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algorithm2e
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\end_header

\begin_body

\begin_layout Section
Iteración de punto fijo
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
sremember{CN}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Dado un espacio métrico 
\begin_inset Formula $(X,d)$
\end_inset

, 
\begin_inset Formula $f:X\to X$
\end_inset

 es 
\series bold
contractiva
\series default
 si existe 
\begin_inset Formula $c\in(0,1)$
\end_inset

 tal que 
\begin_inset Formula $\forall x,y\in X,d(f(x),f(y))\leq cd(x,y)$
\end_inset

, y decimos que 
\begin_inset Formula $c$
\end_inset

 es una 
\series bold
constante de contractividad
\series default
 de 
\begin_inset Formula $f$
\end_inset

.
 [...] 
\series bold
Teorema del punto fijo de Banach:
\series default
 Si 
\begin_inset Formula $(X,d)$
\end_inset

 es completo y 
\begin_inset Formula $f:X\to X$
\end_inset

 es contractiva con constante 
\begin_inset Formula $c$
\end_inset

, existe un único punto fijo 
\begin_inset Formula $\xi\in X$
\end_inset

 y la sucesión dada por 
\begin_inset Formula $x_{0}$
\end_inset

 arbitrario y 
\begin_inset Formula $x_{n+1}=f(x)$
\end_inset

 converge a 
\begin_inset Formula $\xi$
\end_inset

 con
\begin_inset Formula 
\[
d(x_{n},\xi)\leq\frac{c^{n}}{1-c}d(x_{0},x_{1}).
\]

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
eremember
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
sremember{FVV1}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Definimos la norma de una aplicación 
\begin_inset Formula $L:(E,\Vert\cdot\Vert)\rightarrow(F,\Vert\cdot\Vert')$
\end_inset

 como 
\begin_inset Formula 
\[
\Vert L\Vert:=\text{[...]}\sup\{\Vert L(x)\Vert'\}_{x\in E,\Vert x\Vert\leq1}
\]

\end_inset

[...] El 
\series bold
teorema del incremento finito
\series default
 afirma que, sean 
\begin_inset Formula $f:\Omega\subseteq\mathbb{R}^{m}\rightarrow\mathbb{R}^{n}$
\end_inset

, 
\begin_inset Formula $a,b\in\Omega$
\end_inset

 con el segmento 
\begin_inset Formula $[a,b]\subseteq\Omega$
\end_inset

 y 
\begin_inset Formula $M>0$
\end_inset

, si 
\begin_inset Formula $\Vert df(x)\Vert\leq M$
\end_inset

 para todo 
\begin_inset Formula $x\in[a,b]$
\end_inset

 se tiene 
\begin_inset Formula $\Vert f(b)-f(a)\Vert\leq M\Vert b-a\Vert$
\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
eremember
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Sean 
\begin_inset Formula $\Omega\subseteq\mathbb{R}^{n}$
\end_inset

 abierto, 
\begin_inset Formula $f:\Omega\to\mathbb{R}^{n}$
\end_inset

 
\begin_inset Formula ${\cal C}^{1}$
\end_inset

 y 
\begin_inset Formula $x\in\Omega$
\end_inset

 un punto fijo de 
\begin_inset Formula $f$
\end_inset

 con 
\begin_inset Formula $\Vert df(x)\Vert<1$
\end_inset

, existe 
\begin_inset Formula $\delta>0$
\end_inset

 tal que 
\begin_inset Formula $f(\overline{B}(x,\delta))\subseteq\Omega$
\end_inset

, y toda sucesión 
\begin_inset Formula $(x_{n})_{n}$
\end_inset

 dada por 
\begin_inset Formula $x_{0}\in\overline{B}(x,\delta)$
\end_inset

 y 
\begin_inset Formula $x_{k+1}\coloneqq f(x_{k})$
\end_inset

 converge.
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{samepage}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Sean 
\begin_inset Formula $R\coloneqq [a_{1},b_{1}]\times\dots\times[a_{n},b_{n}]$
\end_inset

, 
\begin_inset Formula $f:R\to R$
\end_inset

 diferenciable y 
\begin_inset Formula $K\in(0,1)$
\end_inset

 tal que
\begin_inset Formula 
\[
\left|\frac{\partial f_{i}}{\partial x_{j}}(x)\right|\leq\frac{K}{n}
\]

\end_inset

para cada 
\begin_inset Formula $x\in R$
\end_inset

 e 
\begin_inset Formula $i,j\in\{1,\dots,n\}$
\end_inset

, entonces 
\begin_inset Formula $\Vert df\Vert_{\infty}\leq K$
\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{samepage}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
La 
\series bold
aceleración de Gauss-Seidel
\series default
 de una iteración de punto fijo consiste en considerar, en vez de 
\begin_inset Formula $x_{k+1}\coloneqq g(x_{k})$
\end_inset

,
\begin_inset Formula 
\begin{eqnarray*}
x_{(k+1)1} & := & g_{1}(x_{k1},\dots,x_{kn}),\\
x_{(k+1)2} & := & g_{2}(x_{(k+1)1},x_{k2},\dots,x_{kn}),\\
 & \vdots\\
x_{(k+1)n} & := & g_{n}(x_{(k+1)1},\dots,x_{(k+1)(n-1)},x_{kn}).
\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Section
Método de Newton
\end_layout

\begin_layout Standard
Consideremos una ecuación 
\begin_inset Formula $f(x)=0$
\end_inset

 con 
\begin_inset Formula $f:\mathbb{R}^{n}\to\mathbb{R}^{n}$
\end_inset

 diferenciable.
 Sustituyendo 
\begin_inset Formula $f$
\end_inset

 por su polinomio de Taylor de primer orden en 
\begin_inset Formula $x_{0}$
\end_inset

 tenemos 
\begin_inset Formula $f(x_{0})+df(x_{0})(x-x_{0})=0$
\end_inset

, lo que nos da una aproximación de 
\begin_inset Formula $x$
\end_inset

.
\end_layout

\begin_layout Standard
Como 
\series bold
teorema
\series default
, si 
\begin_inset Formula $f:\Omega\subseteq\mathbb{R}^{n}\to\mathbb{R}^{n}$
\end_inset

 es 
\begin_inset Formula ${\cal C}^{2}$
\end_inset

, 
\begin_inset Formula $\xi\in\Omega$
\end_inset

 es un cero de 
\begin_inset Formula $f$
\end_inset

 y 
\begin_inset Formula $df(\xi)$
\end_inset

 es no singular, existe 
\begin_inset Formula $r>0$
\end_inset

 tal que, para todo 
\begin_inset Formula $x\in B(\xi,r)$
\end_inset

, la sucesión dada por 
\begin_inset Formula $x_{0}\coloneqq x$
\end_inset

 y 
\begin_inset Formula $x_{k+1}\coloneqq x_{k}-df(x_{k})^{-1}f(x_{k})$
\end_inset

 converge a 
\begin_inset Formula $\xi$
\end_inset

, y existe 
\begin_inset Formula $M>0$
\end_inset

 tal que 
\begin_inset Formula 
\[
\Vert x_{k+1}-\xi\Vert\leq M\Vert x_{k}-\xi\Vert^{2}.
\]

\end_inset


\end_layout

\begin_layout Standard

\series bold
Demostración
\series default
 para 
\begin_inset Formula $n=2$
\end_inset


\series bold
:
\series default
 Queremos ver que 
\begin_inset Formula $g(x)\coloneqq x-df(x)^{-1}f(x)$
\end_inset

 es contractiva cerca de 
\begin_inset Formula $\xi$
\end_inset

, para lo que basta ver que es 
\begin_inset Formula ${\cal C}^{1}$
\end_inset

 y que 
\begin_inset Formula $dg(\xi)=0$
\end_inset

.
 Como 
\begin_inset Formula $f$
\end_inset

 es 
\begin_inset Formula ${\cal C}^{2}$
\end_inset

, 
\begin_inset Formula $\det\circ df$
\end_inset

 es continua y diferenciable en 
\begin_inset Formula $\Omega$
\end_inset

, y en particular, en un entorno de 
\begin_inset Formula $\xi$
\end_inset

, las diferenciales tienen inversa
\begin_inset Formula 
\[
df(x)^{-1}\equiv\frac{1}{\det df(x)}\begin{pmatrix}\frac{\partial f_{2}}{\partial x_{2}}(x) & -\frac{\partial f_{1}}{\partial x_{2}}(x)\\
\frac{\partial f_{2}(x)}{\partial x_{1}}(x) & \frac{\partial f_{1}(x)}{\partial x_{1}}(x)
\end{pmatrix}=:\begin{pmatrix}a_{11}(x) & a_{12}(x)\\
a_{21}(x) & a_{22}(x)
\end{pmatrix},
\]

\end_inset

y 
\begin_inset Formula $a_{11},a_{12},a_{21},a_{22}$
\end_inset

 son 
\begin_inset Formula ${\cal C}^{1}$
\end_inset

.
 Entonces
\begin_inset Formula 
\[
g(x,y):=\begin{pmatrix}x-a_{11}(x,y)f_{1}(x,y)-a_{12}(x,y)f_{2}(x,y)\\
y-a_{21}(x,y)f_{1}(x,y)-a_{22}(x,y)f_{2}(x,y)
\end{pmatrix}.
\]

\end_inset

Las derivadas parciales de 
\begin_inset Formula $g_{1}$
\end_inset

 son
\begin_inset Formula 
\begin{eqnarray*}
\frac{\partial g_{1}}{\partial x_{1}}(x) & = & 1-\frac{\partial a_{11}}{\partial x_{1}}(x)f_{1}(x)-a_{11}(x)\frac{\partial f_{1}}{\partial x_{1}}(x)-\frac{\partial a_{12}}{\partial x_{1}}(x)f_{2}(x)-a_{12}(x)\frac{\partial f_{2}}{\partial x_{1}}(x),\\
\frac{\partial g_{1}}{\partial x_{2}}(x) & = & -\frac{\partial a_{11}}{\partial x_{2}}(x)f_{1}(x)-a_{11}(x)\frac{\partial f_{1}}{\partial x_{2}}(x)-\frac{\partial a_{12}}{\partial x_{2}}(x)f_{2}(x)-a_{12}(x)\frac{\partial f_{2}}{\partial x_{2}}(x).
\end{eqnarray*}

\end_inset

Así, como 
\begin_inset Formula $f(\xi)=0$
\end_inset

,
\begin_inset Formula 
\begin{eqnarray*}
\frac{\partial g_{1}}{\partial x_{1}}(\xi) & = & 1-a_{11}(\xi)\frac{\partial f_{1}}{\partial x_{1}}(\xi)-a_{12}(\xi)\frac{\partial f_{2}}{\partial x_{1}}(\xi)\\
 & = & 1-\frac{1}{\det df(x)}\left(\frac{\partial f_{2}}{\partial x_{2}}(\xi)\frac{\partial f_{1}}{\partial x_{1}}(\xi)-\frac{\partial f_{1}}{\partial x_{2}}(\xi)\frac{\partial f_{2}}{\partial x_{1}}(\xi)\right)=1-1=0;\\
\frac{\partial g_{1}}{\partial x_{2}}(\xi) & = & -a_{11}(\xi)\frac{\partial f_{1}}{\partial x_{2}}(\xi)-a_{12}(\xi)\frac{\partial f_{2}}{\partial x_{2}}(\xi)\\
 & = & -\frac{1}{\det df(x)}\left(\frac{\partial f_{2}}{\partial x_{2}}(\xi)\frac{\partial f_{1}}{\partial x_{2}}(\xi)-\frac{\partial f_{1}}{\partial x_{2}}(\xi)\frac{\partial f_{2}}{\partial x_{2}}(\xi)\right)=0.
\end{eqnarray*}

\end_inset

Para 
\begin_inset Formula $g_{2}$
\end_inset

 es análogo, y solo queda ver el orden de convergencia.
 Dados un abierto conexo 
\begin_inset Formula $C\subseteq\mathbb{R}^{n}$
\end_inset

 y 
\begin_inset Formula $f:C\to\mathbb{R}^{n}$
\end_inset

, si existe 
\begin_inset Formula $K>0$
\end_inset

 tal que para 
\begin_inset Formula $x,y\in C$
\end_inset

, 
\begin_inset Formula $\Vert df(x)-df(y)\Vert\leq K\Vert x-y\Vert$
\end_inset

, entonces, para 
\begin_inset Formula $x,y\in C$
\end_inset

, 
\begin_inset Formula 
\[
\Vert f(x)-f(y)-df(y)(x-y)\Vert\leq\frac{K}{2}\Vert x-y\Vert^{2}.
\]

\end_inset

 En efecto, sea 
\begin_inset Formula $\varphi:[0,1]\to C$
\end_inset

 dada por 
\begin_inset Formula $\varphi(t)\coloneqq f(y+t(x-y))$
\end_inset

, por la regla de la cadena, 
\begin_inset Formula $\varphi'(t)=df(y+t(x-y))(x-y)$
\end_inset

, por lo que 
\begin_inset Formula 
\begin{align*}
\Vert\varphi'(t)-\varphi'(0)\Vert & =\Vert(df(y+t(x-y))-df(y))(x-y)\Vert\leq\\
 & \leq\Vert df(y+t(x-y))-df(y)\Vert\Vert x-y\Vert\leq Kt\Vert x-y\Vert^{2},
\end{align*}

\end_inset

y como además 
\begin_inset Formula 
\[
\Delta:=f(x)-f(y)-df(y)(x-y)=\varphi(1)-\varphi(0)-\varphi'(0)=\int_{0}^{1}(\varphi'(t)-\varphi'(0))dt,
\]

\end_inset

queda
\begin_inset Formula 
\[
\Vert\Delta\Vert\leq\int_{0}^{1}\Vert\varphi'(t)-\varphi'(0)\Vert dt\leq K\Vert x-y\Vert^{2}\int_{0}^{1}t\,dt=\frac{K}{2}\Vert x-y\Vert^{2}.
\]

\end_inset

Cuando esto se cumple,
\begin_inset Formula 
\begin{align*}
\Vert x_{k+1}-\xi\Vert & =\Vert(x_{k}-\xi)-df(x_{k})^{-1}f(x_{k})\Vert=\Vert df(x_{k})^{-1}(df(x_{k})(x_{k}-\xi)-f(x_{k}))\Vert\leq\\
 & \leq\Vert df(x_{k})^{-1}\Vert\Vert df(x_{k})(x_{k}-\xi)-f(x_{k})\Vert\overset{f(\xi)=0}{=}\\
 & =\Vert df(x_{k})^{-1}\Vert\Vert f(\xi)-f(x_{k})-df(x_{k})(\xi-x_{k})\Vert\leq\Vert df(x_{k})^{-1}\Vert\frac{K}{2}\Vert x_{k}-\xi\Vert^{2},
\end{align*}

\end_inset

y tomando 
\begin_inset Formula $M\coloneqq \frac{K}{2}\sup_{x\in B(\xi,r)}\Vert df(x)^{-1}\Vert$
\end_inset

 se obtiene la acotación.
\end_layout

\begin_layout Section
Método de Broyden
\end_layout

\begin_layout Standard
El método de Newton requiere calcular la matriz diferencial en cada iteración
 (
\begin_inset Formula $\Theta(n^{2})$
\end_inset

 si se usa derivación numérica y evaluar 
\begin_inset Formula $f$
\end_inset

 en un punto es 
\begin_inset Formula $\Theta(1)$
\end_inset

) y resolver un sistema lineal (
\begin_inset Formula $\Theta(n^{3})$
\end_inset

).
 Para reducir el orden de complejidad, podemos aproximar la diferencial
 con una matriz 
\begin_inset Formula $A_{k}$
\end_inset

 tal que 
\begin_inset Formula $A_{k}(x_{k}-x_{k-1})=f(x_{k})-f(x_{k-1})$
\end_inset

, con lo que solo necesitamos la diferencial para obtener 
\begin_inset Formula $x_{1}$
\end_inset

.
 A cambio, pasamos de orden de convergencia cuadrático a supralineal.
\end_layout

\begin_layout Standard
Podemos usar 
\begin_inset Formula 
\[
A_{k}:=A_{k-1}+\frac{1}{\Vert x_{k}-x_{k-1}\Vert_{2}^{2}}f(x_{k})(x_{k}-x_{k-1})^{t},
\]

\end_inset

tomando 
\begin_inset Formula $A_{0}\coloneqq df(x_{0})$
\end_inset

.
 En efecto, como 
\begin_inset Formula $x_{k}=x_{k-1}-A_{k-1}^{-1}f(x_{k-1})$
\end_inset

, tenemos 
\begin_inset Formula $A_{k-1}(x_{k}-x_{k-1})=f(x_{k-1})$
\end_inset

, y por tanto
\begin_inset Formula 
\begin{multline*}
\left(A_{k-1}+\frac{1}{\Vert x_{k}-x_{k-1}\Vert_{2}^{2}}f(x_{k})(x_{k}-x_{k-1})^{t}\right)(x_{k}-x_{k-1})=\\
=A_{k-1}(x_{k}-x_{k-1})+f(x_{k})=f(x_{k})-f(x_{k-1}).
\end{multline*}

\end_inset


\end_layout

\begin_layout Standard
Para calcular la inversa de 
\begin_inset Formula $A_{k}$
\end_inset

 podemos usar la 
\series bold
fórmula de Sherman-Morrison
\series default
, 
\begin_inset Formula 
\[
(A+vy^{t})^{-1}=A^{-1}-\frac{1}{1+y^{t}A^{-1}v}(A^{-1}v)(y^{t}A^{-1}),
\]

\end_inset

suponiendo que 
\begin_inset Formula $y^{t}A^{-1}x\neq-1$
\end_inset

.
 Esto reduce el orden de complejidad de cada iteración a 
\begin_inset Formula $\Theta(n^{2})$
\end_inset

 en el caso general, asumiendo que evaluar 
\begin_inset Formula $f$
\end_inset

 en un punto es 
\begin_inset Formula $O(n^{2})$
\end_inset

.
\end_layout

\begin_layout Section
Método del descenso rápido
\end_layout

\begin_layout Standard
\begin_inset Float algorithm
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
Entrada{Función $f:
\backslash
mathbb{R}^n
\backslash
to
\backslash
mathbb{R}^n$, aproximación inicial $x
\backslash
in
\backslash
mathbb{R}^n$, margen de error $e>0$ y tolerancia $t>0$.}
\end_layout

\begin_layout Plain Layout


\backslash
Salida{$x$ tal que $f(x)
\backslash
approx 0$, o error indicando que no se puede mejorar la aproximación.}
\end_layout

\begin_layout Plain Layout

Definir $g(x):=
\backslash
sum_{k=1}^n f_k(x)^2$
\backslash
;
\end_layout

\begin_layout Plain Layout


\backslash
Mientras{$g(x)>e$}{
\end_layout

\begin_layout Plain Layout

	$u
\backslash
gets{
\backslash
nabla g(x) 
\backslash
over 
\backslash
Vert
\backslash
nabla g(x)
\backslash
Vert}$
\backslash
;
\end_layout

\begin_layout Plain Layout

	$
\backslash
alpha
\backslash
gets 1$
\backslash
;
\end_layout

\begin_layout Plain Layout

	
\backslash
lMientras{$g(x-
\backslash
alpha u) > g(x)$}{$
\backslash
alpha
\backslash
gets
\backslash
frac12
\backslash
alpha$}
\end_layout

\begin_layout Plain Layout

	
\backslash
lSSi{$|
\backslash
alpha|<t$}{
\backslash
Devolver error}
\end_layout

\begin_layout Plain Layout

	
\backslash
tcp{{
\backslash
rm Interpolar en $(0,g(x)),(
\backslash
frac
\backslash
alpha2,g(x+
\backslash
frac
\backslash
alpha2u)),(
\backslash
alpha,g(x+
\backslash
alpha u))$.}}
\end_layout

\begin_layout Plain Layout

	$g_0
\backslash
gets g(x)$
\backslash
;
\end_layout

\begin_layout Plain Layout

	$g_1
\backslash
gets g(x+
\backslash
frac
\backslash
alpha2u)$
\backslash
;
\end_layout

\begin_layout Plain Layout

	$g_2
\backslash
gets g(x+
\backslash
alpha u)$
\backslash
;
\end_layout

\begin_layout Plain Layout

	$h_0
\backslash
gets {2(g_1 - g_0) 
\backslash
over 
\backslash
alpha}$
\end_layout

\begin_layout Plain Layout

		
\backslash
tcp*{{
\backslash
rm Usamos diferencias divididas de Newton.}}
\end_layout

\begin_layout Plain Layout

	$h_1
\backslash
gets {2(g_2 - g_1) 
\backslash
over 
\backslash
alpha}$
\backslash
;
\end_layout

\begin_layout Plain Layout

	$h_{01}
\backslash
gets {h_2 - h_1 
\backslash
over 
\backslash
alpha}$
\backslash
;
\end_layout

\begin_layout Plain Layout

	
\backslash
tcp{{
\backslash
rm Obtener y usar el vértice de la parábola resultante.}}
\end_layout

\begin_layout Plain Layout

	$
\backslash
alpha_0 
\backslash
gets 
\backslash
frac12 (
\backslash
frac
\backslash
alpha2 - 
\backslash
frac{h_0}{h_{01}})$
\backslash
;
\end_layout

\begin_layout Plain Layout

	
\backslash
lSSi{$g(x+
\backslash
alpha_0u) < g_2$}{$x 
\backslash
gets x + 
\backslash
alpha_0u$}
\end_layout

\begin_layout Plain Layout

	
\backslash
lEnOtroCaso{$x 
\backslash
gets x + 
\backslash
alpha u$}
\end_layout

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "alg:steepest-descent"

\end_inset

Método del descenso rápido.
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
El método del descenso rápido o 
\emph on
\lang english
steepest descent
\emph default
\lang spanish
 es el algoritmo 
\begin_inset CommandInset ref
LatexCommand ref
reference "alg:steepest-descent"
plural "false"
caps "false"
noprefix "false"

\end_inset

, y consiste en minimizar la función 
\begin_inset Formula $g(x)\coloneqq \Vert f(x)\Vert_{2}^{2}$
\end_inset

 desplazándonos, en cada iteración, en la dirección de mayor descenso en
 esta función.
 No es un método muy rápido, pero permite una estimación inicial más lejana
 que los métodos de Newton y Broyden, por lo que se suele usar para obtener
 una aproximación no muy fina que a su vez se usa como punto de partida
 para estos métodos.
\end_layout

\end_body
\end_document