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" "6#*&\"\"\"F$\"\"#!\"\"" }}{PARA 4 "" 0 "" {TEXT -1 12 " --------- " }{TEXT 258 24 "------------------------" }}{PARA 4 "" 0 "" {TEXT 259 17 " | " }{XPPEDIT 18 0 "beta;" "6#%%betaG" }{TEXT 264 10 " " }{XPPEDIT 18 0 "beta;" "6#%%betaG" }{TEXT 265 1 " \+ " }}{PARA 4 "" 0 "" {TEXT -1 8 " donde " }{XPPEDIT 18 0 "alpha;" "6#% &alphaG" }{TEXT -1 6 " y " }{XPPEDIT 18 0 "beta;" "6#%%betaG" } {TEXT -1 47 " son par\341metros arbitrarios " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 40 "(desde aq u\355 es Ejercicio del cap\355tulo 8)" }}{PARA 4 "" 0 "" {TEXT -1 0 " " }}{PARA 4 "" 0 "" {TEXT -1 81 "b) Calc\372lese la funci\363n de esta bilidad para los m\351todos de orden m\341ximo hallados." }}{PARA 4 " " 0 "" {TEXT -1 49 "\277Son A-estables dichos m\351todos de orden m \341ximo?. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 71 "c) B\372squese alg\372n m\351todo de orden 1 de la familia qu e sea A-estable, " }}{PARA 4 "" 0 "" {TEXT -1 50 "describiendo en este caso la regi\363n 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 "y2:=s ubs(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 49 "k1_:=unapply(f (x0+alpha*h,y0+alpha*h*kp1_(h)),h):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "k2_:=unapply(f(x0+h,y0+(1/2)*h*kp1_(h)+(1/2)*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,k p2_(0)=k2_0\},k21(0)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "k 12:=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,kp1_(0)=k1_0,kp2_(0)=k2_0\},k22(0)):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 263 2 "a)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 27 "condicio nes para el orden 1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Fi_0 :=expand(beta*k1_0+beta*k2_0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%F i_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+bet a*k210);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Fi10G,**(%%betaG\"\"\"- -&%\"DG6#F(6#%\"fG6$%#x0G%#y0GF(%&alphaGF(F(**F'F(--&F,6#\"\"#F.F0F(F3 F(-F/F0F(F(*&F'F(F)F(F(*(F'F(F5F(F:F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "cond2a:=(1/2)*coeff(Y2,D[1](f)(x0,y0))=coeff(Fi10,D[1 ](f)(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond2aG/#\"\"\"\" \"#,&*&%%betaGF'%&alphaGF'F'F+F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "cond2b:=(1/2)*coeff(coeff(Y2,D[2](f)(x0,y0)),f(x0,y0) )=coeff(coeff(Fi10,D[2](f)(x0,y0)),f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond2bG/#\"\"\"\"\"#,&*&%%betaGF'%&alphaGF'F'F+F'" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "solucion:=solve(\{cond1a,c ond2a,cond2b\},\{alpha,beta\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%) solucionG<$/%%betaG#\"\"\"\"\"#/%&alphaG\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 262 28 "condiciones para el orden 3 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Fi20:=expand(beta*k120+beta*k220);" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#>%%Fi20G,:*(%%betaG\"\"\"--&%\"DG6$F(F(6#%\"fG6$ %#x0G%#y0GF()%&alphaG\"\"#F(F(*,F5F(F'F(--&F,6$F(F5F.F0F(F3F(-F/F0F(F( **F'F(F3F()F;F5F(--&F,6$F5F5F.F0F(F(*,F5F(F'F(--&F,6#F5F.F0F(F3F(--&F, 6#F(F.F0F(F(*,F5F(F'F()FCF5F(F3F(F;F(F(*&F'F(F)F(F(**F5F(F'F(F7F(F;F(F (*(F'F(F>F(F=F(F(**F'F(FCF(FGF(F4F(F(**F'F(FLF(F4F(F;F(F(*(F'F(FCF(FGF (F(*(F'F(FLF(F;F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "cond 3a:=(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'\" \"#F'*&%%betaGF')%&alphaGF,F'F'F'*&F+F'F.