Interval methods for systems of equations.

*(English)*Zbl 0715.65030
Encyclopedia of Mathematics and its Applications, 37. Cambridge etc.: Cambridge University Press. xvi, 255 p. £35.00; $ 59.50 (1990).

The aim of the book is to present interval arithmetical tools and methods that are needed for the solution of finite dimensional systems of linear and nonlinear interval equations. Here, “solving a problem” is interpreted in a rather rigorous manner and means obtaining a safe verification of existence or uniqueness of a solution and finding guaranteed enclosures with only a small rate of overestimation. The development of interval tools for this purpose has been very vehement during the last decade and has been also affected by several contributions of the author himself. Hence he was extremely authorized to show the state of art in the area of solving systems of interval equations.

There are three points leaping the eye when one casts a first glance at the book. This is, firstly, the immense extent of linear interval algebra and matrix material occurring in the monograph, which is justified by the fact that, finally, almost every numerical method for solving systems of nonlinear equations has to pass through the linear theory, which has turned out to be a bottleneck for the nonlinear theory. Hence large parts of the book are devoted to the theory of M- and H-matrices, which help to steer round the bottleneck. Secondly, it is a recursive view that is featured, which results in creating so-called inclusion algebras. They may be seen as a unifying base for dealing with automatic differentiation, slope arithmetic or with recursive computation of the range of functions, etc. Thirdly, it is the cultured style of presenting mathematics, which seems to be due to the author’s goal “to please both the pure mathematician by being elegant and the applied one being constructive and useful”.

Contents. The first three chapters are dealt with the necessary tools of interval mathematics, that is, Ch. I (Basic properties of interval arithmetic) and Ch. II (Enclosures for the range of functions). Ch. III (Matrices and sublinear mappings) emphasizes M- and H-matrices and investigates the problem of solving systems of linear interval equations. The shape of a solution set is far from being an interval vector and ranges between star-like and butterfly-like. In order to be able to handle such regions within interval arithmetic different kinds of interval hulls of the solution set are introduced. Ch. IV (The solution of square linear systems of equations) starts with the important concept of preconditioning and continues with a thorough discussion of some solution methods such as Krawczyk’s method, interval Gauss-Seidel iteration and interval Gauss elimination. Main topics of Ch. V (Nonlinear systems of equations) are existence, uniqueness or nonexistence of solutions and the computational verification of these properties. Mainly the Newton operator, the Krawczyk operator and the Hansen-Sengupta operator are chosen for the investigation of solution methods. Parameter- dependent equations and the admission of problem domains that are more general than boxes are considered as well. The final Ch. VI (Hull computation) is concerned with the characterization of the hull inverse (one of the hulls of a square system of linear interval equations as mentioned above) and its generation from a finite number of vectors.

There are three points leaping the eye when one casts a first glance at the book. This is, firstly, the immense extent of linear interval algebra and matrix material occurring in the monograph, which is justified by the fact that, finally, almost every numerical method for solving systems of nonlinear equations has to pass through the linear theory, which has turned out to be a bottleneck for the nonlinear theory. Hence large parts of the book are devoted to the theory of M- and H-matrices, which help to steer round the bottleneck. Secondly, it is a recursive view that is featured, which results in creating so-called inclusion algebras. They may be seen as a unifying base for dealing with automatic differentiation, slope arithmetic or with recursive computation of the range of functions, etc. Thirdly, it is the cultured style of presenting mathematics, which seems to be due to the author’s goal “to please both the pure mathematician by being elegant and the applied one being constructive and useful”.

Contents. The first three chapters are dealt with the necessary tools of interval mathematics, that is, Ch. I (Basic properties of interval arithmetic) and Ch. II (Enclosures for the range of functions). Ch. III (Matrices and sublinear mappings) emphasizes M- and H-matrices and investigates the problem of solving systems of linear interval equations. The shape of a solution set is far from being an interval vector and ranges between star-like and butterfly-like. In order to be able to handle such regions within interval arithmetic different kinds of interval hulls of the solution set are introduced. Ch. IV (The solution of square linear systems of equations) starts with the important concept of preconditioning and continues with a thorough discussion of some solution methods such as Krawczyk’s method, interval Gauss-Seidel iteration and interval Gauss elimination. Main topics of Ch. V (Nonlinear systems of equations) are existence, uniqueness or nonexistence of solutions and the computational verification of these properties. Mainly the Newton operator, the Krawczyk operator and the Hansen-Sengupta operator are chosen for the investigation of solution methods. Parameter- dependent equations and the admission of problem domains that are more general than boxes are considered as well. The final Ch. VI (Hull computation) is concerned with the characterization of the hull inverse (one of the hulls of a square system of linear interval equations as mentioned above) and its generation from a finite number of vectors.

Reviewer: H.Ratschek

##### MSC:

65G30 | Interval and finite arithmetic |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

65F05 | Direct numerical methods for linear systems and matrix inversion |

65F10 | Iterative numerical methods for linear systems |

65H10 | Numerical computation of solutions to systems of equations |