{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 236 0 76 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 48 37 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" 0 21 "" 0 1 0 0 0 1 0 0 0 0 2 0 0 0 0 1 }{CSTYLE "" -1 256 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 260 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 265 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 266 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 1 14 0 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{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 "Times" 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 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 4 "" 0 "" {TEXT 261 14 "Ejercicios de " }}{PARA 4 "" 0 "" {TEXT 262 55 "(7) M\351todos de RUNGE-KUTTA impl\355citos y \+ semi-impl\355citos" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 260 50 "Ejercicio 07-06 (del EXAMEN ORDINARIO de 18FEB05)" }} {PARA 4 "" 0 "" {TEXT -1 55 "\nDe entre los m\351todos impl\355citos d e 2 etapas de tablero" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 263 18 " 0 | " }{XPPEDIT 18 0 "alpha;" "6#%&alph aG" }{TEXT 265 9 " 0" }}{PARA 4 "" 0 "" {TEXT 264 18 " 1 \+ | " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT 266 9 " \+ 0" }}{PARA 4 "" 0 "" {TEXT -1 14 " ---------" }{TEXT 256 24 "----- -------------------" }}{PARA 4 "" 0 "" {TEXT 257 17 " | \+ " }{XPPEDIT 18 0 "1/2;" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT 258 10 " \+ " }{XPPEDIT 18 0 "1/2;" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT 259 1 " " }}{PARA 4 "" 0 "" {TEXT -1 35 "entre los que se encuentra, para " } {XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 34 " = 1/2 , el llamado \+ LOBATTO IIIb, " }}{PARA 4 "" 0 "" {TEXT -1 40 "se buscaran los que sea n de mayor orden." }}{PARA 4 "" 0 "" {TEXT -1 1 " " }}{PARA 4 "" 0 "" {TEXT -1 40 "(desde aqu\355 es Ejercicio del cap\355tulo 8)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 73 "Calc\372lese la funci\363n de estabilidad de la familia general de los m\351todos " } }{PARA 4 "" 0 "" {TEXT -1 80 "del tablero y disc\372tase la A-estabili dad del m\351todo LOBATTO IIIb y de los otros" }}}{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 0 21 37 "# desarrollo de la verdadera solucion" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "f0:=f(x,y(x)):" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 44 "f1:=subs(diff(y(x),x)=f(x,y(x)),diff(f0,x)): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "f2:=subs(diff(y(x),x)=f (x,y(x)),diff(f1,x)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "f3 :=subs(diff(y(x),x)=f(x,y(x)),diff(f2,x)):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 43 "f0:=subs(\{y(x)=y0,x=x0\},f0):f0:=expand(f0):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "f1:=subs(\{y(x)=y0,x=x0\},f1 ):f1:=expand(f1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "f2:=su bs(\{y(x)=y0,x=x0\},f2):f2:=expand(f2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "f3:=subs(\{y(x)=y0,x=x0\},f3):f3:=expand(f3):" }}} {EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 23 "# desarrollo del m\351todo" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "c[1]:=0:c[2]:=1:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "a[1,1]:=alpha:a[1,2]:=0:a[2, 