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**Introduction to interval analysis.**
*(English)*
Zbl 1168.65002

Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM) (ISBN 978-0-898716-69-6/pbk; 978-0-89871-771-6/ebook). xi, 223 p. (2009).

The use of interval analysis has steadily increased over the past 40 years. This development was taken into account when writing the present book. It presents the basics in real interval arithmetic and covers elementary methods for verifying and enclosing zeros of functions, global minimizers, solutions of integral and differential equations. It deals with integration of interval functions and shows how interval methods can be applied in various fields of science. Moreover, an introduction into the interval toolbox INTLAB of MATLAB is given in order to understand the many programs which realize the algorithms of the book. Numerous examples illustrate the theory and are spread over more than 220 pages.

The book starts with a short introduction on the necessity of enclosures and on bounding roundoff errors. The interval number system is presented in Chapter 2 including set theoretic operations as well as order relations and operations for intervals, interval vectors, and interval matrices. Some historical references are added. Chapter 3 discusses the problem of computing with inexact initial data and therefore results in first applications of interval arithmetic. It considers outwardly rounded interval arithmetic and gives a first glance to INTLAB. Chapter 4 is devoted to algebraic properties of interval arithmetic, to symmetric intervals and to inclusion isotonicity. Interval functions are introduced in Chapter 5 including elementary interval functions and interval-valued extensions of real functions together with fundamental properties.

The topological side of intervals and of interval arithmetic is considered in Chapter 6. Equipped with the Hausdorff metric the set of intervals turns out to be a complete metric space for which convergence and continuity can be defined in the usual way. Nested interval sequences, finite convergence, refinement of interval extensions, and various centered forms are additional topics of this chapter which is concluded by the Skelboe-Moore algorithm. This algorithm is a prototype of a branch-and-bound algorithm applied in global optimization. Interval matrices form the subject of the succeeding Chapter 7, where linear systems with inexact input data are also discussed. In this connection the Krawczyk method is mentioned as well as the interval Gauss-Seidel method and Gaussian elimination.

Nonlinear equations are studied in Chapter 8. Here the interval Newton method is commented, and cases are presented which imply the necessity of an extended interval arithmetic. For systems of nonlinear equations the Krawczyk method and multivariate interval Newton methods are applied and safe starting intervals are defined. Chapter 9 is devoted to the integration of interval functions, particularly of interval polynomials since for sufficiently smooth functions \(f\) a Taylor expansion is used to construct an enclosure for \(f\). Therefore, automatic differentiation and automatic generation of Taylor coefficients are also handled in order to find enclosures for definite integrals – also multiple ones – over \(f\). A short chapter lists some ideas for verifying and enclosing solutions of integral equations, initial value problems, and boundary value problems. Some literature for partial differential equations is added. The final Chapter 11 gives a first impression of how the tool ‘interval analysis’ is applied in practice. Problems are listed which use computer-assisted proofs based on interval arithmetic. A prototypical algorithm is discussed for global optimization. Numerous examples from engineering are mentioned.

An appendix covers a variety of topics: Sets and functions, a formulary for intervals, hints for selected exercises, internet resources, and INTLAB commands and functions. More than 250 references conclude a wonderful book which is written for all who are interested in scientific computation, in its reliability, and in automatic verification of results.

The book starts with a short introduction on the necessity of enclosures and on bounding roundoff errors. The interval number system is presented in Chapter 2 including set theoretic operations as well as order relations and operations for intervals, interval vectors, and interval matrices. Some historical references are added. Chapter 3 discusses the problem of computing with inexact initial data and therefore results in first applications of interval arithmetic. It considers outwardly rounded interval arithmetic and gives a first glance to INTLAB. Chapter 4 is devoted to algebraic properties of interval arithmetic, to symmetric intervals and to inclusion isotonicity. Interval functions are introduced in Chapter 5 including elementary interval functions and interval-valued extensions of real functions together with fundamental properties.

The topological side of intervals and of interval arithmetic is considered in Chapter 6. Equipped with the Hausdorff metric the set of intervals turns out to be a complete metric space for which convergence and continuity can be defined in the usual way. Nested interval sequences, finite convergence, refinement of interval extensions, and various centered forms are additional topics of this chapter which is concluded by the Skelboe-Moore algorithm. This algorithm is a prototype of a branch-and-bound algorithm applied in global optimization. Interval matrices form the subject of the succeeding Chapter 7, where linear systems with inexact input data are also discussed. In this connection the Krawczyk method is mentioned as well as the interval Gauss-Seidel method and Gaussian elimination.

Nonlinear equations are studied in Chapter 8. Here the interval Newton method is commented, and cases are presented which imply the necessity of an extended interval arithmetic. For systems of nonlinear equations the Krawczyk method and multivariate interval Newton methods are applied and safe starting intervals are defined. Chapter 9 is devoted to the integration of interval functions, particularly of interval polynomials since for sufficiently smooth functions \(f\) a Taylor expansion is used to construct an enclosure for \(f\). Therefore, automatic differentiation and automatic generation of Taylor coefficients are also handled in order to find enclosures for definite integrals – also multiple ones – over \(f\). A short chapter lists some ideas for verifying and enclosing solutions of integral equations, initial value problems, and boundary value problems. Some literature for partial differential equations is added. The final Chapter 11 gives a first impression of how the tool ‘interval analysis’ is applied in practice. Problems are listed which use computer-assisted proofs based on interval arithmetic. A prototypical algorithm is discussed for global optimization. Numerous examples from engineering are mentioned.

An appendix covers a variety of topics: Sets and functions, a formulary for intervals, hints for selected exercises, internet resources, and INTLAB commands and functions. More than 250 references conclude a wonderful book which is written for all who are interested in scientific computation, in its reliability, and in automatic verification of results.

Reviewer: Günter Mayer (Rostock)

### MSC:

65-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to numerical analysis |

65G30 | Interval and finite arithmetic |

65G50 | Roundoff error |

65Fxx | Numerical linear algebra |

65Hxx | Nonlinear algebraic or transcendental equations |

65R20 | Numerical methods for integral equations |

65Lxx | Numerical methods for ordinary differential equations |

65G20 | Algorithms with automatic result verification |

68W30 | Symbolic computation and algebraic computation |