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{SECT 0 {EXCHG {PARA 4 "" 0 "" {TEXT 283 14 "Ejercicios de " }}{PARA 
4 "" 0 "" {TEXT 284 22 "(8) Estabilidad lineal" }}{PARA 4 "" 0 "" 
{TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 260 51 "Ejercicio 08-26   (del E
XAMEN ORDINARIO DE 03JUL10)" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 
4 "" 0 "" {TEXT -1 75 "Consid\351rese la familia de m\351todos impl
\355citos de RUNGE-KUTTA con dos etapas " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 4 "" 0 "" {TEXT 261 5 "     " }{XPPEDIT 18 0 "0;" "6#\"\"
!" }{TEXT 262 13 "       |     " }{XPPEDIT 18 0 "0;" "6#\"\"!" }{TEXT 
263 2 "  " }}{PARA 4 "" 0 "" {TEXT 264 5 "     " }{XPPEDIT 18 0 "C;" "
6#%\"CG" }{TEXT 265 11 "     |     " }{XPPEDIT 18 0 "A[1];" "6#&%\"AG6
#\"\"\"" }{TEXT 271 9 "         " }{XPPEDIT 18 0 "A[2];" "6#&%\"AG6#\"
\"#" }}{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 81 "a) B\372squese el mayor orden alc
anzable por los miembros de esta familia, indicando" }}{PARA 4 "" 0 "
" {TEXT -1 15 "los valores de " }{XPPEDIT 18 0 "C;" "6#%\"CG" }{TEXT 
-1 3 " , " }{XPPEDIT 18 0 "A[1];" "6#&%\"AG6#\"\"\"" }{TEXT -1 4 " y  \+
" }{XPPEDIT 18 0 "A[2];" "6#&%\"AG6#\"\"#" }{TEXT -1 26 "  para los qu
e se consigue" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" 
{TEXT -1 40 "(hasta aqu\355 es Ejercicio del cap\355tulo 7)" }}{PARA 
4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 57 "b) Calc\372lese
 la funci\363n de estabilidad en el caso general" }}{PARA 4 "" 0 "" 
{TEXT -1 63 "c) B\372squese un ejemplo de m\351todo A-estable dentro d
e la familia" }}{PARA 4 "" 0 "" {TEXT -1 24 "de m\351todos del problem
a." }}{PARA 4 "" 0 "" {TEXT -1 63 "d) Compru\351bese si es o no es A-e
stable el m\351todo con valores   " }{XPPEDIT 18 0 "A[1];" "6#&%\"AG6#
\"\"\"" }{TEXT -1 10 "= 2/3  y  " }{XPPEDIT 18 0 "A[2];" "6#&%\"AG6#\"
\"#" }{TEXT -1 2 "=2" }}{PARA 4 "" 0 "" {TEXT -1 20 "de los par\341met
ros ( " }{XPPEDIT 18 0 "C;" "6#%\"CG" }{TEXT -1 70 "  no influye, clar
o) y dib\372jese la regi\363n de estabilidad en este caso." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 26 "interface(labeling=false):" }}}{EXCHG {PARA 4 "" 0 
"" {TEXT -1 2 "a)" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{MPLTEXT 0 21 37 "# desarrollo de la verdadera solucion" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "#y:=unapply(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)),dif
f(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:=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 "# desarrollo del m\351t
odo" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "k10:=unapply(f(x0,y0
),h):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "k20:=unapply(f(x0+
C*h,y0+A[1]*h*kp10(h)+A[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 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)=k
200\},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:=exp
and((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(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 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)=k
210,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)=k1
00,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(f
