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A long note on Mulders’ short product. (English) Zbl 1161.65301
J. Symb. Comput. 37, No. 3, 391-401 (2004); erratum ibid. 66, 111-112 (2015).
Summary: The short product of two power series is the meaningful part of the product of these objects, i.e., \(\sum _{i+j<n}a_{i}b_{j}x^{i+j}\). T. Mulders [Appl. Algebra Eng. Commun. Comput. 11, No. 1, 69–88 (2000; Zbl 0968.68200)] gives an algorithm to compute a short product faster than the full product in the case of Karatsuba’s multiplication [A. A. Karatsuba and Yu. P. Ofman, Dokl. Akad. Nauk SSSR 145, No. 2, 293–294 (1962); per bibl.]. This algorithm works by selecting a cutoff point \(k\) and performing a full \(k\times k\) product and two \((n - k)\times (n - k)\) short products recursively. Mulders also gives a heuristically optimal cutoff point \(\beta n\). We determine the optimal cutoff point in Mulders’ algorithm. We also give a slightly more general description of Mulders’ method.

MSC:
65B10 Numerical summation of series
12Y05 Computational aspects of field theory and polynomials (MSC2010)
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References:
[1] Karatsuba, A.A.; Ofman, Y.P., Multiplication of multiplace numbers by automata, Dokl. akad. nauk SSSR, 145, 2, 293-294, (1962)
[2] Mulders, T., On short multiplications and divisions, Aaecc, 11, 1, 69-88, (2000) · Zbl 0968.68200
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