\documentclass[11pt]{article}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{psfig}
\usepackage{fullpage}
% AmsLatex macros are used.

\newcommand {\norm} [1] { \lVert #1 \rVert}
\newcommand {\Norm} [1] {\Bigl\lVert #1 \Bigr\rVert}
\newcommand {\tnorm} [1] { \lVert #1 \rVert_2}
\newcommand {\fnorm} [1] { \lVert #1 \rVert_F}
\newcommand {\dnorm} [1] { \lVert #1 \rVert_D}
\newcommand {\abs} [1] {\lvert #1 \rvert}
\newcommand {\Abs} [1] {\bigl\lvert #1 \bigr\rvert}
\newcommand {\htr} [1] {#1^*}
\newcommand {\inv} [1] {#1^{-1}}
\newcommand {\invt} [1] {#1^{-T}}
\newcommand {\tr} [1] {#1^{T}}
\newcommand {\ie} {i.e.}
\newcommand {\C} {C}
\newcommand {\CN}{C^N}
\newcommand {\CNN}{C^{N,N}}
\newcommand {\R} {R}

% Greek characters.
\newcommand {\tta} {\theta}
\newcommand {\al} {\alpha}
\newcommand {\LM} {\Lambda}
\newcommand {\lm} {\lambda}
\newcommand {\ee} {\epsilon}
\newcommand {\rr} {\rho}
\newcommand {\nm}[1]{\htr{#1}#1}
\newcommand {\rnm}[1]{#1\htr{#1}}

\newcommand {\diag}{\mathop{\mathrm{diag}}}
\newcommand {\trace}{\mathop{\mathrm{trace}}}
\newcommand {\dist}{\mathop{\mathrm{dist}}}

% floating point notation.
\newcommand{\fplus} {\oplus}
\newcommand{\fminus} {\ominus}
\newcommand{\ftimes} {\otimes}
\newcommand{\fdiv} {\oslash}
\newcommand{\fl}{\mathop{\mathrm{fl}}}

\newcommand{\floor}[1]{\lfloor #1\rfloor}
\newtheorem {thm} {Theorem}[section]
\newtheorem {prop}[thm] {Proposition}
\newtheorem {lem} [thm] {Lemma}
\newtheorem {cor} [thm]  {Corollary}
\newtheorem {conj}{Conjecture}[section]

\theoremstyle {definition}
\newtheorem{defn} {Definition} 
\newtheorem{exmp} {Example} [section]

\theoremstyle {remark}
\newtheorem {rem} {Remark}
\newtheorem* {note} {Note}

\numberwithin{equation}{section}

\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}

\input s0.ltx
\input s1.ltx
\input s2.ltx
\input s3.ltx
\input s4.ltx
\input s5.ltx
\input s6.ltx
\input s7.ltx
\input bib.ltx


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

\end{document}





