NDLI: Improved Parallel Polynomial Division

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Bini, Dario Pan, Victor
Copyright Year	1993
Abstract	The authors compute the first N coefficients of the reciprocal $r(x)$, $r(x)p(x) = 1\bmod x^N $ (given a natural N and a polynomial $p(x)$), $(p(0) \ne 0)$, by using $O(h\log N)$ arithmetic steps and $O(({N / h})(1 + 2^{ - h} \log ^{(h)} N))$ processors, for any h, $h = 1,2, \ldots ,\log ^ * N$, under the PRAM arithmetic models, provided that $O(\log m)$ steps and m processors suffice to perform discrete Fourier transforms on m points and that $\log ^{(0)} N = N$, $\log ^{(h)} N = \log _2 \log ^{(h - 1)} N$, $h = 1, \ldots ,\log ^ * N$, $\log ^ * N = \max \{ {h:\log ^{(h)} N > 0} \}$. The same estimates apply to some other computations, such as the division with a remainder of two polynomials of degrees $O(N)$ and the inversion of an $N \times N$ triangular Toeplitz matrix. This improves the known estimates of ReifTate and Georgiev. The presented techniques are extended to parallel implementation of other recursive processes, such as the evaluation modulo $x^N $ of the mth root $p(x)^{{1 / m}} $ of $p(x)$ (for any fixed natural m), for which we need $O(\log N\log \log N)$ timesteps and $O({N / {\log \log N}})$ processors. The paper demonstrates some new techniques of supereffective slowdown of parallel algebraic computations combined with the technique of stream contraction.
Starting Page	617
Ending Page	626
Page Count	10
File Format	PDF
ISSN	00975397
DOI	10.1137/0222041
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	3
Volume Number	22
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-07-31
Access Restriction	Subscribed
Subject Keyword	parallel algorithms polynomial division Newtons iteration for algebraic computing triangular Toeplitz matrices computational complexity
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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