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Inverse and Implicit Functions in Domain Theory

Abbas Edalat, Dirk Pattinson

Conference or Workshop Paper
IEEE Symposium on Logic in Computer Science(2005)
June, 2005
pp.417–426
IEEE Computer Society
ISBN 0-7695-2266-1
Abstract

We construct a domain-theoretic calculus for differentiable functions, which includes addition, subtraction and composition. Using the domain for differential functions, we develop a domain-theoretic version of the inverse function theorem and implicit function theorem, in which, subject to the usual conditions, the inverse function and the implicit function and their derivatives are obtained as fixed points of Scott continuous functionals and are approximated by step functions. This means that from an increasing sequence of step functions converging to a function and its derivative in the domain of differentiable functions we can effectively obtain an increasing sequence of step functions converging in this domain to the inverse function and its derivative, and also effectively obtain an increasing sequence of polynomial step functions whose lower and upper bounds converge in the $C^1$ norm to the inverse function. A similar result holds for implicit functions. Combined with the domain-theoretic model for computational geometry, this provides a robust technique for construction of curves

and surfaces in geometric modelling and CAD.

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