Abstract
In the search for regions that contain fixed points of a real function of several variables, tests based on interval calculations can be used to establish existence or non-existence of fixed points in regions that are examined in the course of the search. The search can e.g. be performed as a synchronous (sequential) interval iteration: In each iteration step all components of the iterate are calculated based on the previous iterate. In this case it is straight forward to base simple interval existence and non-existence tests on the calculations done in each step of the iteration. The search can also be performed as an asynchronous (parallel) iteration: Only a few components are changed in each step and this calculation is in general based on components from different previous iterates. For the asynchronous iteration it turns out that simple tests of existence and non-existence can be based on the component wise calculations done in the course of the iteration. These component wise tests are useful for parallel implementation of the search, since the tests can then be performed local to each processor and only when a test is successful does a processor communicate this result to other processors.
Similar content being viewed by others
References
Baudet, G.: Asynchronous IterationMethods forMultiprocessors, J. ACM 25 (1978), pp. 226–244.
Chazan, D. and Miranker, W.: Chaotic Relaxation, Linear Algebra Appl. 2 (1969), pp. 199–222.
Frommer, A. and Schwandt, H.: Asynchronous Parallel Methods for Enclosing Solutions of Nonlinear Equations, Journal of Computational and Applied Mathematics 60 (1995), pp. 47–62.
Lamport, L.: Time, Clocks, and the Ordering of Events in a Distributed System, CACM 21 (7) (1978).
Rall, L. B.: A Theory of Interval Iteration, Proceedings of the American Mathematical Society 86 (4) (1982).
Rump, S.M.: Verification Methods for Dense and Sparse Systems of Equations, in: Hertzberger, J. (ed.), Topics in Validated Computations—Studies in Computational Mathematics, Elsevier, Amsterdam, 1994, pp. 63–136.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Caprani, O., Madsen, K. & Stauning, O. Existence Test for Asynchronous Interval Iteration. Reliable Computing 3, 269–275 (1997). https://doi.org/10.1023/A:1009922807454
Issue Date:
DOI: https://doi.org/10.1023/A:1009922807454