Download or read book Algorithms for interpolation of sparse rational functions using polynomial number of evaluations written by Aleksandr L. Chistov and published by . This book was released on 1995 with total page 46 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Algorithms for Interpolation of Sparse Rational Functions Using Polynomial Number of Evaluations written by Alexander L. Chistov and published by . This book was released on 1995 with total page 46 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Computing Polynomial Greatest Common Divisors Using Sparse Interpolation written by Jiaxiong Hu and published by . This book was released on 2018 with total page 146 pages. Available in PDF, EPUB and Kindle. Book excerpt: Computing polynomial greatest common divisors (GCD) plays an important role in Computer Algebra systems because the GCD operation is the bottleneck of many basic applications. For example, to simplify a rational function one divides the numerator and denominator by their GCD. In 1988 Ben-Or and Tiwari introduced the first deterministic polynomial interpolation algorithm which accounts for sparsity. The number of evaluation points needed by the Ben-Or/Tiwari algorithm is linear in the number of non-zero terms in the target polynomial, and moreover, all variables can be interpolated simultaneously hence parallelizing the algorithm is easier. In this thesis, we present modular multivariate polynomial GCD algorithms based on Ben-Or/Tiwari sparse interpolation. They compute the GCD modulo one or more primes. We apply a Kronecker substitution to reduce the number of variables and we modify the Ben-Or/Tiwari evaluation point sequence so that we can use primes of acceptable size (machine primes) as well as gain randomness on the choice of evaluation points to avoid several known issues in polynomial GCD algorithms. Based on several assumptions, we first present a simplified algorithm for GCD computation in Z[x1, . . . , xn] from which we derive some theoretical bounds and convince the reader why it works. Then we present a practical version of the algorithm where those assumptions are dropped. This leads to a more complicated algorithm but it can be shown that it always terminates and it computes the GCD efficiently. In the 1980s, subsequent research in polynomial GCD algorithm mainly focused on polynomials over number fields. In this thesis, we also present a GCD algorithm for multivariate polynomials in Q(_)[x1, . . . , xn] where _ is an algebraic number. With a prime modulus p, all operations are performed in the finite ring Zp(_) where inversions may fail due to zero divisors. We manage to get all necessary bounds to support the correctness of our algorithm.

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.

Download or read book Solving Parametric Systems Using Dixon Resultants and Sparse Interpolation Tools written by Ayoola Isaac Jinadu and published by . This book was released on 2023 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Many elimination techniques such as Grobner bases and Triangular sets have been employed to address the growing demand for solving parametric polynomial systems in practice. However, experiments have shown that these elimination methods when used in computer algebra systems such as Maple and Magma often fail on systems that have many parameters; they can take a very long time to execute or run out of memory. To address this problem, this thesis presents a new interpolation algorithm for solving parametric polynomial systems (systems of n polynomial equations involving n variables and m parameters with rational coefficients) over Q using Dixon resultants. The Dixon resultant R of a parametric polynomial system is a multiple of the unique monic generator of an elimination ideal of a polynomial system, and it can be expressed as the determinant of a matrix M of polynomial entries called the Dixon matrix. Given a black box for the Dixon resultant R = det(M) (we evaluate the Dixon matrix M at integer points modulo primes and compute determinant of integer matrices modulo primes), we present a new Dixon resultant algorithm that interpolates the monic square-free factors Rj of the Dixon resultant R from monic univariate polynomial images of R. This new Dixon resultant algorithm uses our newly developed sparse multivariate rational function interpolation method over Q to interpolate the rational function coefficients of the monic square-free factors modulo primes. It further uses rational number reconstruction and Chinese remaindering to recover the rational coefficients of the Rj 's. We have made a hybrid Maple and C implementation of our Dixon resultant algorithm. Our benchmarks show that our new Dixon resultant algorithm can solve many parametric polynomial systems that other algorithms for computing R are unable to solve. However, the new Dixon resultant algorithm may fail to produce an answer, and even when it is successful, the returned answer might be wrong with provably low probability. Consequently, we identify and classify all the causes of failure in our new algorithm, and we give a detailed failure probability analysis and complexity analysis of our new Dixon resultant algorithm. Furthermore, we consider another related problem. Let Ax = b be a parametric linear system such that the coefficient matrix A is of full rank. In general, the solutions xi will be rational functions in the parameters. We present a new black box algorithm for interpolating iii the entries xi using our new sparse multivariate rational function interpolation method. We present timing results comparing our hybrid Maple and C implementation of our new algorithm with four other algorithms in Maple for solving Ax = b. A failure probability analysis and complexity analysis for our new algorithm is also presented.

Download or read book Algorithms for Sparse Rational Interpolation written by International Computer Science Institute and published by . This book was released on 1991 with total page 14 pages. Available in PDF, EPUB and Kindle. Book excerpt: Abstract: "We present two algorithms on sparse rational interpolation. The first is the interpolation in a sense of the sparse partial fraction representation of rational functions. The second is the algorithm for computing the entier and the remainder of a rational function. The first algorithm works without apriori known bound on the degree of a rational function, the second one is in the class NC provided the degree is known. The presented algorithms complement the sparse interpolation results of [Grigoriev, Karpinski, and Singer 90b]."

Download or read book Computer Algebra in Scientific Computing written by Vladimir P. Gerdt and published by Springer. This book was released on 2017-09-07 with total page 419 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the proceedings of the 19th International Workshop on Computer Algebra in Scientific Computing, CASC 2017, held in Beijing, China, in September 2017. The 28 full papers presented in this volume were carefully reviewed and selected from 33 submissions. They deal with cutting-edge research in all major disciplines of Computer Algebra.

Download or read book Computer Algebra in Scientific Computing written by François Boulier and published by Springer Nature. This book was released on 2022-08-10 with total page 412 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the proceedings of the 24th International Workshop on Computer Algebra in Scientific Computing, CASC 2022, which took place in Gebze, Turkey, in August 2022. The 20 full papers included in this book were carefully reviewed and selected from 32 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.

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 SIAM Journal on Computing written by Society for Industrial and Applied Mathematics and published by . This book was released on 1995 with total page 712 pages. Available in PDF, EPUB and Kindle. Book excerpt:

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 Computer Algebra in Scientific Computing written by Vladimir P. Gerdt and published by Springer. This book was released on 2014-09-01 with total page 515 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the proceedings of the 16th International Workshop on Computer Algebra in Scientific Computing, CASC 2014, held in Warsaw, Poland, in September 2014. The 33 full papers presented were carefully reviewed and selected for inclusion in this book. The papers address issues such as Studies in polynomial algebra are represented by contributions devoted to factoring sparse bivariate polynomials using the priority queue, the construction of irreducible polynomials by using the Newton index, real polynomial root finding by means of matrix and polynomial iterations, application of the eigenvalue method with symmetry for solving polynomial systems arising in the vibration analysis of mechanical structures with symmetry properties, application of Gröbner systems for computing the (absolute) reduction number of polynomial ideals, the application of cylindrical algebraic decomposition for solving the quantifier elimination problems, certification of approximate roots of overdetermined and singular polynomial systems via the recovery of an exact rational univariate representation from approximate numerical data, new parallel algorithms for operations on univariate polynomials (multi-point evaluation, interpolation) based on subproduct tree techniques.

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: Now in its third edition, this highly successful textbook is widely regarded as the 'bible of computer algebra'.

Download or read book Algorithms and Theory of Computation Handbook 2 Volume Set written by Mikhail J. Atallah and published by CRC Press. This book was released on 2022-05-29 with total page 1904 pages. Available in PDF, EPUB and Kindle. Book excerpt: Algorithms and Theory of Computation Handbook, Second Edition in a two volume set, provides an up-to-date compendium of fundamental computer science topics and techniques. It also illustrates how the topics and techniques come together to deliver efficient solutions to important practical problems. New to the Second Edition: Along with updating and revising many of the existing chapters, this second edition contains more than 20 new chapters. This edition now covers external memory, parameterized, self-stabilizing, and pricing algorithms as well as the theories of algorithmic coding, privacy and anonymity, databases, computational games, and communication networks. It also discusses computational topology, computational number theory, natural language processing, and grid computing and explores applications in intensity-modulated radiation therapy, voting, DNA research, systems biology, and financial derivatives. This best-selling handbook continues to help computer professionals and engineers find significant information on various algorithmic topics. The expert contributors clearly define the terminology, present basic results and techniques, and offer a number of current references to the in-depth literature. They also provide a glimpse of the major research issues concerning the relevant topics

Download or read book Symbolic and Algebraic Computation written by Edward W. Ng and published by Lecture Notes in Computer Science. This book was released on 1979-06 with total page 586 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Computer Algebra Handbook written by Johannes Grabmeier and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 656 pages. Available in PDF, EPUB and Kindle. Book excerpt: This Handbook gives a comprehensive snapshot of a field at the intersection of mathematics and computer science with applications in physics, engineering and education. Reviews 67 software systems and offers 100 pages on applications in physics, mathematics, computer science, engineering chemistry and education.