1]:=alpha:a[2,2]:=0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "b[1 ]:=1/2:b[2]:=1/2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "k10:=u napply(f(x0+c[1]*h,y0+a[1,1]*h*kp10(h)+a[1,2]*h*kp20(h)),h):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "k20:=unapply(f(x0+c[2]*h,y0+ a[2,1]*h*kp10(h)+a[2,2]*h*kp20(h)),h):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "k100:=k10(0):k100:=expand(k100):" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 32 "k200:=k20(0):k200:=expand(k200):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Fi0:=expand(b[1]*k100+b[2]*k200):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "k11:=D(k10):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "k21:=D(k20):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 66 "k110:=subs(\{kp10(0)=k100,kp20(0)=k200\},k11(0 )):k110:=expand(k110):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "k 210:=subs(\{kp10(0)=k100,kp20(0)=k200\},k21(0)):k210:=expand(k210):" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Fi1:=expand(b[1]*k110+b[2] *k210):" }}}{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 98 "k120:=subs(\{D(kp10)(0)=k110,D(kp20 )(0)=k210,kp10(0)=k100,kp20(0)=k200\},k12(0)):k120:=expand(k120):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "k220:=subs(\{D(kp10)(0)=k110 ,D(kp20)(0)=k210,kp10(0)=k100,kp20(0)=k200\},k22(0)):k220:=expand(k220 ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Fi2:=expand(b[1]*k120 +b[2]*k220):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "k13:=D(k12) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "k23:=D(k22):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "k130:=subs(\{(D@@2)(kp10)(0 )=k120,(D@@2)(kp20)(0)=k220,D(kp10)(0)=k110,D(kp20)(0)=k210,kp10(0)=k1 00,kp20(0)=k200\},k13(0)):k130:=expand(k130):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "k230:=subs(\{(D@@2)(kp10)(0)=k120,(D@@2)(kp20)( 0)=k220,D(kp10)(0)=k110,D(kp20)(0)=k210,kp10(0)=k100,kp20(0)=k200\},k2 3(0)):k230:=expand(k230):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Fi3:=expand(b[1]*k130+b[2]*k230):" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 24 "# consistencia y orden 1" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 47 "cond1a:=coeff(Fi0,f(x0,y0))=coeff(f0,f(x0,y0));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond1aG/\"\"\"F&" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 29 "# condiciones para el orden 2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "cond2a:=coeff(Fi1,D[1](f)(x0,y0))=c oeff((1/2)*f1,D[1](f)(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'c ond2aG/#\"\"\"\"\"#F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "co nd2b:=coeff(coeff(Fi1,D[2](f)(x0,y0)),f(x0,y0))=coeff(coeff((1/2)*f1,D [2](f)(x0,y0)),f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond2b G/%&alphaG#\"\"\"\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 14 "or den 2 para " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{MPLTEXT 0 21 28 " = 1/2, o sea, LOBATTO IIIb " }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 29 "# condiciones para el orden 3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "cond3a:=coeff(Fi2,D[1,1](f)(x0,y0))=coeff((1/3)*f2,D[ 