Interval methods for systems of equations / Arnold Neumaier.

Includes bibliographical references (pages 231-248) and indexes.

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An interval is a natural way of specifying a number that is specified only within certain tolerances. Interval analysis consists of the tools and methods needed to solve linear and nonlinear systems of equations in the presence of data uncertainties. Applications include the sensitivity analysis of solutions of equations depending on parameters, the solution of global nonlinear problems, and the verification of results obtained by finite-precision arithmetic. In this book emphasis is laid on those aspects of the theory which are useful in actual computations. On the other hand, the theory is developed with full mathematical rigour. In order to keep the book self-contained, various results from linear algebra (Perron-Frobenius theory, M- and H- matrices) and analysis (existence of solutions to nonlinear systems) are proved, often from a novel and more general viewpoint. An extensive bibliography is included.

Cover; Half Title; Series Page; Title; Copyright; CONTENTS; PREFACE; SYMBOL INDEX; 1 Basic properties of interval arithmetic; 1.1 Motivation; 1.2 Intervals; 1.3 Rounded interval arithmetic; 1.4 Interval vectors and arithmetical expressions; 1.5 Algebraic properties of interval operations; 1.6 Rules for midpoint, radius and absolute value; 1.7 Distance and topology; 1.8 Appendix. Input/output representation of intervals; 1.8.1 Syntax; 1.8.2 Interpretation; 1.8.3 Examples; Remarks to Chapter 1; 2 Enclosures for the range of a function; 2.1 Analysis of interval evaluation

2.2 Inclusion algebras and recursive differentiation2.3 The mean value form and other centered forms; 2.4 Interpolation forms; 2.5 Appendix. The extended Horner scheme; Remarks to Chapter 2; 3 Matrices and sublinear mappings; 3.1 Basic facts; 3.2 Norms and spectral radius; 3.3 Distance and topology; 3.4 Linear interval equations; 3.5 Sublinear mappings; 3.6 M-matrices and inverse positive matrices; 3.7 H-matrices; Remarks to Chapter 3; 4 The solution of square linear systems of equations; 4.1 Preconditioning; 4.2 Krawczyk's method and quadratic approximation

4.3 Interval Gauss-Seidel iteration4.4 Linear fixed point equations; 4.5 Interval Gauss elimination; Remarks to Chapter 4; 5 Nonlinear systems of equations; 5.1 Existence and uniqueness; 5.2 Interval iteration; 5.3 Set-valued functions; 5.4 Zeros of continuous functions; 5.5 Local analysis of parameter-dependent nonlinear systems; 5.6 Global problems; Remarks to Chapter 5; 6 Hull computation; 6.1 The equation x = Mlxl + a; 6.2 Characterization and computation of AHb; Remarks to Chapter 6; REFERENCES; AUTHOR INDEX; SUBJECT INDEX

English.

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