Number Theory in Digital Signal Processing
Number Theory in Digital Signal Processing
An Adaptive Hybrid Algorithm for Multiplying Dense Polynomials
EUROCAM '82 Proceedings of the European Computer Algebra Conference on Computer Algebra
Practical fast polynomial multiplication
SYMSAC '76 Proceedings of the third ACM symposium on Symbolic and algebraic computation
A case study in interlanguage communication: Fast LISP polynomial operations written in 'C'
SYMSAC '81 Proceedings of the fourth ACM symposium on Symbolic and algebraic computation
Parallel multiplication and powering of polynomials
Journal of Symbolic Computation
On the bit-complexity of sparse polynomial and series multiplication
Journal of Symbolic Computation
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This paper presents an improved adaptive hybrid algorithm for multiplying dense multivariate polynomials that is both time and space efficient. The hybrid algorithm makes use of two families of univariate algorithms, one Karatsuba based and the other DFT based, which are applied recursively to solve the multivariate problem. The hybrid algorithm is adaptive in that particular univariate algorithms are selected at run time to minimize the time complexity; an order-of-magnitude speedup with respect to classical multiplication is achieved over the entire practical range except for very small problems. Empirical investigation shows that most of the theoretical superiority is maintained in actual implementation. The largest contribution to the space requirements of the total algorithm is determined by the univariate algorithm used for the outermost variable; except for quite small problems, selecting univariate algorithms to minimize run time almost always leads to situations where the space requirements of the total algorithm are extremely close to the space required merely to store the result.