1,1](f)(x0,y0));subs(alpha=1/2,cond3a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3aG/#\"\"\"\"\"##F'\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/#\"\"\"\"\"##F%\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "cond3b:=coeff(coeff(Fi2,D[1,2](f)(x0,y0)),f(x0,y0))=coeff(coeff(( 1/3)*f2,D[1,2](f)(x0,y0)),f(x0,y0));subs(alpha=1/2,cond3b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3bG/%&alphaG#\"\"#\"\"$" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/#\"\"\"\"\"##F&\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "cond3c:=coeff(coeff(Fi2,D[2,2](f)(x0,y0)),f(x0,y0 )^2)=coeff(coeff((1/3)*f2,D[2,2](f)(x0,y0)),f(x0,y0)^2);subs(alpha=1/2 ,cond3c);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3cG/*$)%&alphaG\" \"#\"\"\"#F*\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/#\"\"\"\"\"%#F% \"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "cond3d:=coeff(coe ff(Fi2,D[2](f)(x0,y0)),D[1](f)(x0,y0))=coeff(coeff((1/3)*f2,D[2](f)(x0 ,y0)),D[1](f)(x0,y0));subs(alpha=1/2,cond3d);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3dG/\"\"!#\"\"\"\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"!#\"\"\"\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "cond3e:=coeff(coeff(Fi2,D[2](f)(x0,y0)^2),f(x0,y0))= coeff(coeff((1/3)*f2,D[2](f)(x0,y0)^2),f(x0,y0));subs(alpha=1/2,cond3e );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3eG/,$*&\"\"#\"\"\")%&alp haGF(F)F)#F)\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/#\"\"\"\"\"##F% \"\"$" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 36 "no hay orden 3 para ningun valor de " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{MPLTEXT 0 21 2 " " }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 24 "# funcion de est abilidad" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f:=(x,y)->lambd a*y;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%\"xG%\"yG6\"6$%)ope ratorG%&arrowGF)*&%'lambdaG\"\"\"9%F/F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "ka1:=unapply(f(x0+c[1]*h,y0+a[1,1]*h*kp1(h)+a[1,2] *h*kp2(h)),h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ka1Gf*6#%\"hG6\"6 $%)operatorG%&arrowGF(*&%'lambdaG\"\"\",&%#y0GF.*(%&alphaGF.9$F.-%$kp1 G6#F3F.F.F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "ka2:=u napply(f(x0+c[2]*h,y0+a[2,1]*h*kp1(h)+a[2,2]*h*kp2(h)),h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ka2Gf*6#%\"hG6\"6$%)operatorG%&arrowGF(*& %'lambdaG\"\"\",&%#y0GF.*(%&alphaGF.9$F.-%$kp1G6#F3F.F.F.F(F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "test:=solve(\{k1=subs(\{kp1( h)=k1,kp2(h)=k2\},ka1(h)),k2=subs(\{kp1(h)=k1,kp2(h)=k2\},ka2(h))\},\{ k1,k2\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%testG<$/%#k1G,$*(%'lam bdaG\"\"\"%#y0GF+,&F+!\"\"*(F*F+%&alphaGF+%\"hGF+F+F.F./%#k2GF(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "k1:=subs(test,k1);k2:=subs(t est,k2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k1G,$*(%'lambdaG\"\"\"% #y0GF(,&F(!\"\"*(F'F(%&alphaGF(%\"hGF(F(F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k2G,$*(%'lambdaG\"\"\"%#y0GF(,&F(!\"\"*(F'F(%&alphaG F(%\"hGF(F(F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "y1:=expa nd(y0+h*(b[1]*k1+b[2]*k2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G, &%#y0G\"\"\"**%\"hGF'%'lambdaGF'F&F',&F'!