{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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0 "" }}{PARA 4 "" 0 "" {TEXT 260 51 "Ejercicio 07-10 (del EXAMEN ORDINARIO DE 09FEB09)" }} {PARA 4 "" 0 "" {TEXT -1 75 "Consid\351rese la familia de m\351todos i mpl\355citos de RUNGE-KUTTA con dos etapas " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 263 5 " " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT 264 11 " | " }{XPPEDIT 18 0 "alpha;" "6#%& alphaG" }{TEXT 265 2 " " }}{PARA 4 "" 0 "" {TEXT 266 5 " " } {XPPEDIT 18 0 "2*beta;" "6#*&\"\"#\"\"\"%%betaGF%" }{TEXT 267 9 " | \+ " }{XPPEDIT 18 0 "beta;" "6#%%betaG" }{TEXT 273 11 " " } {XPPEDIT 18 0 "beta;" "6#%%betaG" }}{PARA 4 "" 0 "" {TEXT -1 12 " -- -------" }{TEXT 256 24 "------------------------" }}{PARA 4 "" 0 "" {TEXT 257 19 " | " }{XPPEDIT 18 0 "1/4;" "6#*&\"\"\"F$ \"\"%!\"\"" }{TEXT 258 10 " " }{XPPEDIT 18 0 "3/4;" "6#*&\"\" $\"\"\"\"\"%!\"\"" }{TEXT 259 1 " " }}{PARA 4 "" 0 "" {TEXT -1 78 "B \372squese el mayor orden alcanzable por los miembros de esta familia, indicando" }}{PARA 4 "" 0 "" {TEXT -1 16 "los valores de " } {XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 8 " y de " }{XPPEDIT 18 0 "beta;" "6#%%betaG" }{TEXT -1 26 " para los que se consigue" }} {PARA 4 "" 0 "" {TEXT -1 40 "(desde aqu\355 es Ejercicio del cap\355tu lo 8)" }}{PARA 4 "" 0 "" {TEXT -1 55 "Qued\341ndonos ahora (por simpli cidad) con la situaci\363n " }{XPPEDIT 18 0 "alpha = 0;" "6#/%&alpha G\"\"!" }{TEXT -1 23 " , calc\372lese la funci\363n" }}{PARA 4 "" 0 " " {TEXT -1 40 "de estabilidad del m\351todo en estos casos" }}{PARA 4 "" 0 "" {TEXT -1 58 "Est\372diese la A-estabilidad de estos m\351todos para los que " }{XPPEDIT 18 0 "alpha = 0;" "6#/%&alphaG\"\"!" } {TEXT -1 1 " " }}}{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 verda dera solucion" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "#y:=unappl y(y(x),x):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "#f:=unapply(f (x,y),x,y):" }}}{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 49 "k10: =unapply(f(x0+alpha*h,y0+alpha*h*kp10(h)),h):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "k20:=unapply(f(x0+2*beta*h,y0+beta*h*kp10(h)+bet a*h*kp20(h)),h):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "k100:=k 10(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 35 "Fi0:=expand((1/4)*k100+(3/4)*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 "k210:=subs(\{kp10 (0)=k100,kp20(0)=k200\},k21(0)):k210:=expand(k210):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Fi1:=expand((1/4)*k110+(3/4)*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,k p10(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 35 "Fi2:=expand((1/4)*k120+(3/4) *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)=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)=k2 20,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 35 "Fi3 :=expand((1/4)*k130+(3/4)*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/,&*&\"\"%!\"\"%&alphaG\"\"\"F+*(\"\"$F+\"\"#F)%%betaGF+F+#F+F. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "cond2b:=coeff(coeff(Fi1 ,D[2](f)(x0,y0)),f(x0,y0))=coeff(coeff((1/2)*f1,D[2](f)(x0,y0)),f(x0,y 0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond2bG/,&*&\"\"%!\"\"%&alp haG\"\"\"F+*(\"\"$F+\"\"#F)%%betaGF+F+#F+F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "orden2:=solve(\{cond2a\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'orden2G<$/%&alphaG,&*&\"\"'\"\"\"%%betaGF+!