SIMULATION OF RANDOM PROCESSES ON DIGITAL-COMPUTERS WITH CEBYSEV MIXING TRANSFORMATIONS

The iteration of the Čebyšev polynomial x2 - 2 generates a mixing transformation on the interval x ε [-2, 2]. Extensive computer experiments have demonstrated that this is a convenient method for generating sequences of pseudo-random numbers. Despite the eventual domination of cumulative roundoff errors the asymptotic statistical features of the mixing are preserved. Multiple sequences of stochastically independent variables may be generated by these techniques. In practical computations the Čebyšev mixing eventually terminates in long pseudo-ergodic cycles. These results are linked with the general problem of simulating the stochastic behavior of physical systems by means of functional iteration. Contents. 1. Introduction. 2. Randomness Criteria. 3. Computer Simulation of Čebyšev Mixing-3.1. Single Sequences of Pseudo-random Numbers; 3.2. Computer Simulations: Growth of Round-off Errors; 3.3. Computer Simulations: Free Running and Terminal Cycles; 3.4. Multiple Sequences of Pseudo-random Numbers; 4. Čebyšev Mixing Theorems; Product Transformations; Probabilistic Metrics-Asymptotic Dispersion; Kolmogorov Entropy. 5. Variable Precision Simulations. 6. Terminal Cycles; Simulation of Pre-image Chains; Memory-Dependent Feedback. 7. Other Simulations of Random Processes; Probabilistic Metric for Baker's Transformation; Brolin's Theorem; Šarkovskii's Theorem; Hamiltonian Mechanics; Flow Equation; Non-embeddable Functions. © 1979.