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Book An Interpolation Algorithm for Sparse Polynomials Over Zm

Download or read book An Interpolation Algorithm for Sparse Polynomials Over Zm written by Kai Werther and published by . This book was released on 1993 with total page 30 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Book Finite Fields Theory and Computation

Download or read book Finite Fields Theory and Computation written by Igor Shparlinski and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 532 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is mainly devoted to some computational and algorithmic problems in finite fields such as, for example, polynomial factorization, finding irreducible and primitive polynomials, the distribution of these primitive polynomials and of primitive points on elliptic curves, constructing bases of various types and new applications of finite fields to other areas of mathematics. For completeness we in clude two special chapters on some recent advances and applications of the theory of congruences (optimal coefficients, congruential pseudo-random number gener ators, modular arithmetic, etc.) and computational number theory (primality testing, factoring integers, computation in algebraic number theory, etc.). The problems considered here have many applications in Computer Science, Cod ing Theory, Cryptography, Numerical Methods, and so on. There are a few books devoted to more general questions, but the results contained in this book have not till now been collected under one cover. In the present work the author has attempted to point out new links among different areas of the theory of finite fields. It contains many very important results which previously could be found only in widely scattered and hardly available conference proceedings and journals. In particular, we extensively review results which originally appeared only in Russian, and are not well known to mathematicians outside the former USSR.

Book Sparse Polynomial Interpolation and the Fast Euclidean Algorithm

Download or read book Sparse Polynomial Interpolation and the Fast Euclidean Algorithm written by Su Ko and published by . This book was released on 2012 with total page 152 pages. Available in PDF, EPUB and Kindle. Book excerpt: We introduce an algorithm to interpolate sparse multivariate polynomials with integer coefficients. Our algorithm modifies Ben-Or and Tiwari's deterministic algorithm for interpolating over rings of characteristic zero to work modulo p, a smooth prime of our choice. We present benchmarks comparing our algorithm to Zippel's probabilistic sparse interpolation algorithm, demonstrating that our algorithm makes fewer probes for sparse polynomials. Our interpolation algorithm requires finding roots of a polynomial in GF(p)[x], which in turn requires an efficient polynomial GCD algorithm. Motivated by this observation, we review the Fast Extended Euclidean algorithm for univariate polynomials, which recursively computes the GCD using a divide-and-conquer approach. We present benchmarks for our implementation of the classical and fast versions of the Euclidean algorithm demonstrating a good speedup. We discuss computing resultants as an application of the fast GCD algorithm.

Book Sparse Polynomial Interpolation and the Fast Euclidean Algorithm

Download or read book Sparse Polynomial Interpolation and the Fast Euclidean Algorithm written by Soo Go and published by . This book was released on 2012 with total page 152 pages. Available in PDF, EPUB and Kindle. Book excerpt: We introduce an algorithm to interpolate sparse multivariate polynomials with integer coefficients. Our algorithm modifies Ben-Or and Tiwari's deterministic algorithm for interpolating over rings of characteristic zero to work modulo p, a smooth prime of our choice. We present benchmarks comparing our algorithm to Zippel's probabilistic sparse interpolation algorithm, demonstrating that our algorithm makes fewer probes for sparse polynomials. Our interpolation algorithm requires finding roots of a polynomial in GF(p)[x], which in turn requires an efficient polynomial GCD algorithm. Motivated by this observation, we review the Fast Extended Euclidean algorithm for univariate polynomials, which recursively computes the GCD using a divide-and-conquer approach. We present benchmarks for our implementation of the classical and fast versions of the Euclidean algorithm demonstrating a good speedup. We discuss computing resultants as an application of the fast GCD algorithm.

Book Efficient Algorithms for Computations with Sparse Polynomials

Download or read book Efficient Algorithms for Computations with Sparse Polynomials written by Seyed Mohammad Mahdi Javadi and published by . This book was released on 2011 with total page 252 pages. Available in PDF, EPUB and Kindle. Book excerpt: The problem of interpolating a sparse polynomial has always been one of the central objects of research in the area of computer algebra. It is the key part of many algorithms such as polynomial GCD computation. We present a probabilistic algorithm to interpolate a sparse multivariate polynomial over a finite field, represented with a black box. Our algorithm modifies the Ben-Or/Tiwari algorithm from 1988 for interpolating polynomials over rings with characteristic zero to positive characteristics by doing additional probes. To interpolate a polynomial in n variables with t non-zero terms, Zippel's algorithm interpolates one variable at a time using O(ndt) probes to the black box where d bounds the degree of the polynomial. Our new algorithm does O(nt) probes. We provide benchmarks comparing our algorithm to Zippel's algorithm and the racing algorithm of Kaltofen/Lee. The benchmarks demonstrate that for sparse polynomials our algorithm often makes fewer probes. A key advantage in our new algorithm is, unlike the other two algorithms, it can be parallelized efficiently. Our main application for an efficient sparse interpolation algorithm is computing GCDs of polynomials. We are especially interested in polynomials over algebraic function fields. The best GCD algorithm available is SparseModGcd, presented by Javadi and Monagan in 2006. We further improve this algorithm in three ways. First we prove that we can eliminate the trial divisions in positive characteristic. Trial divisions are the bottleneck of the algorithm for denser polynomials. Second, we give a new (and correct) solution to the normalization problem. Finally we will present a new in-place library of functions for computing GCDs of univariate polynomials over algebraic number fields. Furthermore we present an efficient algorithm for factoring multivariate polynomials over algebraic fields with multiple field extensions and parameters. Our algorithm uses Hensel lifting and extends the EEZ algorithm of Wang which was designed for factorization over rationals. We also give a multivariate p-adic lifting algorithm which uses sparse interpolation. This enables us to avoid using poor bounds on the size of the integer coefficients in the factorization when using Hensel lifting. We provide timings demonstrating the efficiency of our algorithm.

Book A Zero test and an Interpolation Algorithm for the Shifted Sparse Polynomials

Download or read book A Zero test and an Interpolation Algorithm for the Shifted Sparse Polynomials written by Dima Grigorʹev and published by . This book was released on 1992 with total page 10 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Book Early Termination Strategies in Sparse Interpolation Algorithms

Download or read book Early Termination Strategies in Sparse Interpolation Algorithms written by and published by . This book was released on 2001 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: A black box polynomial is an object that takes as input a valuefor each variable and evaluates the polynomial at the given input. The process of determining the coefficients and terms of a blackbox polynomial is the problem of black box polynomialinterpolation. Two major approaches have been addressing suchpurpose: the dense algorithms whose computational complexities aresensitive to the degree of the target polynomial, and the sparsealgorithms that take advantage of the situation when the number ofnon-zero terms in a designate basis is small. In this dissertationwe cover power, Chebyshev, and Pochhammer term bases. However, asparse algorithm is less efficient when the target polynomial isdense, and both approaches require as input an upper bound oneither the degree or the number of non-zero terms. By introducingrandomization into existing algorithms, we demonstrate and developa probabilistic approach which we call 'early termination'. Inparticular we prove that with high probability of correctness theearly termination strategy makes different polynomialinterpolation algorithms 'smart' by adapting to the degree or tothe number of non-zero terms during the process when either is notsupplied as an input. Based on the early termination strategy, wedescribe new efficient univariate algorithms that race a denseagainst a sparse interpolation algorithm in order to exploit thesuperiority of one of them. We apply these racing algorithms asthe univariate interpolation procedure needed in Zippel's multivariate sparse interpolation method. We enhance the earlytermination approach with thresholds, and present insights toother such heuristic improvements. Some potential of the early termination strategy is observed for computing a sparse shift, where a polynomial becomes sparse through shifting the variables by a constant.

Book Interpolation and Approximation of Sparse Multivariate Polynomials Over GF 2

Download or read book Interpolation and Approximation of Sparse Multivariate Polynomials Over GF 2 written by International Business Machines Corporation. Research Division and published by . This book was released on 1990 with total page 37 pages. Available in PDF, EPUB and Kindle. Book excerpt: It is shown that [formula] evaluation points are sufficient for the (deterministic) [epsilon]-approximation of any t-sparse function, and that an order of [formula] points are necessary for this purpose, where [alpha](t, [epsilon]) [greater than or equal to] 0.694 for a large range of t and [epsilon]. Similar bounds hold for the t-term DNF case as well. Finally, a probabilistic polynomial-time algorithm is presented for the [epsilon]-approximation of any t-sparse function."