\"\"\"\"# F+/F,F," }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 29 "# condiciones par a el orden 3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "cond3a:=coe ff(Fi2,D[1,1](f)(x0,y0))=coeff((1/3)*f2,D[1,1](f)(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3aG/,&*&\"\"%!\"\"%&alphaG\"\"#\"\"\" *&\"\"$F,)%%betaGF+F,F,#F,F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "cond3b:=coeff(coeff(Fi2,D[1,2](f)(x0,y0)),f(x0,y0))=coeff(coeff ((1/3)*f2,D[1,2](f)(x0,y0)),f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3bG/,&*&\"\"#!\"\"%&alphaGF(\"\"\"*&\"\"'F+)%%betaGF(F+F+ #F(\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "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);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3cG/,& *&\"\"%!\"\"%&alphaG\"\"#\"\"\"*&\"\"$F,)%%betaGF+F,F,#F,F." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "cond3d:=coeff(coeff(Fi2,D[2 ](f)(x0,y0)),D[1](f)(x0,y0))=coeff(coeff((1/3)*f2,D[2](f)(x0,y0)),D[1] (f)(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3dG/,(*&#\"\"\" \"\"#F)*$)%&alphaGF*F)F)F)*&#\"\"$F*F)*&%%betaGF)F-F)F)F)*&F0F))F2F*F) F)#F)F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "cond3e:=coeff(c oeff(Fi2,D[2](f)(x0,y0)^2),f(x0,y0))=coeff(coeff((1/3)*f2,D[2](f)(x0,y 0)^2),f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3eG/,(*&#\" \"\"\"\"#F)*$)%&alphaGF*F)F)F)*&#\"\"$F*F)*&%%betaGF)F-F)F)F)*&F0F))F2 F*F)F)#F)F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "orden3:=solv e(\{cond2a,cond3a,cond3d\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'ord en3G<$/%%betaG#\"\"\"\"\"$/%&alphaG\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 62 "se obtiene orden 3 para los valores alpha = 0 y b eta = 1/3 " }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 29 "# condiciones \+ para el orden 4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "cond4a:= coeff(Fi3,D[1,1,1](f)(x0,y0))=coeff((1/4)*f3,D[1,1,1](f)(x0,y0));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond4aG/,&*&\"\"%!\"\"%&alphaG\"\"$ \"\"\"*&\"\"'F,)%%betaGF+F,F,#F,F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "cond4b:=coeff(coeff(Fi3,D[1,1,2](f)(x0,y0)),f(x0,y0) )=coeff(coeff((1/4)*f3,D[1,1,2](f)(x0,y0)),f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond4bG/,&*&\"#=\"\"\")%%betaG\"\"$F)F)*(F,F)\" \"%!\"\"%&alphaGF,F)#F,F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "cond4c:=coeff(coeff(Fi3,D[1,2,2](f)(x0,y0)),f(x0,y0)^2)=coeff(coe ff((1/4)*f3,D[1,2,2](f)(x0,y0)),f(x0,y0)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond4cG/,&*&\"#=\"\"\")%%betaG\"\"$F)F)*(F,F)\"\"%! \"\"%&alphaGF,F)#F,F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "c ond4d:=coeff(coeff(Fi3,D[1,2](f)(x0,y0)),D[1](f)(x0,y0))=coeff(coeff(( 1/4)*f3,D[1,2](f)(x0,y0)),D[1](f)(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond4dG/,(*&\"#=\"\"\")%%betaG\"\"$F)F)*&#F,\"\"#F)* $)%&alphaGF,F)F)F)*(\"\"*F))F+F/F)F2F)F)#F,\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "cond4e:=coeff(coeff(coeff(Fi3,D[1,2](f)(x0,y 0)),D[2](f)(x0,y0)),f(x0,y0))=coeff(coeff(coeff((1/4)*f3,D[1,2](f)(x0, y0)),D[2](f)(x0,y0)),f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%' cond4eG/,**&\"\"$\"\"\")%&alphaGF(F)F)*&\"#OF))%%betaGF(F)F)*(\"\"*F)) F/\"\"#F)F+F)F)*&#F1F3F)*&F/F))F+F3F)F)F)#\"\"&\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "cond4f:=coeff(coeff(Fi3,D[2,2,2](f)(x0,y 0)),f(x0,y0)^3)=coeff(coeff((1/4)*f3,D[2,2,2](f)(x0,y0)),f(x0,y0)^3); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond4fG/,&*&\"\"%!