Interval Analysis ：and Automatic Result Verification ( De Gruyter Studies in Mathematics )

Publication subTitle ：and Automatic Result Verification

Publication series ：De Gruyter Studies in Mathematics

Author： Mayer Günter  

Publisher： De Gruyter‎

Publication year： 2017

E-ISBN: 9783110499469

P-ISBN(Paperback): 9783110500639

Subject： O242.29 interval analysis

Keyword： 数值分析,微积分,算法理论,计算机软件

Language： ENG

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Description

The series is devoted to the publication of monographs and high-level textbooks in mathematics, mathematical methods and their applications. Apart from covering important areas of current interest, a major aim is to make topics of an interdisciplinary nature accessible to the non-specialist.

The works in this series are addressed to advanced students and researchers in mathematics and theoretical physics. In addition, it can serve as a guide for lectures and seminars on a graduate level.

The series de Gruyter Studies in Mathematics was founded ca. 30 years ago by the late Professor Heinz Bauer and Professor Peter Gabriel with the aim to establish a series of monographs and textbooks of high standard, written by scholars with an international reputation presenting current fields of research in pure and applied mathematics.
While the editorial board of the Studies has changed with the years, the aspirations of the Studies are unchanged. In times of rapid growth of mathematical knowledge carefully written monographs and textbooks written by experts are needed more than ever, not least to pave the way for the next generation of mathematicians. In this sense the editorial board and the publisher of the Studies are devoted to continue the Studies

Chapter

Preface

Contents

1 Preliminaries

1.1 Notations and basic definitions

1.2 Metric spaces

1.3 Normed linear spaces

1.4 Polynomials

1.5 Zeros and fixed points of functions

1.6 Mean value theorems

1.7 Normal forms of matrices

1.8 Eigenvalues

1.9 Nonnegative matrices

1.10 Particular matrices

2 Real intervals

2.1 Intervals, partial ordering

2.2 Interval arithmetic

2.3 Algebraic properties, χ -function

2.4 Auxiliary functions

2.5 Distance and topology

2.6 Elementary interval functions

2.7 Machine interval arithmetic

3 Interval vectors, interval matrices

3.1 Basics

3.2 Powers of interval matrices

3.3 Particular interval matrices

4 Expressions, P-contraction, ε-inflation

4.1 Expressions, range

4.2 P-contraction

4.3 ε-inflation

5 Linear systems of equations

5.1 Motivation

5.2 Solution sets

5.3 Interval hull

5.4 Direct methods

5.5 Iterative methods

6 Nonlinear systems of equations

6.1 Newton method – one-dimensional case

6.2 Newton method – multidimensional case

6.3 Krawczyk method

6.4 Hansen–Sengupta method

6.5 Further existence tests

6.6 Bisection method

7 Eigenvalue problems

7.1 Quadratic systems

7.2 A Krawczyk-like method

7.3 Lohner method

7.4 Double or nearly double eigenvalues

7.5 The generalized eigenvalue problem

7.6 A method due to Behnke

7.7 Verification of singular values

7.8 An inverse eigenvalue problem

8 Automatic differentiation

8.1 Forward mode

8.2 Backward mode

9 Complex intervals

9.1 Rectangular complex intervals

9.2 Circular complex intervals

9.3 Applications of complex intervals

Final Remarks

Appendix

A Jordan normal form

B Brouwer’s fixed point theorem

C Theorem of Newton–Kantorovich

D The row cyclic Jacobi method

E The CORDIC Algorithm

F The symmetric solution set

G INTLAB

Bibliography

Symbol Index

Author Index

Subject Index

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