Chinese remainder and interpolation algorithms

Lipson, John D.

doi:10.1145/800204.806309

Cited by 30 publications

(4 citation statements)

References 6 publications

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“…The dual problem is to compute the residues of an integer with respect to a set of moduli. Lipson [12] shows that interpolation and the CRA are abstractly equivalent to the CRA for a Euclidean domain and gives a thorough exposition of the classical algorithms for these problems.…”

Section: ~ L Fb B B'mentioning

confidence: 99%

“…The classical Lagrangian and Newtonian algorithms for polynomial interpolation and the CRA require O(N ~) operations (cf. Lipson [12]). Horowitz and Heindel [9] have given a method for computing the integer CRA which has a bound of O(N log ~ N log log N) as a preconditioned algorithm and O(N log z N log log N) as a complete algorithm.…”

Section: Fast Interpolationmentioning

confidence: 99%

“…In the more general context of Euclidean domains (cf. Lipson [12]), the Lagrangian formula for the moduli mi and residues ui is Note that these powers of moduli are none other than the supermoduli computed in Section 5.…”

Section: / \K=n/2+l \I=n/2+l Irmentioning

confidence: 99%

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Fast modular transforms

Borodin

Moenck

1974

Journal of Computer and System Sciences

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It is shown that if division and multiplication in a Euclidean domain can be performed in O(N log ~ N) steps, then the residues of an N precision element in the domain can be computed in O(N log a+l N) steps. A special case of this result is that the residues of an N precision integer can be computed in O(N logS N log log N) total operations. Using a polynomial division algorithm due to Strassen [24], it is shown that a polynomial of degree N-1 can be evaluated at N points in O(N log 2 N) total operations or O(N log N) multiplications. Using the methods of Horowitz [10] and Heindel [9], it is shown that if division and multiplication in a Euclidean domain can be performed in O(N log ~ N) steps, then the Chinese Remainder Algorithm (CRA) can be performed in O(Nlog ~+x N) steps. Special cases are: (a) the integer CRA can be performed in O(N log S N log log N) total operations, and (b) a polynomial of degree N-1 can be interpolated in O(N log 2 N) total operations or O(Nlog N) multiplications. Using these results, it is shown that a polynomial of degree N and all its derivatives can be evaluated at a point in O(N log s N) total operations.

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Section: ~ L Fb B B'mentioning

confidence: 99%

Section: Fast Interpolationmentioning

confidence: 99%

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Fast modular transforms

Borodin

Moenck

1974

Journal of Computer and System Sciences

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“…If the input numbers are very big and each prime number fits into one machine word the basic arithmetic in the homomorphic images is cheap because fixed precision arithmetic can be used. Only in the combination phase, when applying a fold-based Chinese Remainder Algorithm (CRA) (see [10]), expensive arbitrary precision arithmetic has to be used to construct the result values.…”

Section: Case Study: Linsolvmentioning

confidence: 99%

Testing Speculative Work in a Lazy/Eager Parallel Functional Language

Encina

Rodrı́guez

Rubio

2006

Languages and Compilers for Parallel Computing

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Eden is a parallel extension of the functional language Haskell. Eden inherits from Haskell its laziness, which allows it to avoid unnecessary computations. However, in order to enable the parallel execution of processes in Eden, this feature must be disabled when new processes are instantiated. Hence, any newly created process can be speculative, as it is not known whether the computations it performs will actually be required for the overall computation. Therefore, the performance of a program may be affected by the unneeded speculation. In this paper we present a framework to compare the speculated computations of an Eden program with the computations it actually requires. Thus, the programmer is provided with a profiling tool allowing him to produce better programs where speculative work fits better the actual necessities.

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