## Inverse and Implicit Functions in Domain Theory

### BibTeX

@MISC{_inverseand,

author = {},

title = {Inverse and Implicit Functions in Domain Theory},

year = {}

}

### OpenURL

### Abstract

We construct a domain-theoretic calculus for Lipschitz and differentiable functions, which includes addition, subtraction and composition. We then develop a domaintheoretic version of the inverse function theorem for a Lipschitz function, in which the inverse function is obtained as a fixed point of a Scott continuous functional and is approximated by step functions. In the case of a C 1 function, the inverse and its derivative are obtained as the least fixed point of a single Scott continuous functional on the domain of differentiable functions and are approximated by two sequences of step functions, which are effectively computed from two increasing sequences of step functions respectively converging to the original function and its derivative. In this case, we 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, which combined with the domain-theoretic model for computational geometry, provides a robust technique for construction of curves and surfaces. 1.

### Citations

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Generative Modeling for Computer Graphics and CAD
- Snyder
- 1992
(Show Context)
Citation Context ..., where curves and surfaces are usually defined implicitly [2]. Currently, there are no robust methods to approximate an implicit surface and the most reliable technique provided by interval analysis =-=[12]-=- is only able to approximate the implicit surface without approximating its derivative. The paper thus presents a framework for a robust CAD system, where implicitly given surfaces can be effectively ... |

33 | Foundation of a computable solid modeling
- Edalat, Lieutier
- 1999
(Show Context)
Citation Context ... closed connected orientable manifold given by implicit equations such as f(x, y, z) =0when 0 is a regular value of f. Furthermore, the domain-theoretic framework for geometric modelling developed in =-=[6]-=- combined with the results in this work lead to a domain of orientable closed Lipschitz manifolds. This will synthesize the domain-theoretic framework for geometry and that for differential calculus. ... |

20 |
Domain theory and differential calculus (functions of one variable
- Edalat, Lieutier
(Show Context)
Citation Context ... a recursion-theoretic account of the inverse function and the implicit function theorems, which are the main fundamental tools in multi-variable differential calculus and the theory of manifolds. In =-=[7]-=-, a domain-theoretic framework for differential calculus of one variable was developed which in particular provides an effectively given domain for Lipschitz or differentiable functions. Later on, dom... |

19 | A domain theoretic account of Picard’s theorem
- Edalat, Pattinson
- 2004
(Show Context)
Citation Context ...developed which in particular provides an effectively given domain for Lipschitz or differentiable functions. Later on, domain-theoretic techniques for solving initial value problems were obtained in =-=[5, 9]-=-, which enable us to approximate the unique solution of an initial value problem given by a Lipschitz vector field up to the precision required by the user. In [8], the domain-theoretic model was exte... |

13 | Domain-theoretic solution of differential equations (scalar fields
- Edalat, Krznarić, et al.
- 2003
(Show Context)
Citation Context ...developed which in particular provides an effectively given domain for Lipschitz or differentiable functions. Later on, domain-theoretic techniques for solving initial value problems were obtained in =-=[5, 9]-=-, which enable us to approximate the unique solution of an initial value problem given by a Lipschitz vector field up to the precision required by the user. In [8], the domain-theoretic model was exte... |

12 | Numerical Analysis - Balakrishnan, F - 1981 |

10 | A computational model for multi-variable differential calculus
- Edalat, Lieutier, et al.
- 2005
(Show Context)
Citation Context ...l value problems were obtained in [5, 9], which enable us to approximate the unique solution of an initial value problem given by a Lipschitz vector field up to the precision required by the user. In =-=[8]-=-, the domain-theoretic model was extended to multi-variable differential calculus, resulting in the notion of a domain-theoretic derivative, which for Lipschitz functions gives the smallest hyper-rec... |

4 | The constructive implicit function theorem and applications
- Bridges, Calude, et al.
- 1999
(Show Context)
Citation Context ...nstructive proof of the existence of the inverse function (or the implicit function) for a C 1 function is obtained by approximation but no approximations to the derivative of the inverse in provided =-=[4]-=-. In none of these approaches, the inverse or the implicit function is obtained as a fixed point of a functional. 2. Preliminaries Proceedings of the 20 th Annual Symposium on Logic in Computer Scienc... |

4 |
The inverse function theorem for Lipschitz maps, unpublished
- Howard
(Show Context)
Citation Context ...n R n , we write Br = {y ∈ R n |�y� <r} for the open ball around the origin with radius r and denote the closed ball by Br = {y ∈ R n |�y� ≤r}. Theorem 1.1 Inverse Function Theorem for Lipschitz maps =-=[10]-=- Let Br be a closed ball containing the origin in R n and let f : Br → R n with f(0) = 0, so that for some invertible linear map L : R n → R n and some ρ<1 �L −1 f(x2) − L −1 f(x1) − (x2 − x1)� ≤ρ�x2 ... |