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Fast modular transforms

1974
*
Journal of computer and system sciences (Print)
*

Using

doi:10.1016/s0022-0000(74)80029-2
fatcat:ydkoolcujvc75kjfl2p2me6fbi
*a**polynomial**division*algorithm due to Strassen [24], it is shown that*a**polynomial**of*degree N --1 can be*evaluated*at N points in O(N log 2 N) total operations or O(N log N) multiplications. ... Using these results, it is shown that*a**polynomial**of*degree N*and*all its derivatives can be*evaluated*at*a*point in O(N log s N) total operations. ~ l fB B ,B' ... If we assume that*division**with**remainder*requires R(N) = O(N log*a*N) steps, then E(N) is defined*by*E(N) = 2E(N/2) + 2R(N) = 2E(N/2) + 20(N log*a*g),*and*so E(N) = O(N log ~+1 N)*by*the results*of*Section ...##
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Fast In-place Algorithms for Polynomial Operations: Division, Evaluation, Interpolation
[article]

2020
*
arXiv
*
pre-print

We consider space-saving versions

arXiv:2002.10304v3
fatcat:nlfvaxsk6nguxmoy7hewkzttie
*of*several important operations on univariate*polynomials*, namely power series inversion*and**division*,*division**with**remainder*, multi-point*evaluation*,*and**interpolation*... We also provide*a*precise complexity analysis so that all constants are made explicit, parameterized*by*the space usage*of*the underlying multiplication algorithms. ... Acknowledgments We thank Grégoire Lecerf, Alin Bostan*and*Michael Monagan for pointing out the references [7, 16] . ...##
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What Can (and Can't) we Do with Sparse Polynomials?

2018
*
Proceedings of the 2018 ACM on International Symposium on Symbolic and Algebraic Computation - ISSAC '18
*

In this tutorial we examine the state

doi:10.1145/3208976.3209027
dblp:conf/issac/Roche18
fatcat:fygzzsxjwrdk7kfxw4z4g7zsjq
*of*the art for sparse*polynomial*algorithms in three areas: arithmetic,*interpolation*,*and*factorization. ... Simply put,*a*sparse*polynomial*is one whose zero coefficients are not explicitly stored. ... AWD_ID=1319994)*and*1618269 (https://www.nsf.gov/ awardsearch/showAward?AWD_ID=1618269). ...##
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On arithmetical algorithms over finite fields

1989
*
Journal of combinatorial theory. Series A
*

The fast Fourier transform (FFT) is

doi:10.1016/0097-3165(89)90020-4
fatcat:slp47ctwzzatlkzlr2tnaegtha
*a*method for efficiently*evaluating*(or*interpolating*)*a**polynomial**of*degree <n at all*of*the nth roots*of*unity, i.e., on the finite multiplicative subgroups*of*F, ... Such*a**polynomial*is usually obtained*by*choosing it randomly*and*then verifying that it is irreducible, using*a*probabilistic algorithm. If it is not, the procedure is repeated. ... Suppose we start*with**a**polynomial**a*(r)*of*degree <n =pm*and*that we wish to*evaluate*it at each point*of*W,,,. ...##
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On the genericity of the modular polynomial GCD algorithm

1999
*
Proceedings of the 1999 international symposium on Symbolic and algebraic computation - ISSAC '99
*

In this paper we study the generic setting

doi:10.1145/309831.309861
dblp:conf/issac/KaltofenM99
fatcat:b3d3tgpkpneybig62duxhzgp2a
*of*the modular GCD algorithm. We develop the algorithm for multivariate*polynomials*over Euclidean domains which have*a*special kind*of**remainder*function. ... Applying this generic algorithm to*a*GCD problem in Z/(p)[t] [x] where p is small yields an improved asymptotic performance over the usual approach,*and**a*very practical algorithm for*polynomials*over ... The external*and*program committee reviewers saw the strengths*of*this paper through the weaknesses*of*our presentation,*and*encouraged us to improve the latter. We thank all*of*them. ...##
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Polynomial division and its computational complexity

1986
*
Journal of Complexity
*

(ii) Then we accelerate parallel

doi:10.1016/0885-064x(86)90001-4
fatcat:ixnugw7k35bbdo4b6rvi7rhhfm
*division**of*two*polynomials**with*integer coefficients*of*degrees at most m*by**a*factor*of*log m comparing*with*the parallel version*of*the algorithm*of*Sieveking*and*Kung ... (iii) Finally the authors' new algorithm improves the estimates for sequential time complexity*of**division**with**a**remainder**of*two integer*polynomials**by**a*factor*of*log m, m being the degree*of*the dividend ...*By*the definition*of*integer*and**polynomial**division**with**a**remainder*(Knuth, 1981) , 0 5 rx < IWI, r, = s(x) -t(. ...##
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Fast Approximate Polynomial Multipoint Evaluation and Applications
[article]

2016
*
arXiv
*
pre-print

It is well known that, using fast algorithms for

arXiv:1304.8069v2
fatcat:ai6y4i4virhzvm62cpqf65llwm
*polynomial*multiplication*and**division*,*evaluation**of**a**polynomial*F ∈C[x]*of*degree n at n complex-valued points can be done*with*Õ(n) exact field operations ... We complement this result*by*an analysis*of*approximate multipoint*evaluation**of*F to*a*precision*of*L bits after the binary point*and*prove*a*bit complexity*of*Õ(n(L + τ + nΓ)), where 2^τ*and*2^Γ,*with*... Let Q := f div g*and*R := f mod g denote the exact quotient*and**remainder*in the*polynomial**division**of*f*by*g. ...##
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A Simple Method of Interpolation

1943
*
Proceedings of the National Academy of Sciences of the United States of America
*

Or

doi:10.1073/pnas.29.11.397
pmid:16588633
pmcid:PMC1078640
fatcat:tziwdu4dtrhjdi5pohqj3yfnba
*a*slight economy*of*effort may be achieved if the inverse*of*V is worked out once*and*for all so that the weightings*of*the different exponential terms can be easily determined*by*V-'h. ... If the*a*matrix is symmetrical, the latent vector matrix will be orthogonal so that simple transposition will provide the inverse matrix, except for factors*of*proportionality. ...*Of*course,*a*final decisive check is provided*by**evaluating*the resulting*polynomial*to verify that it does go through the prescribed points. This is best done*by*synthetic*division*. IV. ...##
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Sparse polynomial interpolation and division in soft-linear time
[article]

2022
*
arXiv
*
pre-print

Given

arXiv:2202.08106v1
fatcat:xhsvvjausrgrhg4bu5rqozxk6y
*a*way to*evaluate*an unknown*polynomial**with*integer coefficients, we present new algorithms to recover its nonzero coefficients*and*corresponding exponents. ... At the core*of*our results is*a*new Monte Carlo randomized algorithm to recover an integer*polynomial*f(x) given*a*way to*evaluate*f(θ) m for any chosen integers θ*and*m. ... For the Euclidean*division**of*sparse*polynomials*, the case*of*exact*division*(when the*remainder*is known to be zero) was improved*by*similar techniques [19] . is led to the first algorithm that is quasi-linear ...##
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Fast Kötter-Nielsen-Høholdt Interpolation over Skew Polynomial Rings and its Application in Coding Theory
[article]

2022
*
arXiv
*
pre-print

We propose

arXiv:2207.01319v1
fatcat:66mqx2nd5jgdbkuonm5u4kzgs4
*a*fast divide-*and*-conquer variant*of*Kötter-Nielsen-Høholdt (KNH)*interpolation*algorithm: it inputs*a*list*of*linear functionals on skew*polynomial*vectors,*and*outputs*a*reduced Gröbner basis ... Skew*polynomials*are*a*class*of*non-commutative*polynomials*that have several applications in computer science, coding theory*and*cryptography. ... is the*remainder**evaluation*defined in [28, 29] , which generalizes the concept*of**polynomial**evaluation**by**means**of*(right)*division*. ...##
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Page 469 of Mathematics of Computation Vol. 3, Issue 27
[page]

1949
*
Mathematics of Computation
*

Numerical integration is based also upon Lagrange’s

*polynomial**and*the method*of*“‘undetermined coefficients” is described, which merely*means*the finding*of*coefficients*by*solving*a*system*of*linear ...*A*procedure is given for finding the complex roots*of*algebraic equations*with*real coefficients,*by*synthetic*division**by*quadratic factors, the end result being the real quadratic factor that yields ...##
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Evaluation and interpolation over multivariate skew polynomial rings
[article]

2018
*
arXiv
*
pre-print

This allows to define the

arXiv:1710.09606v2
fatcat:os72tj22afebpf3k5xkzvplnqm
*evaluation**of*any skew*polynomial*at any point*by*unique*remainder**division*. ... The concepts*of**evaluation**and**interpolation*are extended from univariate skew*polynomials*to multivariate skew*polynomials*,*with*coefficients over*division*rings. ... DFF-5137-00076B "EliteForsk-Rejsestipendium",*and*Grant No. DFF-7027-00053B). ...##
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Diversification improves interpolation
[article]

2011
*
arXiv
*
pre-print

We consider the problem

arXiv:1101.3682v3
fatcat:zavsjkp4cjaafd77hgav7dmwce
*of**interpolating*an unknown multivariate*polynomial**with*coefficients taken from*a*finite field or as numerical approximations*of*complex numbers. ... Building on the recent work*of*Garg*and*Schost, we improve on the best-known algorithm for*interpolation*over large finite fields*by*presenting*a*Las Vegas randomized algorithm that uses fewer black box ... The comments*and*suggestions*of*the anonymous referees were also very helpful, in particular regarding connections to previous results*and*the proof*of*Theorem 3.1. ...##
###
Decomposition of the infinite companion and interpolation

1995
*
Linear Algebra and its Applications
*

Connections

doi:10.1016/0024-3795(93)00085-e
fatcat:erabx442krhtzg6rvjv7hm6m5a
*with**interpolation*problems*and*partial fraction decompositions*of*rational functions are explained. ... Explicit formulae are given for the inverse*of*the Chinese*remainder*operator*and*for the inverse*of*the confluent Vandermonde matrix. ... Suppose = (X -*a*)x3*with*IZY # 0*and*the operator R = R(x -CY, x3) defined*by*the requirement that u -Ru be*divisible**by*x3, the*polynomial*Ru being*of*the form (X -cu) r(x)*with*at most quadratic r. ...##
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High Degree Toom'n'Half for Balanced and Unbalanced Multiplication

2011
*
2011 IEEE 20th Symposium on Computer Arithmetic
*

The described method generates quite an efficient sequence

doi:10.1109/arith.2011.12
dblp:conf/arith/Bodrato10
fatcat:shpdije5nneyjoreaqe4si5axa
*of*operations*and*the memory footprint is kept low*by*using*a*new strategy: mixing*evaluation*,*interpolation**and*recomposition phases. ... Some hints*and*tricks to automatically obtain high degree Toom-Cook implementations, i.e. functions for integer or*polynomial*multiplication*with**a*reduced complexity. ... Moreover the author thanks the anonymous reviewers for their valuable suggestions*and*corrections. ...
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