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Summary

  1. Express $F(x) = 0$ in the form $x = G(x)$. The fixed-point iteration scheme is

    \begin{displaymath}
x_{n+1} = G(x_{n}).
\end{displaymath}

  2. Fixed-point iteration generates a first order scheme with

    \begin{displaymath}
\epsilon_{n+1} \approx G^{\prime}(r)\epsilon_{n}.
\end{displaymath}

  3. Fixed-point iteration with converge to a root at $x = r$ if

    \begin{displaymath}
\vert G^{\prime}(r)\vert < 1,
\end{displaymath}

    and will not converge if

    \begin{displaymath}
\vert G^{\prime}(r)\vert >1.
\end{displaymath}

    These conditions are local conditions in the sense that they apply if the intial estimate $x_{0}$ is sufficiently close to the root.


Prof. Alan Hood
2000-02-01