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/#\"\"\"\"\"$,&*&%%betaGF')% &alphaG\"\"#F'F'F+F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "co nd3c:=(1/6)*coeff(coeff(Y3,D[2,2](f)(x0,y0)),f(x0,y0)^2)=(1/2)*coeff(c oeff(Fi20,D[2,2](f)(x0,y0)),f(x0,y0)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3cG/#\"\"\"\"\"',&*&#F'\"\"#F'*&%%betaGF')%&alphaGF,F'F'F '*&F+F'F.F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "cond3d:=( 1/6)*coeff(coeff(Y3,D[2](f)(x0,y0)),D[1](f)(x0,y0))=(1/2)*coeff(coeff( Fi20,D[2](f)(x0,y0)),D[1](f)(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3dG/#\"\"\"\"\"',(*&%%betaGF')%&alphaG\"\"#F'F'*&#F'F.F'*&F+ F'F-F'F'F'*&F0F'F+F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 " cond3e:=(1/6)*coeff(coeff(Y3,D[2](f)(x0,y0)^2),f(x0,y0))=(1/2)*coeff(c oeff(Fi20,D[2](f)(x0,y0)^2),f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3eG/#\"\"\"\"\"',(*&%%betaGF')%&alphaG\"\"#F'F'*&#F'F.F'* &F+F'F-F'F'F'*&F0F'F+F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "subs(solucion,[cond1a,cond2a,cond2b,cond3a,cond3b,cond3c,cond3d,co nd3e]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7*/\"\"\"F%/#F%\"\"#F'F&/#F %\"\"'#F%\"\"%/#F%\"\"$F'F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "solve(\{cond1a,cond2a,cond2b,cond3a,cond3b,cond3c,cond3d,cond3 e\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 95 "Es decir, no se alcanza orden 3 . El orden m\341ximo alcanzable es 2 y se alcanza con el m \351todo" }}{PARA 0 "" 0 "" {TEXT 271 19 "dado por el tablero" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 272 18 " 0 \+ | 0" }}{PARA 4 "" 0 "" {TEXT 273 16 " 1 | " } {XPPEDIT 18 0 "1/2;" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT 278 9 " \+ " }{XPPEDIT 18 0 "1/2;" "6#*&\"\"\"F$\"\"#!\"\"" }}{PARA 4 "" 0 "" {TEXT -1 12 " ---------" }{TEXT 274 24 "------------------------" }} {PARA 4 "" 0 "" {TEXT 275 17 " | " }{XPPEDIT 18 0 "1/2; " "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT 276 9 " " }{XPPEDIT 18 0 "1/ 2;" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT 277 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 269 2 "b)" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 266 57 "funci\363n de estabilidad para el m\351todo de or den 2 m\341ximo" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 283 8 "Cuando " } {XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT 284 13 " vale 0 y " } {XPPEDIT 18 0 "beta;" "6#%%betaG" }{TEXT 285 68 " vale 1/2 , o sea, \+ en el \372nico caso de orden m\341ximo 2 , la funci\363n" }}{PARA 0 " " 0 "" {TEXT 282 19 "de estabilidad vale" }}}{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 63 "ka1:=unapply(subs( solucion,f(x0+alpha*h,y0+alpha*h*kp1(h))),h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ka1Gf*6#%\"hG6\"6$%)operatorG%&arrowGF(*&%'lambdaG\" \"\"%#y0GF.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "ka2:=u napply(subs(solucion,f(x0+h,y0+(1/2)*h*kp1(h)+(1/2)*h*kp2(h))),h);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ka2Gf*6#%\"hG6\"6$%)operatorG%&arro wGF(*&%'lambdaG\"\"\",(%#y0GF.*&#F.\"\"#F.*&9$F.-%$kp1G6#F5F.F.F.*&F2F .*&F5F.-%$kp2GF8F.F.F.F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "test:=solve(\{k1=subs(\{kp1(h)=k1,kp2(h)=k2\},ka1(h)),k2=subs( \{kp1(h)=k1,kp2(h)=k2\},ka2(h))\},\{k1,k2\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%testG<$/%#k1G*&%'lambdaG\"\"\"%#y0GF*/%#k2G,$**F)F*F +F*,&\"\"#F**&F)F*%\"hGF*F*F*,&F1!\"\"F2F*F5F5" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 36 "k1:=subs(test,k1);k2:=subs(test,k2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k1G*&%'lambdaG\"\"\"%#y0GF'" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#k2G,$**%'lambdaG\"\"\"%#y0GF(,&\"\"#F(*&F'F(% \"hGF(F(F(,&F+!\"\"F,F(F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "y1:=expand(subs(solucion,y0+h*(beta*k1+beta*k2)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G,*%#y0G\"\"\"**\"\"#!