Elements of Computer Algebra With Applications

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9780471611639: Elements of Computer Algebra With Applications
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Numerical Modeling in Science and Engineering Myron B. Allen, George F. Pinder, and Ismael Herrera Emphasizing applications, this treatment combines three traditionally distinct disciplines—continuum mechanics, differential equations, and numerical analysis—to provide a unified treatment of numerical modeling of physical systems. Covers basic equations of macroscopic systems, numerical methods, steady state systems, dissipative systems, nondissipative systems, and high order, nonlinear, and coupled systems. 1988 (0 471-80635-8) 418 pp. Mathematical Modeling and Digital Simulation for Engineers and Scientists Second Edition Jon M. Smith Totally updated, this Second Edition reflects the many developments in simulation and computer modeling theory and practice that have occurred over the past decade. It includes a new section on the use of modern numerical methods for generating chaos and simulating random processes, a section on simulator verification, and provides applications of these methods for personal computers. Readers will find a wealth of practical fault detection and isolation techniques for simulator verification, fast functions evaluation techniques, and nested parenthetical forms and Chebyshev economization techniques. 1987 (0 471-08599-5) 430 pp. Numerical Analysis 1987 David F. Griffiths and George Alistair Watson An invaluable guide to the direction of current research in many areas of numerical analysis, this volume will be of great interest to anyone involved in software design, curve and surface fitting, the numerical solution of ordinary, partial, and integro-differential equations, and the real-world application of numerical techniques. 1988 (0 470-21012-5) 300 pp.

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Computer algebra refers to the capability of computers to manipulate mathematical expressions in a symbolic, rather than numerical, way--which frees scientists from the painstaking concern for numerical errors (which result from truncation and rounding off). This book describes the best subresultant method for computing polynomial, greatest common divisors and remainder sequences; the importance of Budan's theorem, and its relation to a theorem by Fourier; and the fastest existing method for isolating the real roots of a polynomial equation. Clear and detailed enough to serve as an introductory text, and current and thorough enough to stand as a reference.

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Computer algebra deals mainly with exact numbers and algebraic expressions in terms of their symbolic representations; it differs from numerical analysis in that algebraic expressions are handled symbolically, and floating point arithmetic is not used during computations. Elements of Computer Algebra with Applications covers several important topics in the mathematics of computation with emphasis on efficient algorithms and their historical background. The material in the book is divided into three parts. Part I introduces and explains exactly what computer algebra is. Part II contains the mathematical foundations and basic algorithms. Since computer algebra deals mostly with integers and polynomials with integer coefficients, one chapter details the basic properties of integers while the other covers those of polynomials. Part III applies the concepts developed in Parts I and II—specifically to error-correcting codes and cryptography, computation of polynomial greatest common divisors and polynomial remainder sequences, factorization of polynomials with integer coefficients, and isolation and approximation of the real roots of polynomial equations. There are many &quote;firsts&quote; in this book, including: the best subresultant method for computing polynomial remainder sequences developed in 1986 by the author based on a paper by Sylvester in 1853; Budan’s theorem showing its importance and relation to a theorem by Fourier; and the fastest existing method for isolating the real roots of a polynomial equation developed in 1978 by the author based on a theorem by Vincent in 1836. Elements of Computer Algebra with Applications can be used as a reference source for the researcher and as a text for students in numerical methods in the computer science or mathematics curriculum. It provides an excellent vehicle for teaching both theory and applications of algebra, effectively joining traditional algebra with computer science. At the same time it utilizes a great number of concepts learned in earlier computer science courses. Additionally, it makes students aware of the works of some of the giants in mathematics—including Galois, Hensel, Lagrange, Sturm, Sylvester, and Vincent—and shows how their basic approach to computation cannot be dealt with in numerical analysis, but closely resembles what researchers are now trying to accomplish using computer algebra.

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