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Algorithms for computing the sparsest shifts of polynomials via the Berlekamp/Massey algorithm

Mark Giesbrecht, Erich Kaltofen, Wen-shin Lee
2002 Proceedings of the 2002 international symposium on Symbolic and algebraic computation - ISSAC '02  
Acknowledgements: We thank George Labahn and three anonymous referees for their comments. Note: many of the authors' publications cited below are accessible through links in their Internet homepages.  ...  ., the Berlekamp/Massey algorithm gives us a black box for the discrepancy polynomials.  ...  We give a new class of algorithms for efficiently computing a sparsest or absolutely sparsest shift.  ... 
doi:10.1145/780506.780519 dblp:conf/issac/GiesbrechtKL02 fatcat:stbmtl4tcbewllqivxe5mfpuj4

Sparse polynomial interpolation and Berlekamp/Massey algorithms that correct outlier errors in input values

Matthew T. Comer, Erich L. Kaltofen, Clément Pernet
2012 Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation - ISSAC '12  
Finally, we present the Majority Rule Berlekamp/Massey algorithm, which can recover the unique minimal linear generator of degree t when given bounds T ≥ t and E ≥ e and the initial sequence segment of  ...  We propose algorithms performing sparse interpolation with errors, based on Prony's-Ben-Or's & Tiwari's algorithm, using a Berlekamp/Massey algorithm with early termination.  ...  Acknowledgments: We thank Wen-shin Lee and the referees for their helpful comments.  ... 
doi:10.1145/2442829.2442852 dblp:conf/issac/ComerKP12 fatcat:dnxp6kcrsjd3lljcgekaklcvf4

Algorithms for computing sparsest shifts of polynomials in power, Chebyshev, and Pochhammer bases

Mark Giesbrecht, Erich Kaltofen, Wen-shin Lee
2003 Journal of symbolic computation  
We give a new class of algorithms for computing sparsest shifts of a given polynomial.  ...  Through reformulating as multivariate shifts in a designated set, our algorithms can compute the sparsest shifts that simultaneously minimize the terms of a given set of polynomials.  ...  Acknowledgements This material is based on work supported in part by Natural Sciences and Engineering Research Council of Canada, the Ontario Research & Development Challenge Fund, and Maplesoft (Giesbrecht  ... 
doi:10.1016/s0747-7171(03)00087-7 fatcat:nrvwxfpd3bfirii5xubb6sl2eq

Early termination in sparse interpolation algorithms

Erich Kaltofen, Wen-shin Lee
2003 Journal of symbolic computation  
A probabilistic strategy, early termination, enables different interpolation algorithms to adapt to the degree or the number of terms in the target polynomial when neither is supplied in the input.  ...  . , i } for 1 ≤ i ≤ t, we have det A i = (B i C tB Tr i ) I,I =  ...  We have run each algorithm ten times for each polynomial with different random numbers and taken the average of the number of black box probes needed. The results are listed in Table 2 .  ... 
doi:10.1016/s0747-7171(03)00088-9 fatcat:bna2bi7uzvh5nfvlbh2feg2kka

Page 8152 of Mathematical Reviews Vol. , Issue 2004j [page]

2004 Mathematical Reviews  
Mark Giesbrecht, Erich Kaltofen and Wen-shin Lee, Algo- rithms for computing the sparsest shifts of polynomials via the Berlekamp/Massey algorithm (101-108 (electronic)); Mark van Hoeij and Michael Monagan  ...  in a symbolic computation system (109-115 (electronic)); Elisabetta Fortuna, Patrizia Gianni and Barry Trager, Computation of the radical of polynomial ideals over fields of arbitrary characteristic (  ... 

A Novel Sub-Nyquist Multiband Signal Detection Algorithm for Cognitive Radio [article]

Kai Cao, Peizhong Lu, Yan Zou, Lin Ling
2018 arXiv   pre-print
We exploit the linear recurrent relation between the locations of nonzero frequencies and the DFT of the arithmetic-shifted subsampled signals.  ...  These relations can be uniquely expressed by a series of the so-called frequency locator polynomials (FLPs). The scalar of the relations is related to the bandwidths of the subsignals.  ...  When { } { } 1 1 rank rank T i n = = Y Y , if 2 i r n ≥ 1, 2 i d r n = ≥ , we obtain the unique nontrivial solution of (12) by using Berlekamp-Massey algorithm [27] .  ... 
arXiv:1802.03897v1 fatcat:ivp6b3gj3ffwhkmlodzjy5cxde

What Can (and Can't) we Do with Sparse Polynomials?