\"\"*(F*F'%&alphaGF'F)F'F'F,F ," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "y1:=collect(subs(lambd a=z/h,y1),y0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G*&,&\"\"\"F'*& %\"zGF',&F'!\"\"*&F)F'%&alphaGF'F'F+F+F'%#y0GF'" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 41 "r:=unapply(collect(simplify(y1/y0),z),z);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rGf*6#%\"zG6\"6$%)operatorG%&arrow GF(*&,&\"\"\"!\"\"*&,&%&alphaGF.F.F/F.9$F.F.F.,&F.F/*&F3F.F2F.F.F/F(F( F(" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 43 "# A-estabilidad para e l m\351todo LOBATTO IIIb" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "simplify(subs(alpha=1/2,r(z)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$ *&,&\"\"#\"\"\"%\"zGF'F',&F&!\"\"F(F'F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 267 50 "El dominio de estabilidad, compuesto por los z = " } {XPPEDIT 18 0 "lambda" "6#%'lambdaG" }{TEXT 268 34 " h tales que el \+ m\363dulo de r(z) " }}{PARA 0 "" 0 "" {TEXT 269 88 "sea inferior a 1 , estar\341 formado por los valores de z que disten menos del punto -2" }}{PARA 0 "" 0 "" {TEXT 270 86 "que del punto 2 . Esto correspo nde al semiplano de parte real negativa, por lo que el" }}{PARA 0 "" 0 "" {TEXT 271 33 "m\351todo LOBATTO IIIb es A-estable" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 38 "# A-estabilidad para los otros m\351 todos" }}}{EXCHG {PARA 4 "" 0 "" {TEXT 272 29 "En general, como hemos \+ visto " }}{PARA 4 "" 0 "" {TEXT 274 4 " " }{XPPEDIT 18 0 "r(z) = (1 +(1-alpha)*z)/(1-alpha*z);" "6#/-%\"rG6#%\"zG*&,&\"\"\"F**&,&F*F*%&alp haG!\"\"F*F'F*F*F*,&F*F**&F-F*F'F*F.F." }{TEXT 273 3 " " }}{PARA 4 " " 0 "" {TEXT 276 13 "Supondremos " }{XPPEDIT 18 0 "alpha <> 1/2;" "6# 0%&alphaG*&\"\"\"F&\"\"#!\"\"" }{TEXT 275 42 " y pondremos que z=x+i y . " }}{PARA 4 "" 0 "" {TEXT -1 4 "Como" }}{PARA 4 "" 0 "" {TEXT -1 15 " 1 + ( 1 - " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG " }{TEXT -1 18 " ) z = 1 + ( 1 - " }{XPPEDIT 18 0 "alpha;" "6#%&alpha G" }{TEXT -1 16 " ) x + i ( 1 - " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG " }{TEXT -1 4 " ) y" }}{PARA 4 "" 0 "" {TEXT -1 1 "y" }}{PARA 4 "" 0 " " {TEXT -1 10 " 1 - " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" } {TEXT -1 12 " z = 1 - " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" } {TEXT -1 9 " x + i " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 5 " y , " }}{PARA 4 "" 0 "" {TEXT -1 8 "resulta " }}{PARA 4 "" 0 "" {TEXT -1 17 " | 1 + ( 1 - " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" } {TEXT -1 24 " ) z | ^2 = (1 + ( 1 - " }{XPPEDIT 18 0 "alpha;" "6#%&al phaG" }{TEXT -1 17 " ) x )^2 + ( 1 - " }{XPPEDIT 18 0 "alpha;" "6#%&al phaG" }{TEXT -1 11 " )^2 y^2 " }}{PARA 4 "" 0 "" {TEXT -1 2 "y " }} {PARA 4 "" 0 "" {TEXT -1 12 " | 1 - " }{XPPEDIT 18 0 "alpha;" "6# %&alphaG" }{TEXT -1 17 " z | ^2 = (1 - " }{XPPEDIT 18 0 "alpha;" "6# %&alphaG" }{TEXT -1 11 " x )^2 + " }{XPPEDIT 18 0 "alpha;" "6#%&alph aG" }{TEXT -1 11 " ^2 y^2 " }}{PARA 4 "" 0 "" {TEXT -1 46 "por lo q ue la curva | r(z) |^2 = 1 se escribe" }}{PARA 4 "" 0 "" {TEXT -1 16 " ( 1 + ( 1 - " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 17 " ) x )^2 + ( 1 - " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 20 " )^2 y^2 = ( 1 - " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 11 " x )^2 + " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 10 " ^ 2 y^2 " }}{PARA 4 "" 0 "" {TEXT -1 5 "o sea" }}{PARA 4 "" 0 "" {TEXT -1 11 " ( 1 -2 " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 16 " ) y^2 + ( 1 -2 " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 21 " ) x^2 + 2 x = 0 " }}{PARA 4 "" 0 "" {TEXT -1 12 " x^2 \+ + " }{XPPEDIT 18 0 "2/(1-2*alpha);" "6#*&\"\"#\"\"\",&F%F%*&F$F%%&alp haGF%!\"\"F)" }{TEXT -1 15 " x + y^2 = 0 " }}{PARA 4 "" 0 "" {TEXT -1 59 "La curva l\355mite de la region de estabilidad pasa por (0,0) " }}{PARA 4 "" 0 "" {TEXT 278 29 "Supongamos primeramente que " } {XPPEDIT 18 0 "alpha < 1/2;" "6#2%&alphaG*&\"\"\"F&\"\"#!\"\"" }{TEXT 277 9 " o sea " }{TEXT -1 5 "1 -2 " }{XPPEDIT 18 0 "alpha;" "6#%&alp haG" }{TEXT -1 5 " > 0 " }}{PARA 4 "" 0 "" {TEXT -1 11 "La curva es" } }{PARA 4 "" 0 "" {TEXT -1 12 " ( x + " }{XPPEDIT 18 0 "1/(1-2*alp ha);" "6#*&\"\"\"F$,&F$F$*&\"\"#F$%&alphaGF$!\"\"F)" }{TEXT -1 13 " )^ 2 + y^2 = " }{XPPEDIT 18 0 "1/((1-2*alpha)^2);" "6#*&\"\"\"F$*$,&F$F$* &\"\"#F$%&alphaGF$!\"\"F(F*" }{TEXT -1 3 " " }}{PARA 4 "" 0 "" {TEXT -1 39 "o sea, una circunferencia de centro - " }{XPPEDIT 18 0 " 1/(1-2*alpha);" "6#*&\"\"\"F$,&F$F$*&\"\"#F$%&alphaGF$!\"\"F)" }{TEXT -1 11 " y radio " }{XPPEDIT 18 0 "1/(1-2*alpha);" "6#*&\"\"\"F$,&F$F $*&\"\"#F$%&alphaGF$!\"\"F)" }{TEXT -1 22 " que pasa por (0,0) " }} {PARA 4 "" 0 "" {TEXT -1 39 "Como z=1=(1,0) verifica | r(z) | =| " }{XPPEDIT 18 0 "(2-alpha)/(1-alpha);" "6#*&,&\"\"#\"\"\"%&alphaG!\"\"F &,&F&F&F'F(F(" }{TEXT -1 46 " | > 1 resulta que el dominio de estabil idad " }}{PARA 4 "" 0 "" {TEXT -1 37 "es el interior de la circunferen cia " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 324 "with(plottools):w ith(plots):macro(gris=COLOR(RGB,.8,.8,.8)):exten:=point([-3,-1],symbol =point),point([-3,1],symbol=point),point([0.5,1],symbol=point),point([ 0.5,-1],symbol=point):noAest:= disk([-1,0],1,color=gris):display(exten ,noAest,axes=normal,scaling=constrained,tickmarks=[[-1=\"-1/1-2*a\"],[ 0,0]],axesfont=[COURIER,10]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }{PARA 13 "" 1 "" {GLPLOT2D 380 380 380 {PLOTDATA 2 "6*-%'POINTSG6$7$$ !\"$\"\"!$!\"\"F)-%'SYMBOLG6#%&POINTG-F$6$7$F'$\"\"\"F)F,-F$6$7$$\"\"& F+F3F,-F$6$7$F8F*F,-%)POLYGONSG6$7U7$$F)F)FB7$$!)()H&)y!#5$\"+OBL`7FF7 $$!**QoTJFF$\"+t))*o[#FF7$$!*T^B-(FF$\"+FbC\"o$FF7$$!++K$pB\"FF$\"+Tn` <[FF7$$!+d+$)4>FF$\"+CD&y(eFF7$$!+FPJ5FFF$\"+g5ZXoFF7$$!+/,wDOFF$\"+HC 80xFF7$$!+\\?tTYFF$\"+a#zKW)FF7$$!+#32Au&FF$\"+B0F[!*FF7$$!+i+$)4pFF$ \"+l^c5&*FF7$$!+eo=E\")FF$\"+3D(G#)*FF7$$!+3[4s$*FF$\"+%Gn-)**FF7$$!+? 0zi5!\"*Fhp7$$!+:8Q(=\"F]q$\"+2D(G#)*FF7$$!+%*p,48F]q$\"+j^c5&*FF7$$!+ \"HzdU\"F]q$\"+E0F[!*FF7$$!+&zEe`\"F]q$\"+d#zKW)FF7$$!+*)RUP;F]q$\"+JC 80xFF7$$!+G'o*GF]q$\"+>bC\"o$FF7$$!+hJeo>F]q$\"+l))*o[#FF7$$!+, Z6#*>F]q$\"+IBL`7FF7$$!\"#F)$!+:w1-T!#>7$Fat$!+QBL`7FF7$F\\t$!+t))*o[# FF7$Fgs$!+FbC\"o$FF7$Fbs$!+Sn`<[FF7$F]s$!+?D&y(eFF7$Fhr$!+c5ZXoFF7$Fcr $!+PC80xFF7$$!+%zEe`\"F]q$!+i#zKW)FF7$Fiq$!+H0F[!*FF7$$!+$*p,48F]q$!+m ^c5&*FF7$$!+98Q(=\"F]q$!+4D(G#)*FF7$$!+>0zi5F]q$!+&Gn-)**FF7$$!+*z%4s$ *FF$!+%Gn-)**FF7$$!+]o=E\")FF$!+2D(G#)*FF7$$!+a+$)4pFF$!+i^c5&*FF7$$!+ $32Au&FF$!+C0F[!*FF7$$!+]?tTYFF$!+b#zKW)FF7$F]o$!+HC80xFF7$$!+9PJ5FFF$ !+Z5ZXoFF7$$!+Z+$)4>FF$!+5D&y(eFF7$$!+$>LpB\"FF$!+Hn`<[FF7$$!*P^B-(FF$ !+:bC\"o$FF7$$!*'QoTJFF$!+h))*o[#FF7$$!)')H&)yFF$!+EBL`7FF7$FB$\"+I_8/ #)Fjt-%&COLORG6&%$RGBG$\"\")F+Fh[lFh[l-%*AXESTICKSG6%7#/F+Q)-1/1-2*a6 \"7$F)F)-%%FONTG6$%(COURIERG\"#5-%*AXESSTYLEG6#%'NORMALG-%(SCALINGG6#% ,CONSTRAINEDG" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" }}}}{EXCHG {PARA 4 "" 0 "" {TEXT 279 28 "y el m\351todo no es A-estable." }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 281 22 "Supongamos ahora que " }{XPPEDIT 18 0 "1/2 < alpha;" "6#2*&\"\"\"F%\"\"#!\"\"%&alphaG" } {TEXT 280 9 " o sea " }{TEXT -1 5 "1 -2 " }{XPPEDIT 18 0 "alpha;" "6 #%&alphaG" }{TEXT -1 11 " < 0 y 2 " }{XPPEDIT 18 0 "alpha;" "6#%&alp haG" }{TEXT -1 7 " -1 > 0" }}{PARA 4 "" 0 "" {TEXT -1 11 "La curva es " }}{PARA 4 "" 0 "" {TEXT -1 12 " ( x - " }{XPPEDIT 18 0 "1/(2*al pha-1);" "6#*&\"\"\"F$,&*&\"\"#F$%&alphaGF$F$F$!\"\"F)" }{TEXT -1 13 " )^2 + y^2 = " }{XPPEDIT 18 0 "1/((2*alpha-1)^2);" "6#*&\"\"\"F$*$,&*& \"\"#F$%&alphaGF$F$F$!\"\"F(F*" }{TEXT -1 3 " " }}{PARA 4 "" 0 "" {TEXT -1 37 "o sea, una circunferencia de centro " }{XPPEDIT 18 0 "1/ (2*alpha-1);" "6#*&\"\"\"F$,&*&\"\"#F$%&alphaGF$F$F$!\"\"F)" }{TEXT -1 11 " y radio " }{XPPEDIT 18 0 "1/(2*alpha-1);" "6#*&\"\"\"F$,&*& \"\"#F$%&alphaGF$F$F$!\"\"F)" }{TEXT -1 22 " que pasa por (0,0) " }} {PARA 4 "" 0 "" {TEXT -1 41 "Como z=-1=(-1,0) verifica | r(z) | =| \+ " }{XPPEDIT 18 0 "alpha/(1+alpha);" "6#*&%&alphaG\"\"\",&F%F%F$F%!\"\" " }{TEXT -1 46 " | < 1 resulta que el dominio de estabilidad " }} {PARA 4 "" 0 "" {TEXT -1 37 "es el exterior de la circunferencia " }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 329 "exten:=point([-0.5,-1.1],s ymbol=point),point([-0.5,1.1],symbol=point),point([3,1.1],symbol=point ),point([3,-1.1],symbol=point):plano:= rectangle([-0.5,-1.1], [3,1.1], color=gris):noAest:= disk([1,0],1,color=\"White\"):display(exten,noAes t,plano,axes=normal,scaling=constrained,tickmarks=[[1=\"1/2*a-1\"],[0, 0]],axesfont=[COURIER,10]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} {PARA 13 "" 1 "" {GLPLOT2D 380 380 380 {PLOTDATA 2 "6+-%'POINTSG6$7$$! \"&!\"\"$!#6F)-%'SYMBOLG6#%&POINTG-F$6$7$F'$\"#6F)F,-F$6$7$$\"\"$\"\"! F3F,-F$6$7$F8F*F,-%)POLYGONSG6$7U7$$\"\"#F:$F:F:7$$\"+,Z6#*>!\"*$\"+OB L`7!#57$$\"+hJeo>FI$\"+t))*o[#FL7$$\"+'[w(H>FI$\"+FbC\"o$FL7$$\"+!o1j( =FI$\"+Tn`<[FL7$$\"+%*p,4=FI$\"+CD&y(eFL7$$\"+F'o*G0zi5F I$\"+%Gn-)**FL7$$\"+.[4s$*FLF\\q7$$\"+ao=E\")FL$\"+2D(G#)*FL7$$\"+e+$) 4pFL$\"+j^c5&*FL7$$\"+(32Au&FL$\"+E0F[!*FL7$$\"+a?tTYFL$\"+d#zKW)FL7$$ \"+2,wDOFL$\"+JC80xFL7$$\"+FL$\"+9D&y(eF L7$$\"+&>LpB\"FL$\"+Kn`<[FL7$$\"*Q^B-(FL$\"+>bC\"o$FL7$$\"*(QoTJFL$\"+ l))*o[#FL7$$\")')H&)yFL$\"+IBL`7FL7$FE$!+:w1-T!#>7$$\")()H&)yFL$!+QBL` 7FL7$$\"**QoTJFL$!+t))*o[#FL7$$\"*T^B-(FL$!+FbC\"o$FL7$$\"+*>LpB\"FL$! +Sn`<[FL7$$\"+a+$)4>FL$!+?D&y(eFL7$$\"+BPJ5FFL$!+c5ZXoFL7$$\"+8,wDOFL$ !+PC80xFL7$$\"+h?tTYFL$!+i#zKW)FL7$$\"+%42Au&FL$!+H0F[!*FL7$$\"+m+$)4p FL$!+m^c5&*FL7$$\"+io=E\")FL$!+4D(G#)*FL7$$\"+7[4s$*FL$!+&Gn-)**FL7$$ \"+?0zi5FI$!+%Gn-)**FL7$$\"+:8Q(=\"FI$!+2D(G#)*FL7$$\"+&*p,48FI$!+i^c5 &*FL7$F[p$!+C0F[!*FL7$Ffo$!+b#zKW)FL7$Fao$!+HC80xFL7$$\"+H'o*G " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 282 35 "y los m\351todos son ahora A-estables." }}}}{MARK " 0 1 0" 12 }{VIEWOPTS 1 1 0 3 4 1802 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }