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Parallel Solution of Recurrence Problems

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1 Author(s)
Kogge, P.M. ; IBM Federal Systems Division Electronics Systems Center in Owego, New York 13827, USA

An mth-order recurrence problem is defined as the computation of the sequence x1, ···, xN, where xi = ƒ(ai, xi−1, ···, xi−m) and ai is some vector of parameters. This paper investigates general algorithms for solving such problems on highly parallel computers. We show that if the recurrence function ƒ has associated with it two other functions that satisfy certain composition properties, then we can construct elegant and efficient parallel algorithms that can compute all N elements of the series in time proportional to [log2N]. The class of problems having this property includes linear recurrences of all orders—both homogeneous and inhomogeneous, recurrences involving matrix or binary quantities, and various nonlinear problems involving operations such as computation with matrix inverses, exponentiation, and modulo division.

Note: The Institute of Electrical and Electronics Engineers, Incorporated is distributing this Article with permission of the International Business Machines Corporation (IBM) who is the exclusive owner. The recipient of this Article may not assign, sublicense, lease, rent or otherwise transfer, reproduce, prepare derivative works, publicly display or perform, or distribute the Article.  

Published in:

IBM Journal of Research and Development  (Volume:18 ,  Issue: 2 )