\"\"%\"hGF'%'lambdaGF'F &F'F'**F+F'F,F'F&F',&F)F**&F,F'F+F'F'F*F**,F)F*F,F)F&F'F.F*F+F)F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "y1:=collect(subs(lambda=z/h, y1),y0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G*&,*\"\"\"F'*&\"\"#! \"\"%\"zGF'F'*&F+F',&F)F*F+F'F*F**(F)F*F+F)F-F*F*F'%#y0GF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "r:=unapply(collect(simplify(y1/y0), z),z):rz:=simplify(r(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#rzG,$* &,&\"\"#\"\"\"%\"zGF)F),&F(!\"\"F*F)F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT 286 50 "el dominio de estabilidad, compuesto por los z = " } {XPPEDIT 18 0 "lambda" "6#%'lambdaG" }{TEXT 287 34 " h tales que el \+ m\363dulo de r(z) " }}{PARA 0 "" 0 "" {TEXT 288 88 "sea inferior a 1 , estar\341 formado por los valores de z que disten menos del punto -2" }}{PARA 0 "" 0 "" {TEXT 289 86 "que del punto 2 . Esto correspo nde al semiplano de parte real negativa, por lo que el" }}{PARA 0 "" 0 "" {TEXT 290 30 "m\351todo estudiado es A-estable" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 293 2 "c)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 292 62 "fun ci\363n de estabilidad para un m\351todo de orden exactamente 1" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 294 53 "Tomemos ahora cualquiera de los c asos para los que " }{XPPEDIT 18 0 "beta = 1/2;" "6#/%%betaG*&\"\"\" F&\"\"#!\"\"" }{TEXT 296 26 " , de manera que el orden" }}{PARA 0 "" 0 "" {TEXT 297 48 "sea 1 al menos. Vamos a ensayar con el valor " } {XPPEDIT 18 0 "alpha = 1;" "6#/%&alphaG\"\"\"" }{TEXT 295 27 " , para \+ el que no se tiene " }}{PARA 0 "" 0 "" {TEXT 298 16 "orden 2 (m\341xim o)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "solucion1:=\{alpha=1,beta=1/2\};" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*solucion1G<$/%&alphaG\"\"\"/%%beta G#F(\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f:=(x,y)->lamb da*y;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%\"xG%\"yG6\"6$%)op eratorG%&arrowGF)*&%'lambdaG\"\"\"9%F/F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "ka1:=unapply(subs(solucion1,f(x0+alpha*h,y0+alph a*h*kp1(h))),h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ka1Gf*6#%\"hG6 \"6$%)operatorG%&arrowGF(*&%'lambdaG\"\"\",&%#y0GF.*&9$F.-%$kp1G6#F2F. F.F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "ka2:=unapply( subs(solucion1,f(x0+h,y0+(1/2)*h*kp1(h)+(1/2)*h*kp2(h))),h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ka2Gf*6#%\"hG6\"6$%)operatorG%&arrowGF(*& %'lambdaG\"\"\",(%#y0GF.*&#F.\"\"#F.*&9$F.-%$kp1G6#F5F.F.F.*&F2F.*&F5F .-%$kp2GF8F.F.F.F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "test:=solve(\{k1=subs(\{kp1(h)=k1,kp2(h)=k2\},ka1(h)),k2=subs(\{kp1(h )=k1,kp2(h)=k2\},ka2(h))\},\{k1,k2\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%testG<$/%#k1G,$*(%'lambdaG\"\"\"%#y0GF+,&F+!\"\"*&F*F+%\"hGF+ F+F.F./%#k2GF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "k1:=subs( test,k1);k2:=subs(test,k2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k1G, $*(%'lambdaG\"\"\"%#y0GF(,&F(!\"\"*&F'F(%\"hGF(F(F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k2G,$*(%'lambdaG\"\"\"%#y0GF(,&F(!\"\"*&F'F(%\" hGF(F(F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "y1:=expand(su bs(solucion1,y0+h*(beta*k1+beta*k2)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G,&%#y0G\"\"\"**%\"hGF'%'lambdaGF'F&F',&F'!\"\"*&F*F'F)F'F' F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "y1:=collect(subs(la mbda=z/h,y1),y0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G*&,&\"\"\"F '*&%\"zGF',&F'!\"\"F)F'F+F+F'%#y0GF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "r:=unapply(collect(simplify(y1/y0),z),z):rz:=simplify (r(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#rzG,$*&\"\"\"F',&F'!