Globally convergent polynomial iterative zero-finding using APL
APL '92 Proceedings of the international conference on APL
SCARFS, an efficient polynomial zero-finder system
APL '93 Proceedings of the international conference on APL
Rounding Errors in Algebraic Processes
Rounding Errors in Algebraic Processes
Iterative Algorithms on Heterogeneous Network Computing: Parallel Polynomial Root Extracting
HiPC '02 Proceedings of the 9th International Conference on High Performance Computing
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Aberth's method for the iterative finding of polynomial zeros using simultaneous interacting guesses is seen to be the systematic use of Newton's method under Parallel Anticipatory Implicit Deflation (PAID), changing the single polynomial problem into that for a system of rational functions. It offers the prospect of parallel execution, discourages coincident convergences, and subjects all iterates to the same algorithm, without the uneven roundoff error accumulation due to deflation typical in serial iterative methods. Yet it converges slowly to multiple zeros, and consumes far more total computing power than serial methods. Factors hindering convergence include problem symmetry and clustering of iterates, alleviated by asymmetric initial guesses and an immediate updating strategy, as seen through the speedup in the experimental APL program PAIDAX.