Numerical Polynomial Algebra

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 Methods for Roots of Polynomials - Part II along with Part I (9780444527295) covers most of the traditional methods for polynomial root-finding such as interpolation and methods due to Graeffe, Laguerre, and Jenkins and Traub. It includes many other methods and topics as well and has a chapter devoted to certain modern virtually optimal methods. Additionally, there are pointers to robust and efficient programs. This book is invaluable to anyone doing research in polynomial roots, or teaching a graduate course on that topic. First comprehensive treatment of Root-Finding in several decades with a description of high-grade software and where it can be downloaded Offers a long chapter on matrix methods and includes Parallel methods and errors where appropriate Proves invaluable for research or graduate course

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.

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.

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.

This book introduces the numerical technique of polynomial continuation, which is used to compute solutions to systems of polynomial equations. Originally published in 1987, it remains a useful starting point for the reader interested in learning how to solve practical problems without advanced mathematics. Solving Polynomial Systems Using Continuation for Engineering and Scientific Problems is easy to understand, requiring only a knowledge of undergraduate-level calculus and simple computer programming. The book is also practical; it includes descriptions of various industrial-strength engineering applications and offers Fortran code for polynomial solvers on an associated Web page. It provides a resource for high-school and undergraduate mathematics projects. Audience: accessible to readers with limited mathematical backgrounds. It is appropriate for undergraduate mechanical engineering courses in which robotics and mechanisms applications are studied.