\"\" %\"zGF'F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 299 61 "Ahora, el dominio de estabilidad est\341 compuesto por los z = " }{XPPEDIT 18 0 "lambd a" "6#%'lambdaG" }{TEXT 300 22 " h cuya distancia al" }}{PARA 0 "" 0 "" {TEXT 301 84 "punto 1 sea superior a 1, que es el exterior de \+ la circunferencia de centro 1 y" }}{PARA 0 "" 0 "" {TEXT 302 85 "rad io 1 (que pasa, pues, por el origen). Esta regi\363n contiene al semi plano de parte " }}{PARA 0 "" 0 "" {TEXT 303 59 "real negativa, por lo que el m\351todo estudiado es A-estable" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 304 23 "Otra opcion: tomemos " }{XPPEDIT 18 0 "beta = 1/2;" "6 #/%%betaG*&\"\"\"F&\"\"#!\"\"" }{TEXT 306 8 " , y " }{XPPEDIT 18 0 "alpha = 1/2;" "6#/%&alphaG*&\"\"\"F&\"\"#!\"\"" }{TEXT 305 32 " , par a el que tampoco se tiene " }}{PARA 0 "" 0 "" {TEXT 307 16 "orden 2 (m \341ximo)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "solucion1:=\{alpha=1/2,beta= 1/2\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*solucion1G<$/%&alphaG#\" \"\"\"\"#/%%betaGF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f:=( x,y)->lambda*y;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%\"xG%\"y G6\"6$%)operatorG%&arrowGF)*&%'lambdaG\"\"\"9%F/F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "ka1:=unapply(subs(solucion1,f(x0+al pha*h,y0+alpha*h*kp1(h))),h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ka 1Gf*6#%\"hG6\"6$%)operatorG%&arrowGF(*&%'lambdaG\"\"\",&%#y0GF.*&#F.\" \"#F.*&9$F.-%$kp1G6#F5F.F.F.F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "ka2:=unapply(subs(solucion1,f(x0+h,y0+(1/2)*h*kp1(h)+ (1/2)*h*kp2(h))),h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ka2Gf*6#%\" hG6\"6$%)operatorG%&arrowGF(*&%'lambdaG\"\"\",(%#y0GF.*&#F.\"\"#F.*&9$ F.-%$kp1G6#F5F.F.F.*&F2F.*&F5F.-%$kp2GF8F.F.F.F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "test:=solve(\{k1=subs(\{kp1(h)=k1,k p2(h)=k2\},ka1(h)),k2=subs(\{kp1(h)=k1,kp2(h)=k2\},ka2(h))\},\{k1,k2\} );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%testG<$/%#k1G,$**\"\"#\"\"\"% 'lambdaGF+%#y0GF+,&F*!\"\"*&F,F+%\"hGF+F+F/F//%#k2G,$**\"\"%F+F,F+F-F+ ,(F6F+*(F6F+F,F+F1F+F/*&)F,F*F+)F1F*F+F+F/F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "k1:=subs(test,k1);k2:=subs(test,k2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k1G,$**\"\"#\"\"\"%'lambdaGF(%#y0GF(,&F'! \"\"*&F)F(%\"hGF(F(F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k2G,$** \"\"%\"\"\"%'lambdaGF(%#y0GF(,(F'F(*(F'F(F)F(%\"hGF(!\"\"*&)F)\"\"#F() F-F1F(F(F.F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "y1:=expand( subs(solucion1,y0+h*(beta*k1+beta*k2)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G,(%#y0G\"\"\"**%\"hGF'%'lambdaGF'F&F',&\"\"#!\"\"*&F*F'F)F 'F'F-F-*,F,F'F)F'F*F'F&F',(\"\"%F'*(F1F'F*F'F)F'F-*&)F*F,F')F)F,F'F'F- F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "y1:=collect(subs(lamb da=z/h,y1),y0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G*&,(\"\"\"F'* &%\"zGF',&\"\"#!\"\"F)F'F,F,*(F+F'F)F',(\"\"%F'*&F/F'F)F'F,*$)F)F+F'F' F,F'F'%#y0GF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "r:=unapply (collect(simplify(y1/y0),z),z):rz:=factor(simplify(r(z)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#rzG,$*&\"\"%\"\"\",&\"\"#!\"\"%\"zGF(!\"# F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 308 61 "Ahora, el dominio de estab ilidad est\341 compuesto por los z = " }{XPPEDIT 18 0 "lambda" "6#%'l ambdaG" }{TEXT 309 22 " h cuya distancia al" }}{PARA 0 "" 0 "" {TEXT 310 84 "punto 2 sea superior a 2, que es el exterior de la ci rcunferencia de centro 2 y" }}{PARA 0 "" 0 "" {TEXT 311 93 "radio 2 (que pasa, pues, por el origen). Esta regi\363n tambi\351n contiene a l semiplano de parte " }}{PARA 0 "" 0 "" {TEXT 312 57 "real negativa, \+ por lo que el m\351todo tambi\351n es A-estable" }}}}{MARK "0 1 0" 5 }{VIEWOPTS 1 1 0 3 4 1802 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }