{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 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 1 14 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 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 270 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 271 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 275 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 276 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 277 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 278 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 279 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 280 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 281 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 282 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "T imes" 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 "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 281 14 "Ejercicios de " }}{PARA 4 "" 0 "" {TEXT 282 22 "(8) Estabilidad lineal" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 280 50 "Ejercicio 08-18 (del EX AMEN ORDINARIO de 18FEB05)" }}{PARA 4 "" 0 "" {TEXT -1 55 "\nDe entre \+ los m\351todos impl\355citos de 2 etapas de tablero" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 260 18 " 0 | " } {XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT 262 9 " 0" }}{PARA 4 "" 0 "" {TEXT 261 18 " 1 | " }{XPPEDIT 18 0 "alpha;" "6 #%&alphaG" }{TEXT 263 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 sean de mayor orden." }}{PARA 4 "" 0 "" {TEXT -1 1 " " }}{PARA 4 "" 0 "" {TEXT -1 40 "(desde aqu\355 es Eje rcicio 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 fa milia general de los m\351todos " }}{PARA 4 "" 0 "" {TEXT -1 80 "del t ablero y disc\372tase la A-estabilidad del m\351todo LOBATTO IIIb y d e 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 verdade ra 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\},f 0):f0:=expand(f0):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "f1:=s ubs(\{y(x)=y0,x=x0\},f1):f1:=expand(f1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "f2:=subs(\{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 "# desar rollo 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]:=alp ha: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:=unapply(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:=unapp ly(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,kp 20(0)=k200\},k11(0)):k110:=expand(k110):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "k210:=subs(\{kp10(0)=k100,kp20(0)=k200\},k21(0)):k210 :=expand(k210):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Fi1:=exp and(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(kp1 0)(0)=k110,D(kp20)(0)=k210,kp10(0)=k100,kp20(0)=k200\},k12(0)):k120:=e xpand(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)=k21 0,kp10(0)=k100,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\},k23(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,y 0));" }}{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))=coeff((1/2)*f1,D[1](f)(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond2aG/#\"\"\"\"\"#F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "cond2b:=coeff(coeff(Fi1,D[2](f)(x0,y0)),f(x0,y0))=coe ff(coeff((1/2)*f1,D[2](f)(x0,y0)),f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond2bG/%&alphaG#\"\"\"\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 14 "orden 2 para " }{XPPEDIT 18 0 "alpha;" "6#%&al phaG" }{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%\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "cond3b:=coeff(coeff(Fi2,D[1,2](f)( x0,y0)),f(x0,y0))=coeff(coeff((1/3)*f2,D[1,2](f)(x0,y0)),f(x0,y0));sub s(alpha=1/2,cond3b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3bG/%&a