Velocidad de convergencia

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+
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{xi }i=0 !
x i ≠ x * ∀i &
∞
-
!
,
/!
0
lim
i→∞
!
x i+1 − x *
xi − x *
-.
p
,
β
β
=β
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1
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{xi }i=0
∞
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;
+
4-5 !
!
! 0
4-5
4= 4-.5
x0 ∈ I
&
/!
3 ( 4= 3
!
9
< -.
2 '4
5
/!
/! #
4
& '& 666& > 5 &
!
∞
x
=
g(
x
)
{ i+1
i }i = 0
,
6
,
4-.5 ≠ (
!
!
-.&
)
Velocidad de convergencia (4)
$$
#
;
xi +1 = g(xi ) = g(x * +(xi − x *)) =
0
0
0
(xi − x*)2
(xi − x*)p −1 (p −1
= g(x*) + (xi − x*).g'(x*) +
.g"(x *) + .... +
.g (x *) +
2!
(p − 1)!
(xi − x*)p (p
(xi − x*)p +1 (p +1
.g (x*) +
.g ( x * +θ.(xi − x *) )
+
p!
(p + 1)!
θ ∈ [0,1]
(xi − x*)p (p
(xi − x*)p +1 (p +1
= x* +
.g (x*) +
.g ( x * +θ.(xi − x *) )
p!
(p + 1)!
?
$$
xi+1 = x*
(xi − x*)p (p
(xi − x*)p +1 (p +1
+
.g (x*) +
.g ( x * +θ.(xi − x *) )
p!
(p + 1)!
(xi − x*)p (p
(xi − x*)p +1 (p +1
(xi +1 − x*) =
.g (x*) +
.g ( x * +θ.(xi − x *) )
p!
(p + 1) !
(xi +1 − x*) 1 (p
(xi − x*) (p +1
=
.g
(
x*
)
+
.g ( x * +θ.(xi − x*) )
p
p!
(p + 1)!
(xi − x*)
lim
i →∞
xi +1 − x *
xi − x *
p
=
1 (p
.g ( x*)
p!
=β
c.q.d.
@
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