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 numerical polynomial algebra, an emerging area that falls between classical numerical analysis and classical computer algebra, and which has received surprisingly 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, this book 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, making it more easily accessible.
Hans J. Stetter is Professor Emeritus of Numerical Mathematics at the Vienna University of Technology, Austria. He is the author of more than 90 publications and has been editor or associate editor of Computing, Numerische Mathematik, Transactions on Numerical Software, Mathematics of Computation, and various other journals. He is a member of the German Academy of Natural Scientists Leopoldina.
Preface; Part I. Polynomials and Numerical Analysis: 1. Polynomials; 2. Representations of polynomial ideals; 3. Polynomials with coefficients of limited accuracy; 4. Approximate numerical computation; Part II. Univariate Polynomial Problems: 5. Univariate polynomials; 6. Various tasks with empirical univariate polynomials; Part III. Multivariate Polynomial Problems: 7. One multivariate polynomial; 8. Zero-dimensional systems of multivariate polynomials; 9. Systems of empirical multivariate polynomials; 10. Numerical basis computation; Part IV. Positive-Dimensional Polynomial Systems: 11. Matrix eigenproblems for positive-dimensional systems; Index.
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