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\title{Notes on Newton's Method}
\date{Revised for the Web: July 27, 2004}
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\begin{center}\LARGE\bfseries{}
Notes on Newton's Method
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\begin{center}\large\bfseries{}
Supplementary Material for Honors Calculus\\[0.25\baselineskip] Originally prepared in the Fall Semester 1995
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\large\bfseries{}
Revised for the Web: July 27, 2004
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\medskip
\par{\textbf{Proposition}: Let \(f\) be a function that is differentiable on an interval \(I\) and assume further: \begin{menulist}

\item [{\label{KEY-1}(a)}]
 The derivative \(f^{\prime{}}\) is either positive throughout \(I\) or negative throughout \(I\)\@. \ 

\item [{\label{KEY-2}(b)}]
 The derivative \(f^{\prime{}}\) is either steadily increasing in \(I\) or steadily decreasing in \(I\)\@. \ 

\item [{\label{KEY-3}(c)}]
 There are points \(a, b\) in \(I\) for which \(f(a)\) and \(f(b)\) are both non-zero with opposite sign.  
\end{menulist} Then: \begin{enumerate}

\item  There is one and only one point \(z\) in \(I\) for which \(f(z) \, = \, 0\)\@. \  

\item  If \(x\) is on the \textbf{convex} side of the graph of \(f\) in \(I\)\@,  then so is \(x^{\prime{}} \, = \, x - f(x)/f^{\prime{}}(x)\)\@,  and \(x^{\prime{}}\) lies between \(x\) and \(z\)\@. \  

\item  Successive iterations of Newton's method beginning with a point \(x\) on the \textbf{convex} side of the graph of \(f\) in \(I\) will converge to \(z\)\@. \  

\item  \emph{Error control principle.} \ If \(c\) is any point in \(I\) on the \textbf{concave} side of the graph of \(f\) and \(x\) is on the \textbf{convex} side, then the distance between \(x\) and \(z\) is at most the absolute value of \(f(x)/f^{\prime{}}(c)\)\@. \ 
\end{enumerate}
}
\par{\textsl{Proof:} If \(f^{\prime{}}\) is positive in \(I\) one has \(f(x_{1}) < f(x_{2})\) whenever \(x_{1} < x_{2}\) in \(I\).    If instead \(f^{\prime{}}\) is negative in \(I\), then one has \(f(x_{1}) > f(x_{2})\) for \(x_{1} < x_{2}\)\@. \  For this reason there is \emph{at most} one root \(z\) in \(I\) with \(f(z)\, = \,0\)\@. \   The \emph{Intermediate Value Theorem for Continuous Functions} guarantees that there is at least one root between \(a\) and \(b\)\@. \ 
}
\par{We shall assume that \(f^{\prime{}}\) is positive and increasing.    One may reduce each of the other three cases to this case by reflecting either in the horizontal axis or in the vertical line \(x\, = \,z\) or both.   Under this assumption the convex side of the graph of \(f\) is the right side.    Suppose that \(z < x\): then \(0\, = \,f(z) < f(x)\)\@. \  We apply the Mean Value Theorem to \(f\) on the interval \([z,x]\) to conclude that there is a number \(u\) with \(z < u < x\) for which \[ f(x) - f(z) \ = \  f^{\prime{}}(u)(x - z) \ \ \ . \] Since \(f(z)\, = \,0\) and \(f^{\prime{}}(u) > 0\)\@,  one obtains \[ x - z \ = \  \frac{f(x)}{f^{\prime{}}(u)} \ \ \ . \] Since \(f^{\prime{}}\) is increasing, we find \(f^{\prime{}}(u) < f^{\prime{}}(x)\)\@,  and, therefore, \(f(x)/f^{\prime{}}(x) < f(x)/f^{\prime{}}(u)\)\@. \   Consequently, \(z < x^{\prime{}} < x\)\@. \ 
}
\par{In view of (2) one has \[ z < \ldots{} < x_{n} < \ldots{} < x_{2} < x_{1} \ \ \ . \] Letting \[ x_{*} \ = \  \mbox{inf}_{(n \geq{} 1)} \ \left\{x_{n}\right\} \ , \] one has \[ x_{*} \ = \  \lim _{n \rightarrow{} \infty{}} \ x_{n} \ , \] and, therefore, taking the limit as \,\(n \rightarrow{} \infty{}\)\, on both sides of the relation \[ x_{n+1} \ = \  x_{n} - \frac{f(x_{n})}{f^{\prime{}}(x_{n})} \ , \] one finds that \(f(x_{*}) \, = \, 0\)\@. \   Since by (1) there is only one root of \(f\) in \(I\), it follows that \(x_{*} \, = \, z\)\@. \ 
}
\par{In the proof of (2) we saw that for \(x\) in \(I\) on the right side of \(z\) the distance from \(x\) to \(z\) is \(f(x)/f^{\prime{}}(u)\)\@,  where \(z<u<x\)\@. \   Since \(c\) is on the concave side of the graph of \(f\)\@,  i.e., \(c < z\)\@,  we find also \(c < u\)\@,  hence, \(f^{\prime{}}(c) < f^{\prime{}}(u)\)\@. \  Consequently, \[ \mbox{the error} \ = \  x - z \ = \  \frac{f(x)}{f^{\prime{}}(u)} \leq{}    \frac{f(x)}{f^{\prime{}}(c)} \ \ \ . \]
}
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