Computer Algebra And Polynomials

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Computer Algebra and Polynomials

Jaime Gutierrez,Josef Schicho,Martin Weimann
Computer Algebra and Polynomials Book PDF

Author : Jaime Gutierrez,Josef Schicho,Martin Weimann
Publisher : Springer
Page : 213 pages
File Size : 54,6 Mb
Release : 2015-01-20
Category : Computers
ISBN : 9783319150819

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Computer Algebra and Polynomials by Jaime Gutierrez,Josef Schicho,Martin Weimann Pdf

Algebra and number theory have always been counted among the most beautiful mathematical areas with deep proofs and elegant results. However, for a long time they were not considered that important in view of the lack of real-life applications. This has dramatically changed: nowadays we find applications of algebra and number theory frequently in our daily life. This book focuses on the theory and algorithms for polynomials over various coefficient domains such as a finite field or ring. The operations on polynomials in the focus are factorization, composition and decomposition, basis computation for modules, etc. Algorithms for such operations on polynomials have always been a central interest in computer algebra, as it combines formal (the variables) and algebraic or numeric (the coefficients) aspects. The papers presented were selected from the Workshop on Computer Algebra and Polynomials, which was held in Linz at the Johann Radon Institute for Computational and Applied Mathematics (RICAM) during November 25-29, 2013, at the occasion of the Special Semester on Applications of Algebra and Number Theory.

Polynomial Algorithms in Computer Algebra

Franz Winkler
Polynomial Algorithms in Computer Algebra Book PDF

Author : Franz Winkler
Publisher : Springer Science & Business Media
Page : 284 pages
File Size : 45,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783709165713

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Polynomial Algorithms in Computer Algebra by Franz Winkler Pdf

For several years now I have been teaching courses in computer algebra at the Universitat Linz, the University of Delaware, and the Universidad de Alcala de Henares. In the summers of 1990 and 1992 I have organized and taught summer schools in computer algebra at the Universitat Linz. Gradually a set of course notes has emerged from these activities. People have asked me for copies of the course notes, and different versions of them have been circulating for a few years. Finally I decided that I should really take the time to write the material up in a coherent way and make a book out of it. Here, now, is the result of this work. Over the years many students have been helpful in improving the quality of the notes, and also several colleagues at Linz and elsewhere have contributed to it. I want to thank them all for their effort, in particular I want to thank B. Buchberger, who taught me the theory of Grabner bases nearly two decades ago, B. F. Caviness and B. D. Saunders, who first stimulated my interest in various problems in computer algebra, G. E. Collins, who showed me how to compute in algebraic domains, and J. R. Sendra, with whom I started to apply computer algebra methods to problems in algebraic geometry. Several colleagues have suggested improvements in earlier versions of this book. However, I want to make it clear that I am responsible for all remaining mistakes.

Elimination Methods in Polynomial Computer Algebra

V. Bykov,A. Kytmanov,M. Lazman,Mikael Passare
Elimination Methods in Polynomial Computer Algebra Book PDF

Author : V. Bykov,A. Kytmanov,M. Lazman,Mikael Passare
Publisher : Springer Science & Business Media
Page : 254 pages
File Size : 54,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9789401153027

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Elimination Methods in Polynomial Computer Algebra by V. Bykov,A. Kytmanov,M. Lazman,Mikael Passare Pdf

The subject of this book is connected with a new direction in mathematics, which has been actively developed over the last few years, namely the field of polynomial computer algebra, which lies at the intersection point of algebra, mathematical analysis and programming. There were several incentives to write the book. First of all, there has lately been a considerable interest in applied nonlinear problems characterized by multiple sta tionary states. Practical needs have then in their turn led to the appearance of new theoretical results in the analysis of systems of nonlinear algebraic equations. And finally, the introduction of various computer packages for analytic manipulations has made it possible to use complicated elimination-theoretical algorithms in prac tical research. The structure of the book is accordingly represented by three main parts: Mathematical results driven to constructive algorithms, computer algebra realizations of these algorithms, and applications. Nonlinear systems of algebraic equations arise in diverse fields of science. In particular, for processes described by systems of differential equations with a poly nomial right hand side one is faced with the problem of determining the number (and location) of the stationary states in certain sets.

Algorithms for Computer Algebra

Keith O. Geddes,Stephen R. Czapor,George Labahn
Algorithms for Computer Algebra Book PDF

Author : Keith O. Geddes,Stephen R. Czapor,George Labahn
Publisher : Springer Science & Business Media
Page : 594 pages
File Size : 41,8 Mb
Release : 2007-06-30
Category : Computers
ISBN : 9780585332475

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Algorithms for Computer Algebra by Keith O. Geddes,Stephen R. Czapor,George Labahn Pdf

Algorithms for Computer Algebra is the first comprehensive textbook to be published on the topic of computational symbolic mathematics. The book first develops the foundational material from modern algebra that is required for subsequent topics. It then presents a thorough development of modern computational algorithms for such problems as multivariate polynomial arithmetic and greatest common divisor calculations, factorization of multivariate polynomials, symbolic solution of linear and polynomial systems of equations, and analytic integration of elementary functions. Numerous examples are integrated into the text as an aid to understanding the mathematical development. The algorithms developed for each topic are presented in a Pascal-like computer language. An extensive set of exercises is presented at the end of each chapter. Algorithms for Computer Algebra is suitable for use as a textbook for a course on algebraic algorithms at the third-year, fourth-year, or graduate level. Although the mathematical development uses concepts from modern algebra, the book is self-contained in the sense that a one-term undergraduate course introducing students to rings and fields is the only prerequisite assumed. The book also serves well as a supplementary textbook for a traditional modern algebra course, by presenting concrete applications to motivate the understanding of the theory of rings and fields.

Modern Computer Algebra

Joachim von zur Gathen,Jürgen Gerhard
Modern Computer Algebra Book PDF

Author : Joachim von zur Gathen,Jürgen Gerhard
Publisher : Cambridge University Press
Page : 811 pages
File Size : 44,5 Mb
Release : 2013-04-25
Category : Computers
ISBN : 9781107245259

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Modern Computer Algebra by Joachim von zur Gathen,Jürgen Gerhard Pdf

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.

Computer Algebra Systems

Michael J. Wester
Computer Algebra Systems Book PDF

Author : Michael J. Wester
Publisher : Wiley-Blackwell
Page : 464 pages
File Size : 50,6 Mb
Release : 1999-07-16
Category : Computers
ISBN : UVA:X004339714

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Computer Algebra Systems by Michael J. Wester Pdf

This thorough overview of the major computer algebra (symbolic mathematical) systems compares and contrasts their strengths and weaknesses, and gives tutorial information for using these systems in various ways. * Compares different packages quantitatively using standard 'test suites' * Ideal for assessing the most appropriate package for a particular user or application * Examines the performance and future developments from a user's and developer's viewpoint Internationally recognized specialists overview both the general and special purpose systems and discuss issues such as denesting nested roots, complex number calculations, efficiently computing special polynomials, solving single equations and systems of polynomial equations, computing limits, multiple integration, solving ordinary differential and nonlinear evolution equations, code generation, evaluation and computer algebra in education. The historical origins, computer algebra resources and equivalents for many common operations in seven major packages are also covered. By providing such a comprehensive survey, the experienced user is able to make an informed decision on which system(s) he or she might like to use. It also allows a user new to computer algebra to form an idea of where to begin. Since each system looked at in this book uses a different language, many examples are included to aid the user in adapting to these language differences. These examples can be used as a guide to using the various systems once one understands the basic principles of one CAS. The book also includes contributions which look at the broad issues of the needs of various users and future developments, both from the user's and the developer's viewpoint. The author is a leading figure in the development and analysis of mathematical software and is well known through the 'Wester test suite' of problems which provide a bench mark for measuring the performance of mathematical software systems. The book will help develop our range of titles for applied mathematcians. The book will provide a unique, fully up-to-date and independent assessment of particular systems and will be of interest to users and purchasers of CAS's.

Mathematics for Computer Algebra

Maurice Mignotte
Mathematics for Computer Algebra Book PDF

Author : Maurice Mignotte
Publisher : Springer Science & Business Media
Page : 357 pages
File Size : 44,9 Mb
Release : 2012-12-06
Category : Computers
ISBN : 9781461391715

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Mathematics for Computer Algebra by Maurice Mignotte Pdf

This book corresponds to a mathematical course given in 1986/87 at the University Louis Pasteur, Strasbourg. This work is primarily intended for graduate students. The following are necessary prerequisites : a few standard definitions in set theory, the definition of rational integers, some elementary facts in Combinatorics (maybe only Newton's binomial formula), some theorems of Analysis at the level of high schools, and some elementary Algebra (basic results about groups, rings, fields and linear algebra). An important place is given to exercises. These exercises are only rarely direct applications of the course. More often, they constitute complements to the text. Mostly, hints or references are given so that the reader should be able to find solutions. Chapters one and two deal with elementary results of Number Theory, for example : the euclidean algorithm, the Chinese remainder theorem and Fermat's little theorem. These results are useful by themselves, but they also constitute a concrete introduction to some notions in abstract algebra (for example, euclidean rings, principal rings ... ). Algorithms are given for arithmetical operations with long integers. The rest of the book, chapters 3 through 7, deals with polynomials. We give general results on polynomials over arbitrary rings. Then polynomials with complex coefficients are studied in chapter 4, including many estimates on the complex roots of polynomials. Some of these estimates are very useful in the subsequent chapters.

Some Tapas of Computer Algebra

Arjeh M. Cohen,Hans Cuypers,Hans Sterk
Some Tapas of Computer Algebra Book PDF

Author : Arjeh M. Cohen,Hans Cuypers,Hans Sterk
Publisher : Springer Science & Business Media
Page : 365 pages
File Size : 42,7 Mb
Release : 2013-03-09
Category : Computers
ISBN : 9783662038918

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Some Tapas of Computer Algebra by Arjeh M. Cohen,Hans Cuypers,Hans Sterk Pdf

This book presents the basic concepts and algorithms of computer algebra using practical examples that illustrate their actual use in symbolic computation. A wide range of topics are presented, including: Groebner bases, real algebraic geometry, lie algebras, factorization of polynomials, integer programming, permutation groups, differential equations, coding theory, automatic theorem proving, and polyhedral geometry. This book is a must read for anyone working in the area of computer algebra, symbolic computation, and computer science.

Computer Algebra and Symbolic Computation

Joel S. Cohen
Computer Algebra and Symbolic Computation Book PDF

Author : Joel S. Cohen
Publisher : CRC Press
Page : 472 pages
File Size : 47,6 Mb
Release : 2003-01-03
Category : Computers
ISBN : 9781439863701

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Computer Algebra and Symbolic Computation by Joel S. Cohen Pdf

Mathematica, Maple, and similar software packages provide programs that carry out sophisticated mathematical operations. Applying the ideas introduced in Computer Algebra and Symbolic Computation: Elementary Algorithms, this book explores the application of algorithms to such methods as automatic simplification, polynomial decomposition, and polyno

Computer Algebra

R. Albrecht,B. Buchberger,G.E. Collins,R. Loos
Computer Algebra Book PDF

Author : R. Albrecht,B. Buchberger,G.E. Collins,R. Loos
Publisher : Springer Science & Business Media
Page : 282 pages
File Size : 43,5 Mb
Release : 2013-06-29
Category : Computers
ISBN : 9783709134061

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Computer Algebra by R. Albrecht,B. Buchberger,G.E. Collins,R. Loos Pdf

The journal Computing has established a series of supplement volumes the fourth of which appears this year. Its purpose is to provide a coherent presentation of a new topic in a single volume. The previous subjects were Computer Arithmetic 1977, Fundamentals of Numerical Computation 1980, and Parallel Processes and Related Automata 1981; the topic of this 1982 Supplementum to Computing is Computer Algebra. This subject, which emerged in the early nineteen sixties, has also been referred to as "symbolic and algebraic computation" or "formula manipulation". Algebraic algorithms have been receiving increasing interest as a result of the recognition of the central role of algorithms in computer science. They can be easily specified in a formal and rigorous way and provide solutions to problems known and studied for a long time. Whereas traditional algebra is concerned with constructive methods, computer algebra is furthermore interested in efficiency, in implementation, and in hardware and software aspects of the algorithms. It develops that in deciding effectiveness and determining efficiency of algebraic methods many other tools - recursion theory, logic, analysis and combinatorics, for example - are necessary. In the beginning of the use of computers for symbolic algebra it soon became apparent that the straightforward textbook methods were often very inefficient. Instead of turning to numerical approximation methods, computer algebra studies systematically the sources of the inefficiency and searches for alternative algebraic methods to improve or even replace the algorithms.

Effective Polynomial Computation

Richard Zippel
Effective Polynomial Computation Book PDF

Author : Richard Zippel
Publisher : Springer Science & Business Media
Page : 364 pages
File Size : 55,8 Mb
Release : 2012-12-06
Category : Computers
ISBN : 9781461531883

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Effective Polynomial Computation by Richard Zippel Pdf

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.

Computer Algebra Handbook

Johannes Grabmeier,Erich Kaltofen,Volker Weispfenning
Computer Algebra Handbook Book PDF

Author : Johannes Grabmeier,Erich Kaltofen,Volker Weispfenning
Publisher : Springer Science & Business Media
Page : 656 pages
File Size : 52,5 Mb
Release : 2012-12-06
Category : Computers
ISBN : 9783642558269

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Computer Algebra Handbook by Johannes Grabmeier,Erich Kaltofen,Volker Weispfenning Pdf

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.

Ideals, Varieties, and Algorithms

David A. Cox,John Little,Donal O'Shea
Ideals Varieties and Algorithms Book PDF

Author : David A. Cox,John Little,Donal O'Shea
Publisher : Springer
Page : 664 pages
File Size : 45,5 Mb
Release : 2015-04-30
Category : Mathematics
ISBN : 9783319167213

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Ideals, Varieties, and Algorithms by David A. Cox,John Little,Donal O'Shea Pdf

This text covers topics in algebraic geometry and commutative algebra with a strong perspective toward practical and computational aspects. The first four chapters form the core of the book. A comprehensive chart in the Preface illustrates a variety of ways to proceed with the material once these chapters are covered. In addition to the fundamentals of algebraic geometry—the elimination theorem, the extension theorem, the closure theorem and the Nullstellensatz—this new edition incorporates several substantial changes, all of which are listed in the Preface. The largest revision incorporates a new Chapter (ten), which presents some of the essentials of progress made over the last decades in computing Gröbner bases. The book also includes current computer algebra material in Appendix C and updated independent projects (Appendix D). The book may serve as a first or second course in undergraduate abstract algebra and with some supplementation perhaps, for beginning graduate level courses in algebraic geometry or computational algebra. Prerequisites for the reader include linear algebra and a proof-oriented course. It is assumed that the reader has access to a computer algebra system. Appendix C describes features of MapleTM, Mathematica® and Sage, as well as other systems that are most relevant to the text. Pseudocode is used in the text; Appendix B carefully describes the pseudocode used. Readers who are teaching from Ideals, Varieties, and Algorithms, or are studying the book on their own, may obtain a copy of the solutions manual by sending an email to jlittle@holycross.edu. From the reviews of previous editions: “...The book gives an introduction to Buchberger’s algorithm with applications to syzygies, Hilbert polynomials, primary decompositions. There is an introduction to classical algebraic geometry with applications to the ideal membership problem, solving polynomial equations and elimination theory. ...The book is well-written. ...The reviewer is sure that it will be an excellent guide to introduce further undergraduates in the algorithmic aspect of commutative algebra and algebraic geometry.” —Peter Schenzel, zbMATH, 2007 “I consider the book to be wonderful. ... The exposition is very clear, there are many helpful pictures and there are a great many instructive exercises, some quite challenging ... offers the heart and soul of modern commutative and algebraic geometry.” —The American Mathematical Monthly

Numerical Polynomial Algebra

Hans J. Stetter
Numerical Polynomial Algebra Book PDF

Author : Hans J. Stetter
Publisher : SIAM
Page : 487 pages
File Size : 49,9 Mb
Release : 2004-01-01
Category : Mathematics
ISBN : 0898717973

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Numerical Polynomial Algebra by Hans J. Stetter Pdf

In many important areas of scientific computing, polynomials in one or more variables are employed in the mathematical modeling of real-life phenomena; yet most of classical computer algebra assumes exact rational data. This book is the first comprehensive treatment of the emerging area of numerical polynomial algebra, an area that falls between classical numerical analysis and classical computer algebra but, surprisingly, has received little attention so far. The author introduces a conceptual framework that permits the meaningful solution of various algebraic problems with multivariate polynomial equations whose coefficients have some indeterminacy; for this purpose, he combines approaches of both numerical linear algebra and commutative algebra. For the application scientist, Numerical Polynomial Algebra provides both a survey of polynomial problems in scientific computing that may be solved numerically and a guide to their numerical treatment. In addition, the book provides both introductory sections and novel extensions of numerical analysis and computer algebra, making it accessible to the reader with expertise in either one of these areas.

Computer Algebra in Scientific Computing

Vladimir P. Gerdt,Wolfram Koepf,Werner M. Seiler,Evgenii V. Vorozhtsov
Computer Algebra in Scientific Computing Book PDF

Author : Vladimir P. Gerdt,Wolfram Koepf,Werner M. Seiler,Evgenii V. Vorozhtsov
Publisher : Springer
Page : 379 pages
File Size : 41,9 Mb
Release : 2018-09-03
Category : Computers
ISBN : 9783319996394

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Computer Algebra in Scientific Computing by Vladimir P. Gerdt,Wolfram Koepf,Werner M. Seiler,Evgenii V. Vorozhtsov Pdf

This book constitutes the proceedings of the 20th International Workshop on Computer Algebra in Scientific Computing, CASC 2018, held in Lille, France, in September 2018. The 24 full papers of this volume presented with an abstract of an invited talk and one paper corresponding to another invited talk were carefully reviewed and selected from 29 submissions. They deal with cutting-edge research in all major disciplines of computer algebra in sciences such as physics, chemistry, life sciences, and engineering. Chapter “Positive Solutions of Systems of Signed Parametric Polynomial Inequalities” is available open access under a Creative Commons Attribution 4.0 International License via link.springer.com.