E-methods for fixed point equations f(x)=x
| Autor: | Siegfried M. Rump, E. Kaucher |
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| Rok vydání: | 1982 |
| Předmět: |
Discrete mathematics
Numerical Analysis Computation Interval (mathematics) Fixed point Lipschitz continuity Computer Science Applications Theoretical Computer Science Least fixed point Computational Mathematics Significand Computational Theory and Mathematics Uniqueness Constant (mathematics) Software Mathematics |
| Zdroj: | Computing. 28:31-42 |
| ISSN: | 1436-5057 0010-485X |
| DOI: | 10.1007/bf02237993 |
| Popis: | This paper provides newly implemented [11], [13] and widely applicable methods for, computing inclusion (i. e. a containing interval) (Einschliesung) of the solution of a fixed point equationf(x)=x as well as autmatic verification the existence (Existenz) and uniqueness (Eindeutigkeit) of the solution. These methods make essential use of a new computer arithmetic defined by semimorphisms as developed in [7] and [8]. We call such methods E-Methods in correspondance to the three German words. A priori estimations such as a bound for a Lipschitz constant etc. are not required by the new algorithm. So the algorithm including the a posteriori proof of existence and uniqueness of the fixed point is programmable on computers for linear as well as for nonlinear problems. This is a key feature of our results. The computations produced by E-methods deliver answers the components of which have accuracy better than 10−t+1 (wheret denotes the mantissa length employed in the computer). |
| Databáze: | OpenAIRE |
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