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We present parallel algorithms for fast polynomial interpolation. ... Unlike alternate approaches which use the Lagrange representation, the algorithms described in this paper are based on the fast parallel evaluation of a closed formula for the generalized divided differences ... POLYNOMIAL EVALUATION As mentioned by Egecioglu et al. (1987a) , a fast algorithm for the interpolation would not be very useful unless an algorithm of comparable speed could be designed for the evaluation ...doi:10.1016/0885-064x(89)90018-6 fatcat:lgz2lm44x5a47cvyuibrxgg6ui
We present parallel algorithms for the computation and evaluation of interpolating polynomials. ... The algorithms use parallel prefix techniques for the calculation of divided differences in the Newton representation of the interpolating polynomial. ... The interpolating polynomial in its Newton form can be evaluated by means of a fast parallel algorithm having the same characteristics. ...doi:10.1007/bf02017348 fatcat:orii44aerbedjdh3b4nn6wg4na
Lecture Notes in Computer Science
CUMODP is a CUDA library for exact computations with dense polynomials over finite fields. ... Algorithms combine FFT-based and plain arithmetic, while the implementation strategy emphasizes reducing parallelism overheads and optimizing hardware usage. ... fast algorithm). ...doi:10.1007/978-3-662-44199-2_108 fatcat:3zqkss6555aixjitkprv7vesfa
We engineer an open-source GPU implementation of a distributed algorithm design of Björklund and Kaski [PODC 2016] where (i) the execution of the algorithm can be delegated [Goldwasser, Kalai, and Rothblum ... Footnote 4 for an analysis of the over-provisioning needed to tolerate adversarial errors. ... To expose sufficient parallelism in the design, we look inside the evaluation algorithm for parallelism and implement both the preprocessing and the fast matrix multiplication steps using Yates's algorithm ...doi:10.1137/1.9781611975055.16 dblp:conf/alenex/Kaski18 fatcat:hlfxvpbb2zdmrlf4venyyhipim
Further places in polynomial root-finding algorithms where parallel behaviour can be introduced are described. ... Parallelizations of various different methods for determining the roots of a polynomial are discussed. These include methods which locate a single root only as well as those which find all roots. ... The quotient-difference algorithm tends to be fast as it does not have to perform polynomial evaluations at each step. ...doi:10.1016/0377-0427(87)90056-2 fatcat:z46g2nz6grarbehb7blaepsady
Skrzipek, A Christoffel- Darboux-type formula for Szegd polynomials and polynomial evaluation (203-216); Kurt Suchy, Applications of tensor-valued tri-variate Hermite polynomials and spherical harmonics ... parallel recursive QR factoriza- tion algorithms for SMP systems (120-128); Enric Fontdecaba Baig, José M. ...
In this paper we design and implement algorithms for polynomial multiplication using approaches based the fast Fourier transform (FFT) and the truncated Fourier transform (TFT). ... Polynomial multiplication is a key algorithm underlying computer algebra systems (CAS) and its efficient implementation is crucial for the performance of CAS. ... Recently,  introduced parallel algorithms for sparse polynomial multiplication using heaps, and [11, 16] focused on sparse interpolation for multivariate polynomial and for over finite fields, ...arXiv:1609.01010v1 fatcat:73syghptxja5tjb5ifmfomv7uu
Lecture Notes in Computer Science
the traditional algorithms for polynomial evaluation and interpolation based on subproduct-tree trees, by introducing the notion of a subinverse tree. ... Contributions of this work Summary We propose parallel algorithms for performing subproduct tree construction, evaluation and interpolation and report on their implementation on many-core GPUs We enhance ...doi:10.1007/978-3-319-10515-4_13 fatcat:wnvxdha7hrbwxecz76npmh3bsq
More precisely, we focus on a set of significant examples and prove that: (1) the fastest known polynomial evaluation algorithms are not stable, (2) LUP decomposition cannot be computed stably in polylogarithmic ... parallel time, and (3) reductions among computational problems do not in general constitute a practical tool for solving numerical problems. ... The known O(n log 2 n) methods for polynomial evaluation at many points, which reduces to fast polynomial multiplication and division, are not strongly stable. Proof. ...doi:10.1006/jcom.1996.0016 fatcat:mss7vwdmivdlph7dcpphmfcegm
A simple parallel algorithm for the evaluation of polynomials written in the Chebyshev form is introduced. ... By this method only 2 ⌈log 2 (p − 2)⌉ + ⌈log 2 p⌉ + 4 ⌈N/p⌉ − 7 steps on p processors are needed to evaluate a Chebyshev series of degree N . ... For a general value N the algorithm is similar, but it requires small changes. First, we express the evaluation of the Chebyshev polynomials in parallel. ...doi:10.1137/s1064827596312857 fatcat:en3utck2xzhsbg2toymg62fm2y
We present a novel framework for parallel exact inference in graphical models. ... Our main technical contribution amounts to designing a low-degree polynomial extension of the cutset approach, and then reducing to a univariate polynomial employing techniques recently developed for noninteractive ... Fast Algorithms for Univariate Polynomials. Let us recall the basic toolkit for computing with univariate polynomials (von zur Gathen and Gerhard 2013). ...doi:10.1609/aaai.v34i06.6580 fatcat:avzrzy2ctbavvdz2rj2bjvohpi
In this paper, we introduce a general parallel algorithm for the evaluation of Chebyshev and trigonometric series. ... Several examples, carried out on a Cray T3D, are provided comparing the Forsythe and Clenshaw parallel algorithms. (~) ... The plot, for each polynomial, has first a region of fast decrease, being the slope higher for low degree polynomials. ...doi:10.1016/s0898-1221(99)00289-8 fatcat:2zldehdwejglhobvcq4ewngvqm
This paper develops some new techniques in order to devise fast parallel algorithms for improved parallel computations, including the evaluation of Krylov sequences for Toeplitz-like and Hankel- like matrices ... The results can be further extended to such computational problems as fast parallel evaluation of rank T, of the null space of 7, of the minimum span of a linear recurrence sequence (Berlekamp-Massey problem ...
The algorithm consists of a two-stage recurrence relation for radial polynomials and coefficients of the polynomials, which are specifically derived for fast computation of pseudo-Zernike moments. ... This paper discusses the drawbacks of the existing methods, and proposes an efficient recursive algorithm to compute the pseudo-Zernike moments. ... The algorithm comprises of two recurrence relations that are specifically derived for radial polynomials and coefficients of the polynomials for fast computation of pseudo-Zernike moments. ...doi:10.1142/s0218001403002769 fatcat:7hefdi46r5bzholuktmv3lgs3e
Further results include novel algorithms for counting triangles in sparse graphs, computing the chromatic polynomial of a graph, and computing the Tutte polynomial of a graph. ... Complexity, Report TR16-002, January 2016] with the observation that Merlin's magic is not needed for batch evaluation---mere Knights can prepare the proof, in parallel, and with intrinsic error-correction ... Acknowledgments We thank Ryan Williams for useful discussions and giving us early access to his manuscript describing his batch evaluation framework  of which the present "Camelot" framework forms ...arXiv:1602.01295v1 fatcat:pqozv4nul5e7rakfbsra2al7pu
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