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Fast computation of divided differences and parallel hermite interpolation

Ömer Eǧecioǧlu, E Gallopoulos, Çetin K Koç
1989 Journal of Complexity  
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

A parallel method for fast and practical high-order newton interpolation

Ö. EĞecioĞlu, E. Gallopoulos, Ç. K. Koç
1990 BIT Numerical Mathematics  
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

Dense Arithmetic over Finite Fields with the CUMODP Library [chapter]

Sardar Anisul Haque, Xin Li, Farnam Mansouri, Marc Moreno Maza, Wei Pan, Ning Xie
2014 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

Engineering a Delegatable and Error-Tolerant Algorithm for Counting Small Subgraphs [chapter]

Petteri Kaski
2018 2018 Proceedings of the Twentieth Workshop on Algorithm Engineering and Experiments (ALENEX)  
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

Some parallel methods for polynomial root-finding

A.J. Maeder, S.A. Wynton
1987 Journal of Computational and Applied Mathematics  
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

Page 5714 of Mathematical Reviews Vol. , Issue 2000h [page]

2000 Mathematical Reviews  
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.  ... 

Automatic Library Generation for Modular Polynomial Multiplication [article]

Lingchuan Meng
2016 arXiv   pre-print
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, [30] 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

On the Parallelization of Subproduct Tree Techniques Targeting Many-Core Architectures [chapter]

Sardar Anisul Haque, Farnam Mansouri, Marc Moreno Maza
2014 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

On Speed versus Accuracy: Some Case Studies

Mauro Leoncini
1996 Journal of Complexity  
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 Parallel Algorithm to Evaluate Chebyshev Series on a Message Passing Environment

Roberto Barrio, Javier Sabadell
1998 SIAM Journal on Scientific Computing  
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

Error-Correcting and Verifiable Parallel Inference in Graphical Models

Negin Karimi, Petteri Kaski, Mikko Koivisto
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

Parallel evaluation of Chebyshev and trigonometric series

R. Barrio, J. Sabadell
1999 Computers and Mathematics with Applications  
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

Page 3470 of Mathematical Reviews Vol. , Issue 94f [page]

1994 Mathematical Reviews  
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  ... 

An Efficient Algorithm for Fast Computation of Pseudo-Zernike Moments

Chee-Way Chong, P. Raveendran, R. Mukundan
2003 International journal of pattern recognition and artificial intelligence  
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

How proofs are prepared at Camelot [article]

Andreas Björklund, Petteri Kaski
2016 arXiv   pre-print
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 [35] of which the present "Camelot" framework forms  ... 
arXiv:1602.01295v1 fatcat:pqozv4nul5e7rakfbsra2al7pu
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