0,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))=coeff((1/2)*f1,D[1](f)(x0,y0));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'cond2aG/,$*(\"\"$\"\"\"\"\"%!\"\"%\"CGF)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,y0)
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond2bG/,&*&#\"\"$\"\"%\"\"\"
&%\"AG6#F+F+F+*&F(F+&F-6#\"\"#F+F+#F+F2" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "orden2:=solve(\{cond2a,cond2b\});" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%'orden2G<%/&%\"AG6#\"\"#,&&F(6#\"\"\"!\"\"#F*\"\"$F
./%\"CGF0/F,F," }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 29 "# condicio
nes para el orden 3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "cond
3a:=coeff(Fi2,D[1,1](f)(x0,y0))=coeff((1/3)*f2,D[1,1](f)(x0,y0));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3aG/,$*(\"\"$\"\"\"\"\"%!\"\"%
\"CG\"\"#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#>%'cond3b
G/,&*&#\"\"$\"\"#\"\"\"*&%\"CGF+&%\"AG6#F+F+F+F+*&F(F+*&F-F+&F/6#F*F+F
+F+#F*F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "cond3c:=coeff(
coeff(Fi2,D[2,2](f)(x0,y0)),f(x0,y0)^2)=coeff(coeff((1/3)*f2,D[2,2](f)
(x0,y0)),f(x0,y0)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3cG/,(
*&#\"\"$\"\"%\"\"\"*$)&%\"AG6#F+\"\"#F+F+F+*&#F)F1F+*&F.F+&F/6#F1F+F+F
+*&F(F+*$)F5F1F+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(coe
ff((1/3)*f2,D[2](f)(x0,y0)),D[1](f)(x0,y0));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'cond3dG/,$*&#\"\"$\"\"#\"\"\"*&%\"CGF+&%\"AG6#F*F+F+
F+#F+F)" }}}{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/,&*&#\"
\"$\"\"#\"\"\"*&&%\"AG6#F+F+&F.6#F*F+F+F+*&F(F+*$)F0F*F+F+F+#F+F)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "orden3:=solve(\{cond2a,cond2
b,cond3a,cond3b,cond3c,cond3d,cond3e\});" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%'orden3G<%/%\"CG#\"\"#\"\"$/&%\"AG6#\"\"\"#F/F*/&F-6#F)F0" }}}
{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 36 "se obtiene orden 3 para los v
alores " }{XPPEDIT 18 0 "C;" "6#%\"CG" }{MPLTEXT 0 21 10 " = 2/3 ,  " 
}{XPPEDIT 18 0 "A[1];" "6#&%\"AG6#\"\"\"" }{MPLTEXT 0 21 11 " = 1/3  y
  " }{XPPEDIT 18 0 "A[2];" "6#&%\"AG6#\"\"#" }{MPLTEXT 0 21 9 " = 1/3 \+
  " }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 29 "# condiciones para el \+
orden 4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "cond4a:=coeff(Fi
3,D[1,1,1](f)(x0,y0))=coeff((1/4)*f3,D[1,1,1](f)(x0,y0));" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%'cond4aG/,$*(\"\"$\"\"\"\"\"%!\"\"%\"CGF(F)#F
)F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "cond4b:=coeff(coeff
(Fi3,D[1,1,2](f)(x0,y0)),f(x0,y0))=coeff(coeff((1/4)*f3,D[1,1,2](f)(x0
,y0)),f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond4bG/,&*&#\"
\"*\"\"%\"\"\"*&)%\"CG\"\"#F+&%\"AG6#F+F+F+F+*&F(F+*&F-F+&F16#F/F+F+F+
#\"\"$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(coeff((1/4)*f3,D[1,2,2
](f)(x0,y0)),f(x0,y0)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond4c
G/,(*&#\"\"*\"\"#\"\"\"*(%\"CGF+&%\"AG6#F+F+&F/6#F*F+F+F+*&#F)\"\"%F+*
&F-F+)F.F*F+F+F+*&F4F+*&F-F+)F1F*F+F+F+#\"\"$F5" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 113 "cond4d:=coeff(coeff(Fi3,D[1,2](f)(x0,y0)),D[1
](f)(x0,y0))=coeff(coeff((1/4)*f3,D[1,2](f)(x0,y0)),D[1](f)(x0,y0));" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond4dG/,$*&#\"\"*\"\"#\"\"\"*&)%
\"CGF*F+&%\"AG6#F*F+F+F+#\"\"$\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 145 "cond4e:=coeff(coeff(coeff(Fi3,D[1,2](f)(x0,y0)),D[2]
(f)(x0,y0)),f(x0,y0))=coeff(coeff(coeff((1/4)*f3,D[1,2](f)(x0,y0)),D[2
](f)(x0,y0)),f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond4eG/
,&**\"\"*\"\"\"%\"CGF)&%\"AG6#F)F)&F,6#\"\"#F)F)*(F(F)F*F))F.F0F)F)#\"
\"&\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "cond4f:=coeff(
coeff(Fi3,D[2,2,2](f)(x0,y0)),f(x0,y0)^3)=coeff(coeff((1/4)*f3,D[2,2,2
](f)(x0,y0)),f(x0,y0)^3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond4f
G/,**&#\"\"$\"\"%\"\"\"*$)&%\"AG6#\"\"#F)F+F+F+*&F(F+*$)&F/6#F+F)F+F+F
+*&#\"\"*F*F+*&)F5F1F+F.F+F+F+*&F8F+*&F5F+)F.F1F+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/,&*&#\"\"*\"\"#\"\"\"*(%\"CGF+&%\"AG6#F+F+&F
/6#F*F+F+F+*&F(F+*&F-F+)F1F*F+F+F+#\"\"$\"\"%" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 149 "cond4h:=coeff(coeff(coeff(Fi3,D[2,2](f)(x0,y0))
,D[2](f)(x0,y0)),f(x0,y0)^2)=coeff(coeff(coeff((1/4)*f3,D[2,2](f)(x0,y
0)),D[2](f)(x0,y0)),f(x0,y0)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
'cond4hG/,(*&#\"#F\"\"%\"\"\"*&)&%\"AG6#F+\"\"#F+&F/6#F1F+F+F+*&#F)F1F
+*&F.F+)F2F1F+F+F+*&F(F+*$)F2\"\"$F+F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 113 "cond4i:=coeff(coeff(Fi3,D[1,1](f)(x0,y0)),D[2](f)(
x0,y0))=coeff(coeff((1/4)*f3,D[1,1](f)(x0,y0)),D[2](f)(x0,y0));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond4iG/,$*&#\"\"*\"\"%\"\"\"*&)%\"
CG\"\"#F+&%\"AG6#F/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)=coe
ff(coeff((1/4)*f3,D[1](f)(x0,y0)),D[2](f)(x0,y0)^2);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%'cond4jG/,$*&#\"\"*\"\"#\"\"\"*&%\"CGF+)&%\"AG6#F*
F*F+F+F+#F+\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "cond4k
:=coeff(coeff(Fi3,D[2](f)(x0,y0)^3),f(x0,y0))=coeff(coeff((1/4)*f3,D[2
](f)(x0,y0)^3),f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond4k
G/,&*&#\"\"*\"\"#\"\"\"*$)&%\"AG6#F*\"\"$F+F+F+*&F(F+*&&F/6#F+F+)F.F*F
+F+F+#F+\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "subs(orden
3,\{cond4a,cond4b,cond4c,cond4d,cond4e,cond4f,cond4g,cond4h,cond4i,con
d4j,cond4k\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<'/\"\"\"F%/#\"\"#\"
\"*#F%\"\"%/#F(\"\"$#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 otr
a parte 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 propues
ta" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 68 "Se obtiene orden m\341ximo \+
 3  de una sola manera, para los valores    " }{XPPEDIT 18 0 "C;" "6#%
\"CG" }{TEXT -1 9 "= 2/3 ,  " }{XPPEDIT 18 0 "A[1];" "6#&%\"AG6#\"\"\"
" }{TEXT -1 11 " = 1/3  y  " }{XPPEDIT 18 0 "A[2];" "6#&%\"AG6#\"\"#" 
}{TEXT -1 14 " = 1/3 , o sea" }}{PARA 4 "" 0 "" {TEXT -1 14 "para el m
\351todo" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 268 
22 "    0     |          0" }}{PARA 4 "" 0 "" {TEXT 269 4 "    " }
{XPPEDIT 18 0 "2/3;" "6#*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT 270 14 "     |
        " }{XPPEDIT 18 0 "1/3;" "6#*&\"\"\"F$\"\"$!\"\"" }{TEXT 274 
10 "          " }{XPPEDIT 18 0 "1/3;" "6#*&\"\"\"F$\"\"$!\"\"" }{TEXT 
275 10 "          " }}{PARA 4 "" 0 "" {TEXT -1 12 "   ---------" }
{TEXT 266 44 "--------------------------------------------" }}{PARA 4 
"" 0 "" {TEXT 267 21 "            |        " }{XPPEDIT 18 0 "1/4;" "6#
*&\"\"\"F$\"\"%!\"\"" }{TEXT 272 10 "          " }{XPPEDIT 18 0 "3/4;
" "6#*&\"\"$\"\"\"\"\"%!\"\"" }{TEXT 273 1 " " }}{PARA 4 "" 0 "" 
{TEXT -1 61 "que suele recibir el nombre espec\355fico de RADAU I de 2
 etapas" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 2 "b)" }{MPLTEXT 1 0 0 "" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 0 21 26 "# funcion d
e estabilidad  " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "interface(labeling=fals
e):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f:=(x,y)->lambda*y;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%\"xG%\"yG6\"6$%)operato
rG%&arrowGF)*&%'lambdaG\"\"\"9%F/F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "ka1:=unapply(f(x0,y0),h);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$ka1Gf*6#%\"hG6\"6$%)operatorG%&arrowGF(*&%'lambdaG\"
\"\"%#y0GF.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "ka2:=u
napply(f(x0+C*h,y0+A[1]*h*kp1(h)+A[2]*h*kp2(h)),h);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%$ka2Gf*6#%\"hG6\"6$%)operatorG%&arrowGF(*&%'lambdaG
\"\"\",(%#y0GF.*(&%\"AG6#F.F.9$F.-%$kp1G6#F5F.F.*(&F36#\"\"#F.F5F.-%$k
p2GF8F.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,kp
2(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*&%
\"AG6#F*F*%\"hGF*F*F*,&F*!\"\"*(F)F*&F36#\"\"#F*F5F*F*F7F7" }}}{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,$**%'lambdaG\"\"\"%#y0GF(,&F(F
(*(F'F(&%\"AG6#F(F(%\"hGF(F(F(,&F(!\"\"*(F'F(&F-6#\"\"#F(F/F(F(F1F1" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "y1:=expand(y0+h*((1/4)*k1+
(3/4)*k2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G,*%#y0G\"\"\"**\"
\"%!\"\"%\"hGF'%'lambdaGF'F&F'F'*&#\"\"$F)F'**F+F'F,F'F&F',&F'F**(F,F'
&%\"AG6#\"\"#F'F+F'F'F*F'F**&#F/F)F'*,F,F6F&F'F1F*&F46#F'F'F+F6F'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'*&#\"\"$F)F'*&F+F',&F'F**&F+F'&%\"AG6#\"\"#F'F'F*F'F**&
#F.F)F'*(F+F5F0F*&F36#F'F'F'F*F'%#y0GF'" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 57 "r:=unapply(collect(simplify(y1/y0),z),z):r_general:=r
(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*r_generalG,$*&#\"\"\"\"\"%F
(*&,(F)F(*&,&&%\"AG6#\"\"#!\"\"*&\"\"$F(&F/6#F(F(F(F()%\"zGF1F(F(*&,&*
&F)F(F.F(F2F)F(F(F8F(F(F(,&F(F2*&F8F(F.F(F(F2F(F2" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 43 "r_nulo:=simplify(subs(\{A[2]=3*A[1]\},r(z)))
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'r_nuloG*&,(\"\"\"!\"\"*(\"\"$F
'%\"zGF'&%\"AG6#F'F'F'F+F(F',&F'F(*(F*F'F+F'F,F'F'F(" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 2 "c)
" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 17 "# A-e
stabilidad  " }}}{EXCHG {PARA 4 "" 0 "" {TEXT 276 40 "Estos m\351todos
 (incluyendo el ya citado  " }{TEXT -1 20 "RADAU I de 2 etapas)" }
{TEXT 277 30 "  nunca pueden ser A-estables," }}{PARA 4 "" 0 "" {TEXT 
278 84 "ya que el numerador de  r(z)  es de grado 2 mientras que el de
nominador lo es de  1 " }}{PARA 4 "" 0 "" {TEXT 279 25 "La excepci\363
n podr\355a ser  " }{XPPEDIT 18 0 "A[2] = 3*A[1];" "6#/&%\"AG6#\"\"#*&
\"\"$\"\"\"&F%6#F*F*" }{TEXT 280 53 "  En este \372ltimo caso, la func
i\363n de estabilidad vale" }}{PARA 4 "" 0 "" {TEXT 281 8 "   r(z)=" }
{XPPEDIT 18 0 "(1+(1-3*A[1])*z)/(1-3*A[1]*z);" "6#*&,&\"\"\"F%*&,&F%F%
*&\"\"$F%&%\"AG6#F%F%!\"\"F%%\"zGF%F%F%,&F%F%*(F)F%&F+6#F%F%F.F%F-F-" 
}}{PARA 4 "" 0 "" {TEXT 282 43 "y es en principio de consecuencias dud
osas." }{TEXT -1 0 "" }}}{EXCHG {PARA 4 "" 0 "" {TEXT 285 21 "Si hacem
os entonces  " }{XPPEDIT 18 0 "A[1];" "6#&%\"AG6#\"\"\"" }{TEXT 286 
44 " =1/3  obtenemos como funci\363n de estabilidad" }}{PARA 4 "" 0 "
" {TEXT 288 8 "   r(z)=" }{XPPEDIT 18 0 "1/(1-z);" "6#*&\"\"\"F$,&F$F$
%\"zG!\"\"F'" }{TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 287 66 " y ya sab
emos que dicho m\351todo es A-estable. Se trata del m\351todo  " }}
{PARA 4 "" 0 "" {TEXT 294 15 "      0   |   0" }}{PARA 4 "" 0 "" 
{TEXT 291 6 "      " }{XPPEDIT 18 0 "C;" "6#%\"CG" }{TEXT 295 7 "   | \+
  " }{XPPEDIT 18 0 "1/3;" "6#*&\"\"\"F$\"\"$!\"\"" }{TEXT 296 8 "    1
   " }}{PARA 4 "" 0 "" {TEXT -1 12 "   ---------" }{TEXT 289 10 "-----
-----" }}{PARA 4 "" 0 "" {TEXT 290 14 "          |   " }{XPPEDIT 18 0 
"1/4;" "6#*&\"\"\"F$\"\"%!\"\"" }{TEXT 292 4 "    " }{XPPEDIT 18 0 "3/
4;" "6#*&\"\"$\"\"\"\"\"%!\"\"" }{TEXT 293 1 " " }{TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "r_ejemplo1:=simplify(subs(\{
A[1]=1/3\},r_nulo));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+r_ejemplo1G
,$*&\"\"\"F',&F'!\"\"%\"zGF'F)F)" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 
2 "d)" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 6 "Para  \+
" }{XPPEDIT 18 0 "A[1];" "6#&%\"AG6#\"\"\"" }{TEXT -1 10 "= 2/3  y  " 
}{XPPEDIT 18 0 "A[2];" "6#&%\"AG6#\"\"#" }{TEXT -1 40 "= 2 , que entra
 dentro de la situaci\363n  " }{TEXT 297 1 " " }{XPPEDIT 18 0 "A[2] = \+
3*A[1];" "6#/&%\"AG6#\"\"#*&\"\"$\"\"\"&F%6#F*F*" }{TEXT -1 6 ",  la \+
" }{TEXT 298 25 "funci\363n de estabilidad es" }}{PARA 4 "" 0 "" 
{TEXT 299 8 "   r(z)=" }{XPPEDIT 18 0 "(z-1)/(2*z-1);" "6#*&,&%\"zG\"
\"\"F&!\"\"F&,&*&\"\"#F&F%F&F&F&F'F'" }{TEXT -1 0 "" }}{PARA 4 "" 0 "
" {TEXT -1 14 "              " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 53 "r_ejemplo2:=simplify(subs(\{A[1]=2/3,A[1]=2\}
,r_nulo));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+r_ejemplo2G*&,&\"\"\"
!\"\"%\"zGF'F',&F'F(*&\"\"#F'F)F'F'F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 4 "" 0 "" {TEXT 300 21 "Pongamos   z=x+i y . " }{TEXT -1 
4 "Como" }}{PARA 4 "" 0 "" {TEXT -1 39 "     | 2 z - 1 |^2 = (2 x -1)^
2 + 4 y^2" }}{PARA 4 "" 0 "" {TEXT -1 4 "y   " }}{PARA 4 "" 0 "" 
{TEXT -1 33 "     | z - 1 |^2 = (x -1)^2 + y^2" }}{PARA 4 "" 0 "" 
{TEXT -1 78 "la frontera  | r(z) | = 1  o mejor  | r(z) |^2 = 1  del d
ominio de estabilidad" }}{PARA 4 "" 0 "" {TEXT -1 23 "es la curva de e
cuaci\363n" }}{PARA 4 "" 0 "" {TEXT -1 39 "    (2 x -1)^2 + 4 y^2 = (x
 -1)^2 + y^2" }}{PARA 4 "" 0 "" {TEXT -1 39 "que, tras operar, es la c
ircunferencia " }}{PARA 4 "" 0 "" {TEXT -1 9 "    (x - " }{XPPEDIT 18 
0 "1/3;" "6#*&\"\"\"F$\"\"$!\"\"" }{TEXT -1 13 " )^2 + y^2 = " }
{XPPEDIT 18 0 "1/9;" "6#*&\"\"\"F$\"\"*!\"\"" }{TEXT -1 2 "  " }}
{PARA 4 "" 0 "" {TEXT -1 10 "de centro " }{XPPEDIT 18 0 "1/3;" "6#*&\"
\"\"F$\"\"$!\"\"" }{TEXT -1 11 "  y radio  " }{XPPEDIT 18 0 "1/3;" "6#
*&\"\"\"F$\"\"$!\"\"" }{TEXT -1 56 " , que pasa por  ( 0 , 0 ) . El va
lor  z= - 1  queda en " }}{PARA 4 "" 0 "" {TEXT -1 79 "el exterior de \+
la circunferencia, y, para \351l, r(-1)= 2/3 , loego  -1  pertenece" }
}{PARA 4 "" 0 "" {TEXT -1 53 "al dominio de estabilidad.  El dominio d
e estabilidad" }}{PARA 4 "" 0 "" {TEXT -1 63 "es el exterior de esa ci
rcunferencia, y el m\351todo es  A-estable" }}}}{MARK "0 3 0" 15 }
{VIEWOPTS 1 1 0 3 4 1802 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }