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### Fast modular transforms

A. Borodin, R. Moenck
1974 Journal of computer and system sciences (Print)
Using a polynomial division algorithm due to Strassen , 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  ...

### Fast In-place Algorithms for Polynomial Operations: Division, Evaluation, Interpolation [article]

Pascal Giorgi, Bruno Grenet, Daniel S. Roche
2020 arXiv   pre-print
We consider space-saving versions 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] .  ...

### 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.  ...  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).  ...

### On arithmetical algorithms over finite fields

David G Cantor
1989 Journal of combinatorial theory. Series A
The fast Fourier transform (FFT) is 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,,,.  ...

### On the genericity of the modular polynomial GCD algorithm

Erich Kaltofen, Michael B. Monagan
1999 Proceedings of the 1999 international symposium on Symbolic and algebraic computation - ISSAC '99
In this paper we study the generic setting 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.  ...

### Polynomial division and its computational complexity

Dario Bini, Victor Pan
1986 Journal of Complexity
(ii) Then we accelerate parallel 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(.  ...

### Fast Approximate Polynomial Multipoint Evaluation and Applications [article]

Alexander Kobel, Michael Sagraloff
2016 arXiv   pre-print
It is well known that, using fast algorithms for 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.  ...

### A Simple Method of Interpolation

P. A. Samuelson
1943 Proceedings of the National Academy of Sciences of the United States of America
Or 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.  ...

### Sparse polynomial interpolation and division in soft-linear time [article]

Pascal Giorgi, Bruno Grenet, Armelle Perret du Cray, Daniel S. Roche
2022 arXiv   pre-print
Given 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  . is led to the first algorithm that is quasi-linear  ...

### Fast Kötter-Nielsen-Høholdt Interpolation over Skew Polynomial Rings and its Application in Coding Theory [article]

Hannes Bartz and Thomas Jerkovits and Johan Rosenkilde
2022 arXiv   pre-print
We propose 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.  ...

### 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  ...

### Evaluation and interpolation over multivariate skew polynomial rings [article]

Umberto Martínez-Peñas, Frank R. Kschischang
2018 arXiv   pre-print
This allows to define the 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).  ...

### Diversification improves interpolation [article]

Mark Giesbrecht, Daniel S. Roche
2011 arXiv   pre-print
We consider the problem 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

Vlastimil Pták
1995 Linear Algebra and its Applications
Connections 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.  ...

### High Degree Toom'n'Half for Balanced and Unbalanced Multiplication

Marco Bodrato
2011 2011 IEEE 20th Symposium on Computer Arithmetic
The described method generates quite an efficient sequence 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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