\"\"%&alphaG \"\"$\"\"\"*&\"\"'F,)%%betaGF+F,F,#F,F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "cond4g:=coeff(coeff(coeff(Fi3,D[2,2](f)(x0,y0)),D[1] (f)(x0,y0)),f(x0,y0))=coeff(coeff(coeff((1/4)*f3,D[2,2](f)(x0,y0)),D[1 ](f)(x0,y0)),f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond4gG/ ,(*&\"#=\"\"\")%%betaG\"\"$F)F)*&#F,\"\"#F)*$)%&alphaGF,F)F)F)*(\"\"*F ))F+F/F)F2F)F)#F,\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 149 " cond4h:=coeff(coeff(coeff(Fi3,D[2,2](f)(x0,y0)),D[2](f)(x0,y0)),f(x0,y 0)^2)=coeff(coeff(coeff((1/4)*f3,D[2,2](f)(x0,y0)),D[2](f)(x0,y0)),f(x 0,y0)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond4hG/,**(\"\"*\"\" \")%%betaG\"\"#F)%&alphaGF)F)*&#F(\"\"%F)*&F+F))F-F,F)F)F)*&F/F)*$)F- \"\"$F)F)F)*&\"#FF))F+F6F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "cond4i:=coeff(coeff(Fi3,D[1,1](f)(x0,y0)),D[2](f)(x0,y0))=coe ff(coeff((1/4)*f3,D[1,1](f)(x0,y0)),D[2](f)(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond4iG/,(*&\"\"*\"\"\")%%betaG\"\"$F)F)*&#F,\"\" %F)*$)%&alphaGF,F)F)F)*&#F(F/F)*&F+F))F2\"\"#F)F)F)#F)F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "cond4j:=coeff(coeff(Fi3,D[1](f)(x0 ,y0)),D[2](f)(x0,y0)^2)=coeff(coeff((1/4)*f3,D[1](f)(x0,y0)),D[2](f)(x 0,y0)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond4jG/,**&#\"\"$\"\" #\"\"\"*$)%&alphaGF)F+F+F+*&\"\"*F+)%%betaGF)F+F+*&#F0F*F+*&F2F+)F.F*F +F+F+*&F4F+*&)F2F*F+F.F+F+F+#F+\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "cond4k:=coeff(coeff(Fi3,D[2](f)(x0,y0)^3),f(x0,y0))= coeff(coeff((1/4)*f3,D[2](f)(x0,y0)^3),f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond4kG/,**&#\"\"$\"\"#\"\"\"*$)%&alphaGF)F+F+F+*&\" \"*F+)%%betaGF)F+F+*&#F0F*F+*&F2F+)F.F*F+F+F+*&F4F+*&)F2F*F+F.F+F+F+#F +\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "subs(orden3,\{con d4a,cond4b,cond4c,cond4d,cond4e,cond4f,cond4g,cond4h,cond4i,cond4j,con d4k\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<'/#\"\"\"\"\"$#F&\"\"%/F&F &/#\"\"#F'#F'F)/#F-\"\"*F(/#F)F'#\"\"&F)" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 81 "y no se puede alcanzar orden 4 , lo que por otra par te era obvio ya que el m\351todo" }}{PARA 0 "" 0 "" {MPLTEXT 0 21 62 " de GAUSS de 2 etapas no est\341 entre los de la familia propuesta" }}} {EXCHG {PARA 4 "" 0 "" {TEXT -1 66 "Se obtiene orden m\341ximo 3 de \+ una sola manera, para los valores " }{XPPEDIT 18 0 "alpha = 0;" "6#/% &alphaG\"\"!" }{TEXT -1 5 " y " }{XPPEDIT 18 0 "beta = 1/3;" "6#/%%b etaG*&\"\"\"F&\"\"$!\"\"" }{TEXT -1 9 " , o sea" }}{PARA 4 "" 0 "" {TEXT -1 14 "para el m\351todo" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 4 "" 0 "" {TEXT 270 22 " 0 | 0" }}{PARA 4 "" 0 " " {TEXT 271 4 " " }{XPPEDIT 18 0 "2/3;" "6#*&\"\"#\"\"\"\"\"$!\"\" " }{TEXT 272 14 " | " }{XPPEDIT 18 0 "1/3;" "6#*&\"\"\"F$\" \"$!\"\"" }{TEXT 276 10 " " }{XPPEDIT 18 0 "1/3;" "6#*&\"\"\" F$\"\"$!\"\"" }{TEXT 277 10 " " }}{PARA 4 "" 0 "" {TEXT -1 12 " ---------" }{TEXT 268 44 "------------------------------------- -------" }}{PARA 4 "" 0 "" {TEXT 269 21 " | " } {XPPEDIT 18 0 "1/4;" "6#*&\"\"\"F$\"\"%!\"\"" }{TEXT 274 10 " \+ " }{XPPEDIT 18 0 "3/4;" "6#*&\"\"$\"\"\"\"\"%!\"\"" }{TEXT 275 1 " " }}{PARA 4 "" 0 "" {TEXT -1 61 "que suele recibir el nombre espec\355fi co de RADAU I de 2 etapas" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {MPLTEXT 0 21 30 "# fu ncion de estabilidad para " }{XPPEDIT 18 0 "alpha = 0;" "6#/%&alphaG\" \"!" }{MPLTEXT 0 21 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 " restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "interface(label ing=false):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "alpha:=0:" }} }{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 9 "alpha:=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&alphaG \"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "ka1:=unapply(f(x0+ alpha*h,y0+alpha*h*kp1(h)),h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$k a1Gf*6#%\"hG6\"6$%)operatorG%&arrowGF(*&%'lambdaG\"\"\"%#y0GF.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "ka2:=unapply(f(x0+2*beta* h,y0+beta*h*kp1(h)+beta*h*kp2(h)),h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ka2Gf*6#%\"hG6\"6$%)operatorG%&arrowGF(*&%'lambdaG\"\"\",(%#y0GF .*(%%betaGF.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*&%'lambd aG\"\"\"%#y0GF*/%#k2G,$**F)F*F+F*,&F*F**(F)F*%%betaGF*%\"hGF*F*F*,&F*! \"\"F1F*F5F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "k1:=subs(te st,k1);k2:=subs(test,k2);simplify(k1+3*k2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k1G*&%'lambdaG\"\"\"%#y0GF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k2G,$**%'lambdaG\"\"\"%#y0GF(,&F(F(*(F'F(%%betaGF(% \"hGF(F(F(,&F(!\"\"F+F(F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*,\" \"#\"\"\"%'lambdaGF&%#y0GF&,&F%F&*(F'F&%%betaGF&%\"hGF&F&F&,&F&!\"\"F* F&F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "y1:=expand(y0+h*( (1/4)*k1+(3/4)*k2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G,*%#y0G \"\"\"**\"\"%!\"\"%\"hGF'%'lambdaGF'F&F'F'*.\"\"$F'F)F*F+F'F,F'F&F',&F 'F**(F,F'%%betaGF'F+F'F'F*F**0F.F'F)F*F,\"\"#F&F'F/F*F1F'F+F3F*" }}} {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'**\"\"$F'F)F*F+F',&F'F**&F+F'%%betaGF'F'F*F**,F-F'F)F*F+ \"\"#F.F*F0F'F*F'%#y0GF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "r:=unapply(collect(simplify(y1/y0),z),z):r_general:=r(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*r_generalG,$*(\"\"#!\"\",(F'\"\"\"*&)%\"z GF'F*%%betaGF*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 0 21 21 "# A-estabilidad para " } {XPPEDIT 18 0 "alpha = 0;" "6#/%&alphaG\"\"!" }{MPLTEXT 0 21 1 " " }}} {EXCHG {PARA 4 "" 0 "" {TEXT 278 40 "Estos m\351todos (incluyendo el y a citado " }{TEXT -1 19 "RADAU I de 2 etapas" }{TEXT 279 30 " nunca \+ pueden ser A-estables," }}{PARA 4 "" 0 "" {TEXT 280 84 "ya que el nume rador de r(z) es de grado 2 mientras que el denominador lo es de 1 \+ " }}{PARA 4 "" 0 "" {TEXT 281 25 "La excepci\363n podr\355a ser " } {XPPEDIT 18 0 "beta = 0;" "6#/%%betaG\"\"!" }{TEXT 282 53 " En este \+ \372ltimo caso, la funci\363n de estabilidad vale" }}{PARA 4 "" 0 "" {TEXT 283 11 " r(z)=1+z" }}{PARA 4 "" 0 "" {TEXT 284 88 "por lo que \+ tampoco se tiene A-estabilidad. Se observar\341 que este \372ltimo m \351todo concreto," }}{PARA 4 "" 0 "" {TEXT 285 13 "para el que " } {XPPEDIT 18 0 "alpha = 0;" "6#/%&alphaG\"\"!" }{TEXT 286 6 " y " } {XPPEDIT 18 0 "beta = 0;" "6#/%%betaG\"\"!" }{TEXT 287 28 " es el m \351todo de tablero " }}{PARA 4 "" 0 "" {TEXT 293 15 " 0 | \+ 0" }}{PARA 4 "" 0 "" {TEXT 290 24 " 0 | 0 0 " }}{PARA 4 "" 0 "" {TEXT -1 12 " ---------" }{TEXT 288 10 "----------" }} {PARA 4 "" 0 "" {TEXT 289 14 " | " }{XPPEDIT 18 0 "1/4;" "6 #*&\"\"\"F$\"\"%!\"\"" }{TEXT 291 4 " " }{XPPEDIT 18 0 "3/4;" "6#*& \"\"$\"\"\"\"\"%!\"\"" }{TEXT 292 1 " " }}{PARA 4 "" 0 "" {TEXT -1 57 "que es de hecho el m\351todo expl\355cito de EULER (de 1 etapa)" }}}} {MARK "0 0 0" 7 }{VIEWOPTS 1 1 0 3 4 1802 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }