Fixed Point and Steffensen's Acceleration
Author
zekeriya
Last Updated
7 years ago
License
Creative Commons CC BY 4.0
Abstract
Math336 Project
Math336 Project
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\title[Fixed Point and Steffensen's Acceleration]{MATH 336 Presentation \\Fixed Point and Steffensen's Acceleration Method}
\author{Zekeriya Ünal}
\institute{Boğaziçi University}
\date{27/05/2015}
\begin{document}
\begin{frame}
\titlepage
\end{frame}
\begin{frame}[plain]
\tableofcontents
\end{frame}
% Uncomment these lines for an automatically generated outline.
%\begin{frame}{Outline}
% \tableofcontents
%\end{frame}
\section{Fixed Point Iteration Method}
\begin{frame}{Fixed Point Iteration Method}
\begin{block}{Definition}<1->
A point p is a \textbf{fixed point} for a given function g if g(p)= p.
\end{block}
\begin{block}{Remark}<2->
Fixed point problems and root finding problems are infact equivalent.
\begin{itemize}
\item<1->if p is a fixed point of the function g, then p is a root of the function
$$f(x)=[g(x)-x]h(x)$$ [as long as $h(x)\in \mathbb{R}$]
\item<2->if p is a root of the function of f, then p is a fixed point of the function
$$g(x)=x-h(x)f(x)$$ [as long as $h(x)\in \mathbb{R}$]
\end{itemize}
\end{block}
\end{frame}
\begin{frame}{Fixed Point Iteration Method}
\begin{block}{Definition}<1->
Let U be a subset of a metric space X.
\\A function g:U $\rightarrow$X called \textbf{Lipschitz continuous} provided there exists a constant $\lambda\ge$ 0 (called Lipschitz constant)
\\such that $\forall$ x,y$\in$U d(g(x),g(y))$\le$d(x,y)
\\if $\lambda\in$[0,1], then g is called \textbf{contraction} (with contraction constant $\lambda$).
\end{block}
\begin{block}{Theorem (A Fixed Point Theorem)}<2->
Suppose $g:[a,b]\rightarrow[a,b]$ is continuous. Then g has a fixed point.
\end{block}
\end{frame}
\begin{frame}{Fixed Point Iteration Method}
\begin{block}{Lemma}<1->
A contraction has at most one fixed point.
\end{block}
\begin{block}{Corollary}<2->
Suppose $g:[a,b]\rightarrow[a,b]$ is continuous and $\lambda:=sup\mid g'(x)\mid<1$ for $x\in (a,b)$
\\Then g is a contraction with contraction constant $\lambda.$
\end{block}
\vskip 1cm
\end{frame}
\begin{frame}{Graphical determination of the existence of a fixed point for the function $g(x)= \frac{x^2-3}{2}$}
\includegraphics[width=355px, height= 200px]{Figure2.jpg}
\end{frame}
\subsection{Banach Fixed Point Theorem}
\begin{frame}{Banach Fixed Point Theorem}
\begin{theorem}[Banach Fixed Point Theorem]
\begin{itemize}
\item[]<1-> Let U be a complete subset of a metric space X, and let g:U$\rightarrow$U be a contraction with contraction constant $\lambda$.
\\Then \begin{itemize}
\item<2-> g has a unique fixed point, say p.
\item<2->For any sequence $\left\{x_{n}\right\}$ defined by $x_n$=g($x_{n-1}$), n=1,2,,...
\\converges to this unique fixed point p.
\item[]<3->Moreover, we have the \textbf{a priori} error estimate
$$ \mid x_n-p\mid\le\frac{\lambda^n}{1-\lambda}\mid x_1-x_0\mid$$
\item[]<4->and the \textbf{a posteriori} error estimate
$$\mid x_n-p\mid\le\frac{\lambda}{1-\lambda}\mid x_n-x_{n-1}\mid$$
\end{itemize}
\end{itemize}
\end{theorem}
\vskip 1cm
\end{frame}
\begin{frame}{Banach Fixed Point Theorem}
\begin{block}{Proof}
For $n > m$ , we have
\begin{align*}
\mid x_n-x_m\mid&=\mid x_n-x_{n-1}+x_{n-1}-x_{n-2}+...+x_{m+1}-x_{m}\mid\\
&\le\mid x_n-x_{n-1}\mid + \mid x_{n-1}-x_{n-2}\mid+...+\mid x_{m+1}-x_{m}\mid\\
&by *\\
&\le(\lambda^{n-1}+\lambda^{n-2}+...+\lambda^{m})\mid x_1-x_0\mid\\
&=\lambda^{m}(\lambda^{n-m-1}+\lambda^{n-m-2}+...+1)\mid x_1-x_0\mid\\
&=\lambda^{m}\frac{1-\lambda^{n-m}}{1-\lambda}\mid x_1-x_0\mid\le \frac{\lambda^m}{1-\lambda}\mid x_1-x_0\mid
\end{align*}
so that ${x_n}$ is Cauchy sequence in U.
\\Since U is complete, ${x_n}$ converges to a point p $\in$ U
$$*\color{red}\mid x_k-x_{k-1}\mid=\mid g(x_{k-1})-g(x_{k-2})\mid\le\lambda\mid x_{k-1}-x_{k-2}\mid\le..\le\lambda^{k-1}\mid x_1-x_0\mid\color{black}$$
\end{block}
\vskip 1cm
\end{frame}
\begin{frame}
\begin{proof}[Continue]
\begin{itemize}
\item[]<1->Now,since g being contraction is continuous, we have
$$p=\lim_{n\to\infty}x_{n}=\lim_{n\to\infty}g(x_{n-1})=g(lim_{n\to\infty}x_{n-1})=g(p)$$
\\so that p is fixed point of g.
\\By the lemma p is the unique fixed point of g.
\item[]<2->Since $$\mid x_n-x_m\mid\le \frac{\lambda^m}{1-\lambda}\mid x_1-x_0\mid,$$
\\letting n$\rightarrow\infty$
\item[]<3->we get $$\mid p-x_m\mid\le \frac{\lambda^m}{1-\lambda}\mid x_1-x_0\mid$$
\\for $y_0=x_{n-1},$ $y_1= x_n$
$$\mid y_1- p\mid \le \frac{\lambda}{1-\lambda}\mid y_1-y_0\mid$$
\end{itemize}
\end{proof}
\vskip 1cm
\end{frame}
\subsection{The Fixed Point Algorithm}
\begin{frame}{The Fixed Point Algorithm}
\begin{block}{The Fixed Point Algorithm}<1->
If g has a fixed point p, then the fixed point algorithm generates a sequence $\left\{x_n\right\} $ defined as
\\$x_0$: arbitrary but fixed,
\\$x_n=g(x_{n-1})$, n=1,2,3,... to approximate p.
\end{block}
\end{frame}
\subsection{Fixed Point The Case Where Multiple Derivatives Are Zero at The Fixed Point}
\begin{frame}{Fixed Point The Case Where Multiple Derivatives Are Zero at The Fixed Point}
\begin{block}{Theorem}
\begin{itemize}
\item[]<1->Let g be a continuous function on the closed internal $[a,b]$ with $\alpha>1$ continuous derivatives on the internal (a,b).
\\Further, Let p $\in $(a,b) be a fixed point of g.
\item[]<2->if $$g'(p)=g''(p)=...=g^{(\alpha-1)}(p)=0$$ but $g^{(\alpha)}(p)\neq 0$,
\item[]<3->then there exist a $\delta>0$ such that for any $p_0\in[p-\delta,p+\delta]$, the sequence $p_n=g(p_{n-1})$ converges to the fixed point p of order $\alpha$ with asymtotic error constant $$\lim_{n\to\infty}\frac{\mid e_{n+1}\mid}{\mid e_n\mid^\alpha}=\frac{\mid g^{(\alpha)}(p)\mid}{\alpha!}$$.
\end{itemize}
\end{block}
\end{frame}
\begin{frame}
\begin{block}{Proof}
\begin{itemize}
\item[]<1->Let$'$s start by establishing the existence of $\delta>0$ such that for any $p_0\in [p-\delta,p+\delta]$, the sequence $p_n=g(p_{n-1})$ converges to the fixed point p.
\item[]<2->Let $\lambda<1$. Since $g'(p)=0$ and $g'$ is continuous, it follows that there exists a $\delta>0$ such that$\mid g'(x)\mid\le \lambda<1$ for all $x\in I \equiv[p-\delta,p+\delta]$ From this, it follows that g:I$\rightarrow$I; for if x$\in$ I then,
\item[]<3->
\begin{align*}
\mid g(x)-p\mid&=\mid g(x)-g(p)\mid\\
&=\mid g'(\xi)\mid\mid x-p\mid\\
&\le \lambda\mid x-p\mid<\mid x-p\mid\\
&\le\delta
\end{align*}
\item[]<4->Therefore by the a fixed point theorem established earlier, the sequence $p_n=g(p_{n-1})$ converges to the fixed point p for any $p_0\in [p-\delta,p+\delta]$.
\end{itemize}
\end{block}
\vskip 1cm
\end{frame}
\begin{frame}
\begin{block}{Continue}
\begin{itemize}
\item[]<1->To establish the order of convergence, let x$\in$ I and expand the iteration function g into a Taylor series about x=p:
$$g(x)=g(p)+g'(p)(x-p)+...+\frac{g^{\alpha-1}(p)}{(\alpha-1)!}(x-p)^{\alpha-1}+\frac{g^{\alpha}(\xi)}{(\alpha)!}(x-p)^{\alpha}$$
\\where $\xi$ is between x and p.
\item[]<2->Using the hypotheses regarding the value of $g^{(k)}(p)$ for $1\le k\le \alpha-1$ and letting $x=p_n$, the Taylor series expansion simplifies to $$p_n+1-p=\frac{g^(\alpha)(\xi)}{\alpha!}(p_n-p)^\alpha$$
\\where $\xi$ is now between $p_n$ and p.
\end{itemize}
\end{block}
\vskip 1cm
\end{frame}
\begin{frame}
\begin{proof}[Continue]
\begin{itemize}
\item[]<1->The definitions of fixed point iteration scheme and of a fixed point have been used to replace $g(p_n)$ with $p_{n+1}$ and g(p) with p.
\item[]<2->Finally, let $n\rightarrow\infty$.Then $p_n\rightarrow p$, forcing $\xi\rightarrow$ p also.
Hence $$\lim_{n\to\infty}\frac{\mid e_{n+1}\mid}{\mid e_n\mid^\alpha}=\frac{\mid g^{(\alpha)}(p)\mid}{\alpha!}$$
\\or $p_n\rightarrow$ p of order $\alpha$.
\end{itemize}
\end{proof}
\end{frame}
\section{Steffensen's Acceleration Method}
\subsection{Aitken's $\Delta^2$ Method}
\begin{frame}
\begin{theorem}[Aitken's $\Delta^2$ method]
\begin{itemize}
\item[] <1-> Suppose that
\begin{itemize}
\item $\left\{x_{n}\right\}$ is a sequence with $x_{n}\neq$ p for all n $\in \mathbb{N}$
\item there is a constant c $\in \Re\setminus \left\{ \mp 1 \right\}$ and a sequece $\left\{ \delta_{n} \right\}$
such that
\begin{itemize}
\item $\lim_{n\rightarrow\infty}\delta_{n}=0$
\item $x_{n+1}-p=(c+\delta_{n})(x_{n}-p)$ for all n $\in \mathbb{N}$
\end{itemize}
\end{itemize}
\item[] <2->Then
\begin{itemize}
\item $\left\{x_{n}\right\}$ converges to p iff $\left|c\right|<1$
\item if $\left|c\right|<1$ , then
\[
y_{n}=\frac{x_{n+2}x_{n}-x_{n+1}^2}{x_{n+2} - 2x_{n+1}+x_{n}}= x_n-\frac{(x_{n+1}-x_n)^2}{x_{n+2}-2x_{n+1}+x_n}
\]
\end{itemize}
\ is well-defined for all sufficiently large n.
\item[]<3-> Moreover $\left\{y_{n}\right\}$ converges to p faster than $\left\{x_{n}\right\}$ in the sense that
\[
\lim_{n\rightarrow\infty} \frac{y_{n}-p}{x_{n}-p}=0
\]
\end{itemize}
\end{theorem}
\vskip 1cm
\end{frame}
\begin{frame}
\begin{proof}
\begin{itemize}
\item[]<1-> Let $e_{n} = x_{n}-p$
\item[]<2-> $y_{n}-p=\frac{x_{n+2}x_{n}-x_{n+1}^2}{x_{n+2} - 2x_{n+1}+x_{n}}-p$
\item[]<3-> $y_{n}-p$=$\frac{(e_{n+2}+p)(e_{n}+p)-(e_{n+1}+p)^2}{(e_{n+2}+p)-2(e_{n+1}+p)+(e_{n}+p)}$
\item[]<4-> $y_{n}-p$=$\frac{e_{n+2}e_{n}-e_{n}^2}{e_{n+2}-2e_{n+1}+e_{n}}$
\item[]<5-> since we have
\item[]<5-> $x_{n+1}-p= (c+\delta_{n})(x_{n}-p)$
\ and $e_{n+1}=(c+\delta_{n}e_{n})$
\item[]<6-> $y_{n}-p$=$\frac{(c+\delta_{n+1})(c+\delta_{n})e_{n}e_{n}-(c+\delta_{n})^2e_{n}^2}{(c+\delta_{n+1})(c+\delta_{n})e_{n}-2(c+\delta_{n})e_{n}+e_{n}}$
\item[]<7-> $y_{n}-p$=$\frac{(c+\delta_{n+1})(c+\delta_{n})-(c+\delta_{n})^2}{(c+\delta_{n+1})(c+\delta_{n})-2(c+\delta_{n})+1}$ $(x_{n}-p)$
\item[]<8-> $y_{n}-p$=$\frac{(c+\delta_{n})(\delta_{n+1}-\delta_{n})}{(c-1)^+c(\delta_{n+1}+\delta_{n})+\delta_{n}(\delta_{n+1}-2)}$
\item[]<9-> Therefore $\lim_{n\rightarrow\infty} \frac{y_{n}-p}{x_{n}-p}=0$
\end{itemize}
\end{proof}
\end{frame}
\subsection{Steffensen's Acceleration Method}
\begin{frame}
\begin{block}{Steffensen's Acceleration Method}
\begin{itemize}
\item[]<1->Steffensen’s Method is a combination of fixed-point iteration and the Aitken’s $\Delta^2$ method:
\item[]<2->Suppose we have a fixed point iteration: $$x_0,x_1=g(x_0),x_2=g(x_1),...$$
\item[]<3->Once we have $x_0, x_1,$ and $x_2,$ we can compute $$ y_0=x_0-\frac{(x_1-x_0)^2}{x_2-2x_1+x_0}$$
At this point we "restart" the fixed point iteration with $x_0=y_0$
\item[]<4->e.g.\color{brown} $$x_3=y_0, x_4=g(x_3), x_5=g(x_4),$$\color{black}
and compute \color{brown} $$y_3=x_3-\frac{(x_4-x_3)^2}{x_5-2x_4+x_3}$$ \color{black}
\end{itemize}
\end{block}
\end{frame}
\section{Comparison With Fixed point iteration and Steffensen's Acceleration Method}
\begin{frame}{Comparison with Fixed Point Iteration and Steffensen's Acceleration Method}
\begin{block}{EXAMPLE}
\begin{itemize}
\item[]<1-> Use the Fixed Point iteration method to find a solution to $f(x) = x^2-2x-3$ using $x_0=0$, tolerance =$10^{-1}$ and compare the approximations with those given by Steffensen's Acceleration method with $x_0=0$, tolerance = $10^{-2}$.
\item<2-> We can see that my MATLAB code while Fixed Point iteration method reaches the root by 788 iteration, Steffensen's Acceleration method reaches the root by only 3 iterations.
\end{itemize}
\end{block}
\end{frame}
\end{document}