Book Computer Algebra in Scientific Computing

Download or read book Computer Algebra in Scientific Computing written by François Boulier and published by Springer Nature. This book was released on 2023-08-23 with total page 441 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the refereed proceedings of the 25th International Workshop on Computer Algebra in Scientific Computing, CASC 2023, which took place in Havana, Cuba, during August 28-September 1, 2023. The 22 full papers included in this book were carefully reviewed and selected from 29 submissions. They focus on the theory of symbolic computation and its implementation in computer algebra systems as well as all other areas of scientific computing with regard to their benefit from or use of computer algebra methods and software.

Book Effective Polynomial Computation

Download or read book Effective Polynomial Computation written by Richard Zippel and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 364 pages. Available in PDF, EPUB and Kindle. Book excerpt: Effective Polynomial Computation is an introduction to the algorithms of computer algebra. It discusses the basic algorithms for manipulating polynomials including factoring polynomials. These algorithms are discussed from both a theoretical and practical perspective. Those cases where theoretically optimal algorithms are inappropriate are discussed and the practical alternatives are explained. Effective Polynomial Computation provides much of the mathematical motivation of the algorithms discussed to help the reader appreciate the mathematical mechanisms underlying the algorithms, and so that the algorithms will not appear to be constructed out of whole cloth. Preparatory to the discussion of algorithms for polynomials, the first third of this book discusses related issues in elementary number theory. These results are either used in later algorithms (e.g. the discussion of lattices and Diophantine approximation), or analogs of the number theoretic algorithms are used for polynomial problems (e.g. Euclidean algorithm and p-adic numbers). Among the unique features of Effective Polynomial Computation is the detailed material on greatest common divisor and factoring algorithms for sparse multivariate polynomials. In addition, both deterministic and probabilistic algorithms for irreducibility testing of polynomials are discussed.

Book Early Termination Strategies in Sparse Interpolation Algorithms

Download or read book Early Termination Strategies in Sparse Interpolation Algorithms written by Wen-shin Lee and published by . This book was released on 2001 with total page 107 pages. Available in PDF, EPUB and Kindle. Book excerpt: Keywords: Sparse polynomial, Black box polynomial, Interpolation, Sparse interpolation, Randomized algorithm, Chebyshev basis, Pochhammer basis, Early termination, Racing two algorithms, Sparse shift, Zippel's algorithm, Ben-Or's and Tiwari's algorithm.

Book Efficient Approximation Algorithms for Sparse Polynomials Over Finite Fields

Download or read book Efficient Approximation Algorithms for Sparse Polynomials Over Finite Fields written by International Computer Science Institute and published by . This book was released on 1994 with total page 20 pages. Available in PDF, EPUB and Kindle. Book excerpt: Abstract: "We obtain new lower bounds on the number of non zeros of sparse polynomials and give a fully polynomial time ([eta], [delta]) approximation algorithm for the number of non-zeros of multivariate sparse polynomials over a finite field of q elements and degree less than q - 1. This answers partially to an open problem of D. Grigoriev and M. Karpinski. Also, probabilistic and deterministic algorithms for testing identity to zero of a sparse polynomial given by a 'black-box' are given. Finally, we propose an algorithm to estimate the size of the image of a univariate sparse polynomial."

Book Sparse Interpolation from Multiple Derivatives

Download or read book Sparse Interpolation from Multiple Derivatives written by International Computer Science Institute and published by . This book was released on 1993 with total page 7 pages. Available in PDF, EPUB and Kindle. Book excerpt: Abstract: "In this note, we consider the problem of interpolating a sparse function from the values of its multiple derivatives at some given point. We give efficient algorithms for reconstructing sparse Fourier series and sparse polynomials over Sturm-Liouville bases. In both cases, the number of evaluations is linear in the sparsity."