Evaluating parametric holonomic sequences using rectangular splitting
Fredrik Johansson
- 23 Jul 2014
- pp 256-263
TL;DR: The rectangular splitting technique is adapted to the problem of evaluating terms in holonomic sequences that depend on a parameter to allow computing the n-th term in a recurrent sequence of suitable type using O(n1/2) "expensive" operations at the cost of an increased number of "cheap" operations.
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Abstract: We adapt the rectangular splitting technique of Paterson and Stockmeyer to the problem of evaluating terms in holonomic sequences that depend on a parameter. This approach allows computing the n-th term in a recurrent sequence of suitable type using O(n1/2) "expensive" operations at the cost of an increased number of "cheap" operations.Rectangular splitting has little overhead and can perform better than either naive evaluation or asymptotically faster algorithms for ranges of n encountered in applications. As an example, fast numerical evaluation of the gamma function is investigated. Our work generalizes two previous algorithms of Smith.
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References
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TL;DR: This highly successful textbook, widely regarded as the “bible of computer algebra”, gives a thorough introduction to the algorithmic basis of the mathematical engine in computer algebra systems.
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