Accuracy and Stability of Numerical Algorithms
Accuracy and Stability of Numerical Algorithms
The iRRAM: Exact Arithmetic in C++
CCA '00 Selected Papers from the 4th International Workshop on Computability and Complexity in Analysis
RealLib: An efficient implementation of exact real arithmetic
Mathematical Structures in Computer Science
VALENCIA-IVP: A Comparison with Other Initial Value Problem Solvers
SCAN '06 Proceedings of the 12th GAMM - IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics
Domain Theoretic Solutions of Initial Value Problems for Unbounded Vector Fields
Electronic Notes in Theoretical Computer Science (ENTCS)
A domain theoretic account of euler's method for solving initial value problems
PARA'04 Proceedings of the 7th international conference on Applied Parallel Computing: state of the Art in Scientific Computing
Hi-index | 0.00 |
We present an implementation of the domain-theoretic Picard method for solving initial value problems (IVPs) introduced by Edalat and Pattinson [1]. Compared to Edalat and Pattinson's implementation, our algorithm uses a more efficient arithmetic based on an arbitrary precision floating-point library. Despite the additional overestimations due to floating-point rounding, we obtain a similar bound on the convergence rate of the produced approximations. Moreover, our convergence analysis is detailed enough to allow a static optimisation in the growth of the precision used in successive Picard iterations. Such optimisation greatly improves the efficiency of the solving process. Although a similar optimisation could be performed dynamically without our analysis, a static one gives us a significant advantage: we are able to predict the time it will take the solver to obtain an approximation of a certain (arbitrarily high) quality.