Interval Methods for Fixed-Point Problems.
Technical summary rept.,
WISCONSIN UNIV-MADISON MATHEMATICS RESEARCH CENTER
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Scientific computation is devoted largely to processes for numerical solution of ordinary and partial differential equations, integral equations, and finite systems of linear and nonlinear equations, all problems which can be formulated abstractly as a fixed-point problem in some suitable space. Since the computation is carried out on a computer, and not in the space in which the problem is posed, methods of discrete approximation are used, and ordinarily detailed analysis is required to estimate the reliability and accuracy of the flood of floating-point numbers produced, as well as their interpretation in terms of the original problem. In this report, it is shown that interval methods can be used to overcome many of the difficulties of this kind which arise in scientific computation. The goal is to have the computation itself provide the required information about the reliability and accuracy of the computed results. It is shown that interval iteration can be used to prove existence or nonexistence of solutions in given regions, and to obtain lower and upper bounds for solutions in those regions. The transition to actual computation consists simply of identification of the floating-point numbers available on a computer with a finite set of elements of the space in question, for example, as step functions with floating-point values, and directed rounding from the space to this finite subspace.
- Theoretical Mathematics