Symbolic Algebraic Methods and Verification Methods

ISBN-13: 9783211835937 / Angielski / Miękka / 2001 / 266 str.

Gotz Alefeld; Jiri Rohn; S. Rump

Opis

The usual usual "implementation" "implementation" ofreal numbers as floating point numbers on exist iing ng computers computers has the well-known disadvantage that most of the real numbers are not exactly representable in floating point. Also the four basic arithmetic operations can usually not be performed exactly. For numerical algorithms there are frequently error bounds for the computed approximation available. Traditionally a bound for the infinity norm is estima ted using ttheoretical heoretical ccoonncceeppttss llike ike the the condition condition number number of of a a matrix matrix for for example. example. Therefore Therefore the error bounds are not really available in practice since their com putation requires more or less the exact solution of the original problem. During the last years research in different areas has been intensified in or der to overcome these problems. As a result applications to different concrete problems were obtained. The LEDA-library (K. Mehlhorn et al.) offers a collection of data types for combinatorical problems. In a series of applications, where floating point arith metic fails, reliable results are delivered. Interesting examples can be found in classical geometric problems. At the Imperial College in London was introduced a simple principle for "exact arithmetic with real numbers" (A. Edalat et al.), which uses certain nonlinear transformations. Among others a library for the effective computation of the elementary functions already has been implemented.

Spis treści

Introduction (G. Alefeld, J. Rohn, S. Rump, T. Yamamoto).- Topological Concepts for Hierarchies of Variables, Types and Controls (R. Albrecht).- Modifications of the Oettli-Prager Theorem with Application to the Eigenvalue Problem (G. Alefeld, V. Kreinovich, G. Mayer).- Symbolic-Numeric Algorithms for Polynomials. Some Recent Results (R. Corless).- Symbolic-Numeric QD-Algorithms with Applications in Function Theory and Linear Algebra (A. Cuyt).- On the Isoefficiency of the Parallel Dscartes Method (Th. Decker, W. Krandick).- Matrix Methods for Solving Algebraic Systems (I. Z. Emiris).- A Feasibility Result for Interval Gaussian Elimination Relying on Graph Structure (A. Frommer).- Solution of Systems of Polynomial Equations by Using Bernstein Expansion (J. Garloff, A. P. Smith).- Symbolic-Algebraic Computations in Modeling Language for Mathematical Programming (D. M. Gay).- Translation of Taylor Series into LFT Expansions (R. Heckmann).- Quasi Convex-Concave Extensions (Chr. Jansson).- Rewriting, Induction and Decision Procedures: A Case Study of Presburger Arithmetic (D. Kapur).- Derivative-Based Subdivision in Multi-dimensional Verified Gaussian Quadrature (B. Lang).- On the Shape of the Fixed Points of [f]([x]) = [A][x] [b].- Exact Computation with leda_real – Theory and Geometric Applications (K. Mehlhorn, St. Schirra).- Numerical Verification Method for Solutions of Nonlinear Hyperbolic Equations (T. Minamoto).- Geometric Series Bounds for the Local Errors of Taylor Methods for Linear n-th-Order ODEs (M. Neher).- Save Numerical Error Bounds for Solutions of Nonlinear Elliptic Boundary Value Problems (M. Plum).- Fast Verification Algorithms in MATLAB (S. Rump).- The Linear Complementarity Problem with Interval Data (U. Schäfer).- Some Numerical Methods for Nonlinear Least Squares Problems (St. Shakhno).- A New Insight of the Shortley-Weller Approximation for Dirichlet Problems (T. Yamamoto).- How Orthogonality is Lost in Krylov Methods (J. Zemke).