lphaG#\"\"#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/#\"\"\"\"\"##F&\" \"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "cond3c:=coeff(coeff (Fi2,D[2,2](f)(x0,y0)),f(x0,y0)^2)=coeff(coeff((1/3)*f2,D[2,2](f)(x0,y 0)),f(x0,y0)^2);subs(alpha=1/2,cond3c);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3cG/*$)%&alphaG\"\"#\"\"\"#F*\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/#\"\"\"\"\"%#F%\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "cond3d:=coeff(coeff(Fi2,D[2](f)(x0,y0)),D[1](f)(x0,y 0))=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,y 0));subs(alpha=1/2,cond3e);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond 3eG/,$*&\"\"#\"\"\")%&alphaGF(F)F)#F)\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/#\"\"\"\"\"##F%\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 36 "no hay orden 3 para ningun valor de " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{MPLTEXT 0 21 2 " " }}}{EXCHG {PARA 0 "" 0 " " {MPLTEXT 0 21 24 "# funcion de estabilidad" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f:=(x,y)->lambda*y;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF)*&%'lambd aG\"\"\"9%F/F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "ka1:= unapply(f(x0+c[1]*h,y0+a[1,1]*h*kp1(h)+a[1,2]*h*kp2(h)),h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ka1Gf*6#%\"hG6\"6$%)operatorG%&arrowGF(*& %'lambdaG\"\"\",&%#y0GF.*(%&alphaGF.9$F.-%$kp1G6#F3F.F.F.F(F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "ka2:=unapply(f(x0+c[2]*h,y0+ a[2,1]*h*kp1(h)+a[2,2]*h*kp2(h)),h);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%$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,$*(%'lambdaG\"\"\"%#y0GF+,&F+!\" \"*(F*F+%&alphaGF+%\"hGF+F+F.F./%#k2GF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "k1:=subs(test,k1);k2:=subs(test,k2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k1G,$*(%'lambdaG\"\"\"%#y0GF(,&F(!\"\"*(F'F(%&a lphaGF(%\"hGF(F(F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k2G,$*(%'la mbdaG\"\"\"%#y0GF(,&F(!\"\"*(F'F(%&alphaGF(%\"hGF(F(F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "y1:=expand(y0+h*(b[1]*k1+b[2]*k2)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G,&%#y0G\"\"\"**%\"hGF'%'lamb daGF'F&F',&F'!\"\"*(F*F'%&alphaGF'F)F'F'F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "y1:=collect(subs(lambda=z/h,y1),y0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G*&,&\"\"\"F'*&%\"zGF',&F'!\"\"*&F)F'%& alphaGF'F'F+F+F'%#y0GF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 " r:=unapply(collect(simplify(y1/y0),z),z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rGf*6#%\"zG6\"6$%)operatorG%&arrowGF(*&,&\"\"\"!\" \"*&,&%&alphaGF.F.F/F.9$F.F.F.,&F.F/*&F3F.F2F.F.F/F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 43 "# A-estabilidad para el m\351todo LO BATTO IIIb" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "simplify(subs (alpha=1/2,r(z)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&\"\"#\"\" \"%\"zGF'F',&F&!\"\"F(F'F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 264 50 " El dominio de estabilidad, compuesto por los z = " }{XPPEDIT 18 0 "la mbda" "6#%'lambdaG" }{TEXT 265 34 " h tales que el m\363dulo de r(z ) " }}{PARA 0 "" 0 "" {TEXT 266 88 "sea inferior a 1 , estar\341 form ado por los valores de z que disten menos del punto -2" }}{PARA 0 " " 0 "" {TEXT 267 86 "que del punto 2 . Esto corresponde al semiplano \+ de parte real negativa, por lo que el" }}{PARA 0 "" 0 "" {TEXT 268 33 "m\351todo LOBATTO IIIb es A-estable" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 38 "# A-estabilidad para los otros m\351todos" }}} {EXCHG {PARA 4 "" 0 "" {TEXT 269 29 "En general, como hemos visto " }} {PARA 4 "" 0 "" {TEXT 271 4 " " }{XPPEDIT 18 0 "r(z) = (1+(1-alpha) *z)/(1-alpha*z);" "6#/-%\"rG6#%\"zG*&,&\"\"\"F**&,&F*F*%&alphaG!\"\"F* F'F*F*F*,&F*F**&F-F*F'F*F.F." }{TEXT 270 3 " " }}{PARA 4 "" 0 "" {TEXT 273 13 "Supondremos " }{XPPEDIT 18 0 "alpha <> 1/2;" "6#0%&alph aG*&\"\"\"F&\"\"#!\"\"" }{TEXT 272 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#%&alphaG" }{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#%&alphaG" }{TEXT -1 17 " ) x )^2 + ( 1 - " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{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#%&alphaG" }{TEXT -1 11 " ^2 y^2 " }}{PARA 4 "" 0 "" {TEXT -1 46 "por lo que 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$F%%&alphaGF%!\"\"F)" } {TEXT -1 15 " x + y^2 = 0 " }}{PARA 4 "" 0 "" {TEXT -1 59 "La curva \+ l\355mite de la region de estabilidad pasa por (0,0)" }}{PARA 4 "" 0 "" {TEXT 275 29 "Supongamos primeramente que " }{XPPEDIT 18 0 "alpha \+ < 1/2;" "6#2%&alphaG*&\"\"\"F&\"\"#!\"\"" }{TEXT 274 9 " o sea " } {TEXT -1 5 "1 -2 " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 5 " \+ > 0 " }}{PARA 4 "" 0 "" {TEXT -1 11 "La curva es" }}{PARA 4 "" 0 "" {TEXT -1 12 " ( x + " }{XPPEDIT 18 0 "1/(1-2*alpha);" "6#*&\"\"\" F$,&F$F$*&\"\"#F$%&alphaGF$!\"\"F)" }{TEXT -1 13 " )^2 + y^2 = " } {XPPEDIT 18 0 "1/((1-2*alpha)^2);" "6#*&\"\"\"F$*$,&F$F$*&\"\"#F$%&alp haGF$!\"\"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$%&alphaG F$!\"\"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 estabilidad " }}{PARA 4 "" 0 "" {TEXT -1 37 "es el interior de la circunferencia " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 324 "with(plottools):with(plots) :macro(gris=COLOR(RGB,.8,.8,.8)):exten:=point([-3,-1],symbol=point),po int([-3,1],symbol=point),point([0.5,1],symbol=point),point([0.5,-1],sy mbol=point):noAest:= disk([-1,0],1,color=gris):display(exten,noAest,ax es=normal,scaling=constrained,tickmarks=[[-1=\"-1/1-2*a\"],[0,0]],axes font=[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$$!**QoTJF F$\"+t))*o[#FF7$$!*T^B-(FF$\"+FbC\"o$FF7$$!++K$pB\"FF$\"+Tn`<[FF7$$!+d +$)4>FF$\"+CD&y(eFF7$$!+FPJ5FFF$\"+g5ZXoFF7$$!+/,wDOFF$\"+HC80xFF7$$!+ \\?tTYFF$\"+a#zKW)FF7$$!+#32Au&FF$\"+B0F[!*FF7$$!+i+$)4pFF$\"+l^c5&*FF 7$$!+eo=E\")FF$\"+3D(G#)*FF7$$!+3[4s$*FF$\"+%Gn-)**FF7$$!+?0zi5!\"*Fhp 7$$!+:8Q(=\"F]q$\"+2D(G#)*FF7$$!+%*p,48F]q$\"+j^c5&*FF7$$!+\"HzdU\"F]q $\"+E0F[!*FF7$$!+&zEe`\"F]q$\"+d#zKW)FF7$$!+*)RUP;F]q$\"+JC80xFF7$$!+G 'o*GF]q$\"+>bC\"o$FF7$$!+hJeo>F]q$\"+l))*o[#FF7$$!+,Z6#*>F]q$\" +IBL`7FF7$$!\"#F)$!+:w1-T!#>7$Fat$!+QBL`7FF7$F\\t$!+t))*o[#FF7$Fgs$!+F bC\"o$FF7$Fbs$!+Sn`<[FF7$F]s$!+?D&y(eFF7$Fhr$!+c5ZXoFF7$Fcr$!+PC80xFF7 $$!+%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$$!*P^B-(FF$!+:bC\"o$FF 7$$!*'QoTJFF$!+h))*o[#FF7$$!)')H&)yFF$!+EBL`7FF7$FB$\"+I_8/#)Fjt-%&COL ORG6&%$RGBG$\"\")F+Fh[lFh[l-%*AXESSTYLEG6#%'NORMALG-%*AXESTICKSG6%7#/F +Q)-1/1-2*a6\"7$F)F)-%%FONTG6$%(COURIERG\"#5-%(SCALINGG6#%,CONSTRAINED G" 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 276 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 278 22 "Supongamos ahora que " } {XPPEDIT 18 0 "1/2 < alpha;" "6#2*&\"\"\"F%\"\"#!\"\"%&alphaG" }{TEXT 277 9 " o sea " }{TEXT -1 5 "1 -2 " }{XPPEDIT 18 0 "alpha;" "6#%&alp haG" }{TEXT -1 11 " < 0 y 2 " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" } {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*alpha- 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*alp ha-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],sy mbol=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],c olor=gris):noAest:= disk([1,0],1,color=\"White\"):display(exten,noAest ,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$$\"\"$\"\"! 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