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The Ontario Research Centre for Computer Algebra
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Abstract
In this paper, in apparent contradiction to the standard theory of
information-based complexity for the numerical solution of initial value
problems for ordinary differential equations, I prove that adaption is
better than non-adaption, and that there are algorithms of cost polynomial
in the number of digits of accuracy requested, for the solution of IVP for
ODE. Although I use some new methods, derived from a new result on
equidistribution, the real reason these results are different from the
standard theory is that I have modified the hypotheses of the standard
theory. The standard theory of the computational complexity of the solution
of IVP for ODE is correct as it stands, working from the standard
hypotheses. However, it is my belief that the standard hypotheses, while
plausible, are not good models of what happens in practice. What is observed
in practice is that, generally, adaption is better than non-adaption, and
that solution methods are efficient enough for practical purposes (and hence
exponential cost is not a reasonable model). This paper should provoke some
useful discussion, and perhaps further development of this neglected theory.
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