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Fast algorithms for the characteristics polynomial

1985
*
Theoretical Computer Science
*

Computing

doi:10.1016/0304-3975(85)90049-0
fatcat:4gsko26vq5byrjx7cheryfidxm
*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. ...##
###
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

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. ...

##
###
Rings : An efficient Java/Scala library for polynomial rings

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] ). ...

##
###
Fast computation of special resultants

2006
*
Journal of symbolic computation
*

We propose

doi:10.1016/j.jsc.2005.07.001
fatcat:qqx7snatwzbu5gyvwwlxkvfpi4
*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. ...##
###
Fast Polynomial Factorization and Modular Composition

2011
*
SIAM journal on computing (Print)
*

We show that modular composition

doi:10.1137/08073408x
fatcat:vm32on54svfizhtnvgx6h7zfmm
*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] . ...##
###
A fast spherical harmonics transform algorithm

2001
*
Mathematics of Computation
*

In this paper, we propose

doi:10.1090/s0025-5718-01-01386-2
fatcat:5x7cn6uuerfozlrfazpwyu7nga
*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 ...##
###
Chapter 10: Algebraic Algorithms
[article]

2013
*
arXiv
*
pre-print

Diaz-Herrera, editors, covers Algebraic

arXiv:1311.3731v1
fatcat:whtgwztbmbgqbl44s4e663oulu
*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 ...##
###
Accelerated tower arithmetic

2019
*
Journal of Complexity
*

In this paper we design

doi:10.1016/j.jco.2019.03.002
fatcat:v7ddzq2no5g3ngv3egcnhg5sg4
*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. ...##
###
Towards factoring bivariate approximate polynomials

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*. ...

##
###
EFFICIENT IMPLEMENTATION OF BIT PARALLEL FINITE FIELD MULTIPLIERS

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. ...

##
###
On a family of preimage-resistant functions

2010
*
Tatra Mountains Mathematical Publications
*

First we define

doi:10.2478/v10127-010-0028-3
fatcat:4pqc3u5f65cnxhf4marsv7nxqi
*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. ...##
###
Algebraic and Numerical Algorithms
[chapter]

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*. ...

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

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). ...

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

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. ...

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