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Fast algorithms for the characteristics polynomial

Walter Keller-Gehrig
1985 Theoretical Computer Science  
Computing the coefficients of the characteristic polynomial is about as hard as matrix multiplication.  ...  Acknowledgment The contents of this paper constitutes part of the author's Master's Thesis (Diplomarbeit, Institut fiir Angewandte Mathematik, Universitiit of Ziirich, 1982).  ...  The author is indebted to Prof. V. Strassen who directed this thesis.  ... 
doi:10.1016/0304-3975(85)90049-0 fatcat:4gsko26vq5byrjx7cheryfidxm

Page 2218 of Mathematical Reviews Vol. , Issue 93d [page]

1993 Mathematical Reviews  
A nearly optimal algorithm for computing character- istic sets is also described; if n is the number of variables and d the maximum degree of the s input polynomials, then this algo- rithm has a sequential  ...  This paper surveys the theory of Wu-Ritt characteristic sets for polynomial ideals.  ... 

Fast beampattern evaluation by polynomial rooting

P. Häcker, S. Uhlich, B. Yang
2011 Advances in Radio Science  
The algorithm works for arrays having the sensors on a uniformly spaced grid. We use a general version of the gcd (greatest common divisor) function in order to write the problem as a polynomial.  ...  In this paper, a novel algorithm is proposed to find all maxima of an array's beampattern fast and reliably, leading to accelerated array optimizations.  ...  This research has been conducted in cooperation with the Robert Bosch Corporation, to which the authors would like to express their gratitude.  ... 
doi:10.5194/ars-9-145-2011 fatcat:g4pmzkke7bc3vg2moem7lkbwli

Rings : An efficient Java/Scala library for polynomial rings

Stanislav Poslavsky
2018 Computer Physics Communications  
Polynomial arithmetic, GCDs, polynomial factorization and Gr\"obner bases are implemented with the use of modern asymptotically fast algorithms.  ...  The use of the Scala language brings a quite novel powerful, strongly typed functional programming model allowing to write short, expressive, and fast code for applications.  ...  For polynomials over nite elds it switches between classical Euclid algorithm (for small polynomials) and a fast Half-GCD algorithm (for medium and large polynomials, see Sec. 11 in [13] ).  ... 
doi:10.1016/j.cpc.2018.09.005 fatcat:nrizpb23h5bsfomwqvr6z25f2e

Fast computation of special resultants

Alin Bostan, Philippe Flajolet, Bruno Salvy, Éric Schost
2006 Journal of symbolic computation  
We propose fast algorithms for computing composed products and composed sums, as well as diamond products of univariate polynomials.  ...  These operations correspond to special multivariate resultants, that we compute using power sums of roots of polynomials, by means of their generating series.  ...  We wish to thank the referees of an earlier version of this article for their helpful scholarly comments.  ... 
doi:10.1016/j.jsc.2005.07.001 fatcat:qqx7snatwzbu5gyvwwlxkvfpi4

Fast Polynomial Factorization and Modular Composition

Kiran S. Kedlaya, Christopher Umans
2011 SIAM journal on computing (Print)  
We show that modular composition and multipoint evaluation of multivariate polynomials are essentially equivalent, in the sense that an algorithm for one achieving exponent α implies an algorithm for the  ...  We then give two new algorithms that solve the problem near-optimally: an algebraic algorithm for fields of characteristic at most n o(1) , and a nonalgebraic algorithm that works in arbitrary characteristic  ...  We thank Swastik Kopparty and Madhu Sudan for some references mentioned in Section 5, and Ronald de Wolf and the FOCS 2008 referees for helpful comments on the conference paper [KU08] .  ... 
doi:10.1137/08073408x fatcat:vm32on54svfizhtnvgx6h7zfmm

A fast spherical harmonics transform algorithm

Reiji Suda, Masayasu Takami
2001 Mathematics of Computation  
In this paper, we propose a fast approximate algorithm for the associated Legendre transform.  ...  Our algorithm evaluates the transform by means of polynomial interpolation accelerated by the Fast Multipole Method (FMM).  ...  This research is partly supported by the Japan Society for Promotion of Science (Computational Science and Engineering for Global Scale Flow Systems Project), Grant-in-Aid #11450038 of the Ministry of  ... 
doi:10.1090/s0025-5718-01-01386-2 fatcat:5x7cn6uuerfozlrfazpwyu7nga

Chapter 10: Algebraic Algorithms [article]

Ioannis Z. Emiris, Victor Y. Pan, Elias P. Tsigaridas
2013 arXiv   pre-print
Diaz-Herrera, editors, covers Algebraic Algorithms, both symbolic and numerical, for matrix computations and root-finding for polynomials and systems of polynomials equations.  ...  Our Chapter in the upcoming Volume I: Computer Science and Software Engineering of Computing Handbook (Third edition), Allen Tucker, Teo Gonzales and Jorge L.  ...  We can use just O(n log 2 n) ops if we apply fast multipoint polynomial evaluation algorithms based of fast FFT based polynomial division [1, 21, 25, 178, 190] , but then we would face numerical stability  ... 
arXiv:1311.3731v1 fatcat:whtgwztbmbgqbl44s4e663oulu