Daniel S. Roche
2018 Proceedings of the 2018 ACM on International Symposium on Symbolic and Algebraic Computation - ISSAC '18  
In this tutorial we examine the state of the art for sparse polynomial algorithms in three areas: arithmetic, interpolation, and factorization.  ...  However, with this compact storage comes new algorithmic challenges, as fast algorithms for dense polynomials may no longer be efficient.  ...  ACKNOWLEDGMENTS This work was performed while the author was graciously hosted by the Laboratoire Jean Kuntzmann at the Université Grenoble Alpes.  ... 
doi:10.1145/3208976.3209027 dblp:conf/issac/Roche18 fatcat:fygzzsxjwrdk7kfxw4z4g7zsjq

msolve: A Library for Solving Polynomial Systems [article]

Jérémy Berthomieu
2021 arXiv   pre-print
The core algorithmic framework of msolve relies on Gröbner bases and linear algebra based algorithms for polynomial system solving.  ...  exploit the computational capabilities of this algorithmic framework.  ...  Faugère for his advices and support and for providing us the rational parametrizations of Katsuran for 15 ≤ n ≤ 17, and A. Bostan for his comments on this paper.  ... 
arXiv:2104.03572v1 fatcat:nqbrm5mijfd7pgtw33dd3bdm24

Computational linear algebra over finite fields [article]

Jean-Guillaume Dumas, Clément Pernet (INRIA Grenoble Rhône-Alpes / LIG Laboratoire d'Informatique de Grenoble)
2012 arXiv   pre-print
We present here algorithms for efficient computation of linear algebra problems over finite fields.  ...  Definition 10 1 . 1 The Krylov matrix of order d for a vector ; 2 : 2 Use the Berlekamp-Massey algorithm to compute the minimal polynomial of the scalar sequence S; Definition 17 We extend Euler's totient  ...  System solving and the Lanczos algorithm For the solution of a linear system Ax = b, one could compute the minimal polynomial Π A,b and then derive a solution of the system as a linear combination of the  ... 
arXiv:1204.3735v1 fatcat:a3j26roivfd55fsf3sk7ahxgk4

A unified approach to sparse signal processing

Farokh Marvasti, Arash Amini, Farzan Haddadi, Mehdi Soltanolkotabi, Babak Khalaj, Akram Aldroubi, Saeid Sanei, Jonathon Chambers
2012 EURASIP Journal on Advances in Signal Processing  
The Error Locator Polynomial (ELP) and iterative methods are shown to work quite effectively for both sampling and coding applications.  ...  For each of these fields, various algorithms and techniques, which have been developed to leverage sparsity, are described succinctly.  ...  We are also thankful to the students of the Multimedia Lab and members of ACRI at Sharif University of Technology for their invaluable help and simulations. We are specifically indebted to A.  ... 
doi:10.1186/preaccept-1686979482577015 fatcat:7zjru2zjpne3rceluc5empoanq

A unified approach to sparse signal processing

Farokh Marvasti, Arash Amini, Farzan Haddadi, Mahdi Soltanolkotabi, Babak Hossein Khalaj, Akram Aldroubi, Saeid Sanei, Janathon Chambers
2012 EURASIP Journal on Advances in Signal Processing  
The Error Locator Polynomial (ELP) and iterative methods are shown to work quite effectively for both sampling and coding applications.  ...  For each of these fields, various algorithms and techniques, which have been developed to leverage sparsity, are described succinctly.  ...  We are also thankful to the students of the Multimedia Lab and members of ACRI at Sharif University of Technology for their invaluable help and simulations. We are specifically indebted to A.  ... 
doi:10.1186/1687-6180-2012-44 fatcat:sbby5qh65bh5jd57i567br7el4

A Unified Approach to Sparse Signal Processing [article]

F. Marvasti, A. Amini, F. Haddadi, M. Soltanolkotabi, B. H. Khalaj, A. Aldroubi, S. Holm, S. Sanei, J. Chambers
2009 arXiv   pre-print
Specifically, the relations of the approach of Prony to an annihilating filter and Error Locator Polynomials in coding are emphasized; the Pisarenko and MUSIC methods are further improvements of the Prony  ...  For each of these fields, various algorithms and techniques, which have been developed to leverage sparsity, are described succinctly.  ...  The Berlekamp-Massey method for real numbers is sensitive to noise and will not be discussed here [111] .  ... 
arXiv:0902.1853v1 fatcat:lxd3crtgd5fl5ofshryzm5pkfi

Design of large polyphase filters in the Quadratic Residue Number System

Gian Carlo Cardarilli, Alberto Nannarelli, Yann Oster, Massimo Petricca, Marco Re
2010 2010 Conference Record of the Forty Fourth Asilomar Conference on Signals, Systems and Computers  
Mp6-6 3:55 pM A Low Energy High Speed Reed-Solomon Decoder Using Decomposed inversionless Berlekamp-Massey Algorithm Hazem A.  ...  Fahmy, Cairo university This paper proposes an area efficient, low energy, high speed pipelined architecture for a Reed-Solomon decoder based on Decomposed Inversionless Berlekamp-Massey Algorithm, where  ... 
doi:10.1109/acssc.2010.5757589 fatcat:ccxnu5owr5fyrcjcqukumerueq

Low Complexity Decoding of Non-binary Low-density Parity-check Codes

WIJEKOON MUDIYANSELAGEDARA VIDURANGA B WIJEKOON
2021
But they are not widely used in practice due to the high computational complexity in decoding.  ...  The thesis presents some novel strategies to reduce the said decoding complexity of NB-LDPC codes so that more practical applications could benefit from their impressive error-correcting performance.  ...  In an overwhelming majority of the applications where RS codes are used, they are decoded via the popular Berlekamp-Massey Lemma 5 . 1 . 51 Consider the (n, k) RS code over F q .  ... 
doi:10.26180/14854416 fatcat:i2achqumzfhsfilaklhab6aqyu