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\hyphenation{pse-udo-spec-tra}

\hyphenation{di-men-sion-al} 
\title{A Conditioning Function for the
Convergence of Numerical ODE Solvers and Lyapunov's Theory of
Stability}

\author {Divakar Viswanath \thanks{ 
Research at MSRI is supported
in part by NSF grant DMS-9701755}}
\date{22 December 1998}
\begin{document}
\maketitle
\begin{abstract}
For the ordinary differential equation (ODE) $\dot{x}(t) = f(t,x)$,
$x(0) = x_0$, $t\geq 0$, $x\in R^d$, assume $f$ to be at least
continuous in $t$ and locally Lipshitz in $x$, and if necessary,
several times continuously differentiable in $t$ and $x$.
We associate a conditioning function $E(t)$ with each solution $x(t)$
which captures the accumulation of global error in a numerical
approximation in the following sense: if $\tilde{x}(t;h)$ is an 
approximation derived from a single step method of time step $h$ and
order $r$ then $\norm{\tilde{x}(t;h) - x(t)} < K(E(t)+\epsilon)h^r$
for $0\leq t\leq T$, any $\epsilon > 0$, sufficiently small $h$,
and a constant $K>0$.

Using techniques from the stability theory of differential equations,
this paper gives conditions on $x(t)$ for $E(t)$ to be upper bounded
linearly or by a constant for $t\geq 0$. More concretely, these
techniques give constant or linear bounds on $E(t)$ when $x(t)$ is
a trajectory of a dynamical system which falls into a stable, hyperbolic
fixed point; or into a stable, hyperbolic cycle; or into a normally hyperbolic 
and contracting manifold with quasiperiodic flow on the manifold.
\end{abstract}

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\noindent
Mathematical Sciences Research Institute (MSRI)\\
1000 Centennial Drive\\
Berkeley, CA 94720.\\
divakar@cs.cornell.edu

\end{document}