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.
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.
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.
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.
Download or read book Modern Computer Algebra written by Joachim von zur Gathen and published by Cambridge University Press. This book was released on 2013-04-25 with total page 811 pages. Available in PDF, EPUB and Kindle. Book excerpt: Computer algebra systems are now ubiquitous in all areas of science and engineering. This highly successful textbook, widely regarded as the 'bible of computer algebra', gives a thorough introduction to the algorithmic basis of the mathematical engine in computer algebra systems. Designed to accompany one- or two-semester courses for advanced undergraduate or graduate students in computer science or mathematics, its comprehensiveness and reliability has also made it an essential reference for professionals in the area. Special features include: detailed study of algorithms including time analysis; implementation reports on several topics; complete proofs of the mathematical underpinnings; and a wide variety of applications (among others, in chemistry, coding theory, cryptography, computational logic, and the design of calendars and musical scales). A great deal of historical information and illustration enlivens the text. In this third edition, errors have been corrected and much of the Fast Euclidean Algorithm chapter has been renovated.
Download or read book Fast Computation of the Rational Interpolation Table and Toeplitz Systems of Equations Via the Fast Extended Euclidean Algorithm written by F. G. Gustavson and published by . This book was released on 1979 with total page 9 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Fast parallel algorithms for sparse multivariate polynomial interpolation over finite fields written by Dima J. Grigorʹev and published by . This book was released on 1988 with total page 20 pages. Available in PDF, EPUB and Kindle. Book excerpt:
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:
Download or read book A Fast Algorithm for Rational Interpolation Via Orthogonal Polynomials written by Ömer Nuri Eğecioğlu and published by . This book was released on 1987 with total page 43 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Fast Algorithms for Polynomial Interpolation Integration and Differentiation written by Yale University. Dept. of Computer Science and published by . This book was released on 1993 with total page 37 pages. Available in PDF, EPUB and Kindle. Book excerpt: Abstract: "For functions tabulated at Chebyshev nodes on an interval, spectral interpolation, integration and differentiation can be performed stably and efficiently via the fast Fourier transform. In this paper, a group of algorithms is presented for the efficient evaluation of Lagrange polynomial interpolants at multiple points on the line, and for the rapid spectral integration and differentiation of functions tabulated at nodes other than Chebyshev. The interpolation scheme requires O(N [multiplied by] log(1/[epsilon])) arithmetic operations, and O(N [multiplied by] log N + N [multiplied by] log(1/[epsilon])) operations are required for the integration and differentiation schemes, where [epsilon] is the precision of computations and N is the numer of nodes. The algorithms utilize efficient versions of the fast multipole method which have been designed specifically for one-dimensional problems; these are also described in the present paper. Several experiments are included to illustrate the numerical performance of the approach."
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:
Download or read book Polynomial and Matrix Computations Fundamental algorithms written by Dario Bini and published by . This book was released on 1994 with total page 474 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Fast intepolation algorithms for sparse polynomials with respect to the size of coefficients written by Aleksandr L. Chistov and published by . This book was released on 1994 with total page 14 pages. Available in PDF, EPUB and Kindle. Book excerpt:
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.
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.
Download or read book Polynomial and Matrix Computations written by Dario Bini and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 433 pages. Available in PDF, EPUB and Kindle. Book excerpt: Our Subjects and Objectives. This book is about algebraic and symbolic computation and numerical computing (with matrices and polynomials). It greatly extends the study of these topics presented in the celebrated books of the seventies, [AHU] and [BM] (these topics have been under-represented in [CLR], which is a highly successful extension and updating of [AHU] otherwise). Compared to [AHU] and [BM] our volume adds extensive material on parallel com putations with general matrices and polynomials, on the bit-complexity of arithmetic computations (including some recent techniques of data compres sion and the study of numerical approximation properties of polynomial and matrix algorithms), and on computations with Toeplitz matrices and other dense structured matrices. The latter subject should attract people working in numerous areas of application (in particular, coding, signal processing, control, algebraic computing and partial differential equations). The au thors' teaching experience at the Graduate Center of the City University of New York and at the University of Pisa suggests that the book may serve as a text for advanced graduate students in mathematics and computer science who have some knowledge of algorithm design and wish to enter the exciting area of algebraic and numerical computing. The potential readership may also include algorithm and software designers and researchers specializing in the design and analysis of algorithms, computational complexity, alge braic and symbolic computing, and numerical computation.
Download or read book Proceedings of the Ninth Annual ACM SIAM Symposium on Discrete Algorithms written by Howard Karloff and published by SIAM. This book was released on 1998-01-01 with total page 726 pages. Available in PDF, EPUB and Kindle. Book excerpt: This symposium is jointly sponsored by the ACM Special Interest Group on Algorithms and Computation Theory and the SIAM Activity Group on Discrete Mathematics.