Accelerated tower arithmetic

Joris van der Hoeven, Grégoire Lecerf
2019 Journal of Complexity  
In this paper we design the first algorithm for general ground fields with a complexity exponent that can be made arbitrarily close to one from the asymptotic point of view.  ...  Nowadays, asymptotically fast algorithms are widely used in computer algebra for computations in towers of algebraic field extensions of small height.  ...  Based on these results, Poteaux and Schost have derived [49] fast algorithms for multiplication, division, traces and primitive element representations for separable towers over finite fields.  ... 
doi:10.1016/j.jco.2019.03.002 fatcat:v7ddzq2no5g3ngv3egcnhg5sg4

Towards factoring bivariate approximate polynomials

Robert M. Corless, Mark W. Giesbrecht, Mark van Hoeij, Ilias S. Kotsireas, Stephen M. Watt
2001 Proceedings of the 2001 international symposium on Symbolic and algebraic computation - ISSAC '01  
A new algorithm is presented for factoring bivariate approximate polynomials over C[a:, y].  ...  Given a particular polynomial, the method constructs a nearby composite polynomial, if one exists, and its irreducible factors.  ...  of bivariate polynomials. • A fast method for bivariate approximate polynomial division.  ... 
doi:10.1145/384101.384114 dblp:conf/issac/CorlessGHKW01 fatcat:weqd4dmozfef7pt7oryzqp5d44

EFFICIENT IMPLEMENTATION OF BIT PARALLEL FINITE FIELD MULTIPLIERS

Ajitha.S.S .
2014 International Journal of Research in Engineering and Technology  
The LUT complexity is evaluated on FPGA by using Xilinx ISE 8.1i.Furthermore,the experimental results on FPGAs for bit parallel Karatsuba Multiplier and Classical Multiplier were shown and the comparison  ...  Arithmetic in Finite/Galois field is a major aspect for many applications such as error correcting code and cryptography.  ...  It is the fast multiplication algorithm and at the same time it is the fast computational algorithm. It uses the technique divide and conquer technique.  ... 
doi:10.15623/ijret.2014.0303122 fatcat:ejsisfgphrai7lep2lojhj3bti

On a family of preimage-resistant functions

Attila Bérczes, János Folláth, Attila Pethő
2010 Tatra Mountains Mathematical Publications  
First we define a large family of polynomials over finite fields and we prove that the members of this family are nearly permutational polynomials.  ...  In the present paper we define a new hash function, based on inhomogeneous polynomials.  ...  The authors would like to thank L a j o s Ró n y a i for his valuable remarks about the complexity of the root finding problem.  ... 
doi:10.2478/v10127-010-0028-3 fatcat:4pqc3u5f65cnxhf4marsv7nxqi

Algebraic and Numerical Algorithms [chapter]

Ioannis Emiris, Victor Pan, Elias Tsigaridas
2009 Algorithms and Theory of Computation Handbook, Second Edition, Volume 1  
For fast algorithms for polynomial shifts, the reader may refer to (27; 114; 115) .  ...  Notice that the sturm algorithm does not assume a square-free polynomial.  ... 
doi:10.1201/9781584888239-c17 fatcat:khegroceujdbpc3ukvlv6s3j4i

Polynomial Factorization over Finite Fields By Computing Euler-Poincare Characteristics of Drinfeld Modules [article]

Anand Kumar Narayanan
2016 arXiv   pre-print
The first algorithm estimates the degree of an irreducible factor of a polynomial from Euler-Poincare characteristics of random Drinfeld modules.  ...  As a consequence, the problem of factoring polynomials over finite fields in time nearly linear in the degree is reduced to finding Euler-Poincare characteristics of random Drinfeld modules with high probability  ...  The obvious procedure to compute χ φ,h for a given a square free h ∈ A and a rank 2 Drinfeld module φ, is to compute the characteristic polynomial of the (F q linear) φ action on F q [t]/(h).  ... 
arXiv:1504.07697v2 fatcat:irqqjhiyt5haxdrdbplyvdb4hy

Polynomial factorization over finite fields by computing Euler–Poincaré characteristics of Drinfeld modules

Anand Kumar Narayanan
2018 Finite Fields and Their Applications  
The first algorithm estimates the degree of an irreducible factor of a polynomial from Euler-Poincare characteristics of random Drinfeld modules.  ...  As a consequence, the problem of factoring polynomials over finite fields in time nearly linear in the degree is reduced to finding Euler-Poincare characteristics of random Drinfeld modules with high probability  ...  Acknowledgement I thank Lior Bary-Soroker, Zeyo Guo, Ming-Deh Huang and Chris Umans for valuable discussions.  ... 
doi:10.1016/j.ffa.2018.08.003 fatcat:ulwspxyzurgbxerjt2w2crfnty
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