(simplify (subs(solucion,r(z))),z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r1Gf*6 #%\"zG6\"6$%)operatorG%&arrowGF(,$*&,&\"\"#\"\"\"9$F0F0,&F/!\"\"F1F0F3 F3F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 288 2 "c)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 287 18 "La A-estabilidad " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 289 8 "Cuando " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT 290 6 " y " }{XPPEDIT 18 0 "beta;" "6#%%betaG" }{TEXT 291 69 " val en 1/2 , o sea, en el \372nico caso de orden m\341ximo 2 , la funci \363n" }}{PARA 0 "" 0 "" {TEXT 292 20 "de estabil\361idad vale" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "r1:=unapply(simplify(subs(so lucion,r(z))),z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r1Gf*6#%\"zG6 \"6$%)operatorG%&arrowGF(,$*&,&\"\"#\"\"\"9$F0F0,&F/!\"\"F1F0F3F3F(F(F (" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 281 50 "el dominio de estabilidad, \+ compuesto por los z = " }{XPPEDIT 18 0 "lambda" "6#%'lambdaG" }{TEXT 282 35 " h tales que el m\363dulo de r3(z) " }}{PARA 0 "" 0 "" {TEXT 283 88 "sea inferior a 1 , estar\341 formado por los valores de z que disten menos del punto -2" }}{PARA 0 "" 0 "" {TEXT 284 86 "q ue del punto 2 . Esto corresponde al semiplano de parte real negativa , por lo que el" }}{PARA 0 "" 0 "" {TEXT 285 30 "m\351todo estudiado e s A-estable" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 297 52 "Tomemos ahora cu alquiera de los casos para los que " }{XPPEDIT 18 0 "alpha = 2*beta; " "6#/%&alphaG*&\"\"#\"\"\"%%betaGF'" }{TEXT 298 25 " , lo que reduce \+ a 1 el" }}{PARA 0 "" 0 "" {TEXT 299 62 "numerador de la funci\363n d e estabilidad. Por ejemplo, hagamos " }{XPPEDIT 18 0 "alpha = 1;" "6# /%&alphaG\"\"\"" }{TEXT 300 6 " y " }{XPPEDIT 18 0 "beta = 1/2;" "6 #/%%betaG*&\"\"\"F&\"\"#!\"\"" }{TEXT 301 13 " . Para ese " }}{PARA 0 "" 0 "" {TEXT 302 36 "caso, la funci\363n de estabilidad vale" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 303 14 " \+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "solucion2:=\{alph a=1,beta=1/2\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*solucion2G<$/%%b etaG#\"\"\"\"\"#/%&alphaGF)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "r2:=unapply(simplify(subs(solucion2,r(z))),z);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#r2Gf*6#%\"zG6\"6$%)operatorG%&arrowGF(,$*&\"\"\"F. ,&F.!\"\"9$F.F0F0F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 304 61 "Ahora , el dominio de estabilidad est\341 compuesto por los z = " } {XPPEDIT 18 0 "lambda" "6#%'lambdaG" }{TEXT 305 22 " h cuya distanci a al" }}{PARA 0 "" 0 "" {TEXT 306 84 "punto 1 sea superior a 1, que es el exterior de la circunferencia de centro 1 y" }}{PARA 0 "" 0 " " {TEXT 307 85 "radio 1 (que pasa, pues, por el origen). Esta regi \363n contiene al semiplano de parte " }}{PARA 0 "" 0 "" {TEXT 308 59 "real negativa, por lo que el m\351todo estudiado es A-estable" }}}} {MARK "0 0 0" 12 }{VIEWOPTS 1 1 0 3 4 1802 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }