Numerical Polynomial Algebra

Numerical Polynomial Algebra
Author: Hans J. Stetter
Publisher: SIAM
Total Pages: 475
Release: 2004-05-01
Genre: Mathematics
ISBN: 0898715571

This book is the first comprehensive treatment of numerical polynomial algebra, an area which so far has received little attention.

Numerically Solving Polynomial Systems with Bertini

Numerically Solving Polynomial Systems with Bertini
Author: Daniel J. Bates
Publisher: SIAM
Total Pages: 372
Release: 2013-11-08
Genre: Science
ISBN: 1611972698

This book is a guide to concepts and practice in numerical algebraic geometry ? the solution of systems of polynomial equations by numerical methods. Through numerous examples, the authors show how to apply the well-received and widely used open-source Bertini software package to compute solutions, including a detailed manual on syntax and usage options. The authors also maintain a complementary web page where readers can find supplementary materials and Bertini input files. Numerically Solving Polynomial Systems with Bertini approaches numerical algebraic geometry from a user's point of view with numerous examples of how Bertini is applicable to polynomial systems. It treats the fundamental task of solving a given polynomial system and describes the latest advances in the field, including algorithms for intersecting and projecting algebraic sets, methods for treating singular sets, the nascent field of real numerical algebraic geometry, and applications to large polynomial systems arising from differential equations. Those who wish to solve polynomial systems can start gently by finding isolated solutions to small systems, advance rapidly to using algorithms for finding positive-dimensional solution sets (curves, surfaces, etc.), and learn how to use parallel computers on large problems. These techniques are of interest to engineers and scientists in fields where polynomial equations arise, including robotics, control theory, economics, physics, numerical PDEs, and computational chemistry.

Computer Algebra and Polynomials

Computer Algebra and Polynomials
Author: Jaime Gutierrez
Publisher: Springer
Total Pages: 222
Release: 2015-01-20
Genre: Computers
ISBN: 3319150812

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.

A Polynomial Approach to Linear Algebra

A Polynomial Approach to Linear Algebra
Author: Paul A. Fuhrmann
Publisher: Springer Science & Business Media
Total Pages: 368
Release: 2012-10-01
Genre: Mathematics
ISBN: 1441987347

A Polynomial Approach to Linear Algebra is a text which is heavily biased towards functional methods. In using the shift operator as a central object, it makes linear algebra a perfect introduction to other areas of mathematics, operator theory in particular. This technique is very powerful as becomes clear from the analysis of canonical forms (Frobenius, Jordan). It should be emphasized that these functional methods are not only of great theoretical interest, but lead to computational algorithms. Quadratic forms are treated from the same perspective, with emphasis on the important examples of Bezoutian and Hankel forms. These topics are of great importance in applied areas such as signal processing, numerical linear algebra, and control theory. Stability theory and system theoretic concepts, up to realization theory, are treated as an integral part of linear algebra. Finally there is a chapter on Hankel norm approximation for the case of scalar rational functions which allows the reader to access ideas and results on the frontier of current research.

Numerical Polynomial Algebra

Numerical Polynomial Algebra
Author: Hans J. Stetter
Publisher: SIAM
Total Pages: 487
Release: 2004-01-01
Genre: Mathematics
ISBN: 9780898717976

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.

Numerical Mathematics

Numerical Mathematics
Author: Alfio Quarteroni
Publisher: Springer
Total Pages: 669
Release: 2017-01-26
Genre: Mathematics
ISBN: 0387227504

The purpose of this book is to provide the mathematical foundations of numerical methods, to analyze their basic theoretical properties and to demonstrate their performances on examples and counterexamples. Within any specific class of problems, the most appropriate scientific computing algorithms are reviewed, their theoretical analyses are carried out and the expected results are verified using the MATLAB software environment. Each chapter contains examples, exercises and applications of the theory discussed to the solution of real-life problems. While addressed to senior undergraduates and graduates in engineering, mathematics, physics and computer sciences, this text is also valuable for researchers and users of scientific computing in a large variety of professional fields.

Solving Systems of Polynomial Equations

Solving Systems of Polynomial Equations
Author: Bernd Sturmfels
Publisher: American Mathematical Soc.
Total Pages: 162
Release: 2002
Genre: Mathematics
ISBN: 0821832514

Bridging a number of mathematical disciplines, and exposing many facets of systems of polynomial equations, Bernd Sturmfels's study covers a wide spectrum of mathematical techniques and algorithms, both symbolic and numerical.

Interpolation and Approximation by Polynomials

Interpolation and Approximation by Polynomials
Author: George M. Phillips
Publisher: Springer Science & Business Media
Total Pages: 325
Release: 2006-04-06
Genre: Mathematics
ISBN: 0387216820

In addition to coverage of univariate interpolation and approximation, the text includes material on multivariate interpolation and multivariate numerical integration, a generalization of the Bernstein polynomials that has not previously appeared in book form, and a greater coverage of Peano kernel theory than is found in most textbooks. There are many worked examples and each section ends with a number of carefully selected problems that extend the student's understanding of the text. The author is well known for his clarity of writing and his many contributions as a researcher in approximation theory.

The Numerical Solution Of Systems Of Polynomials Arising In Engineering And Science

The Numerical Solution Of Systems Of Polynomials Arising In Engineering And Science
Author: Andrew J Sommese
Publisher: World Scientific
Total Pages: 425
Release: 2005-03-21
Genre: Mathematics
ISBN: 9814480886

Written by the founders of the new and expanding field of numerical algebraic geometry, this is the first book that uses an algebraic-geometric approach to the numerical solution of polynomial systems and also the first one to treat numerical methods for finding positive dimensional solution sets. The text covers the full theory from methods developed for isolated solutions in the 1980's to the most recent research on positive dimensional sets.

Solving Polynomial Equations

Solving Polynomial Equations
Author: Alicia Dickenstein
Publisher: Springer Science & Business Media
Total Pages: 433
Release: 2005-04-27
Genre: Computers
ISBN: 3540243267

This book provides a general introduction to modern mathematical aspects in computing with multivariate polynomials and in solving algebraic systems. It presents the state of the art in several symbolic, numeric, and symbolic-numeric techniques, including effective and algorithmic methods in algebraic geometry and computational algebra, complexity issues, and applications ranging from statistics and geometric modelling to robotics and vision. Graduate students, as well as researchers in related areas, will find an excellent introduction to currently interesting topics. These cover Groebner and border bases, multivariate resultants, residues, primary decomposition, multivariate polynomial factorization, homotopy continuation, complexity issues, and their applications.