Interval Computations: Introduction, Uses, and Resources

Interval analysis is a broad field in which rigorous mathematics is associated with with scientific computing. A number of researchers worldwide have produced a voluminous literature on the subject. This article introduces interval arithmetic and its interaction with established mathematical theory. The article provides pointers to traditional literature collections, as well as electronic resources. Some successful scientific and engineering applications are listed. 1 What is Interval Arithmetic, and Why is it Considered? Interval arithmetic is an arithmetic defined on sets of intervals, rather than sets of real numbers. A form of interval arithmetic perhaps first appeared in 1924 and 1931 in [8, 104], then later in [98]. Modern development of interval arithmetic began with R. E. Moore’s dissertation [64]. Since then, thousands of research articles and numerous books have appeared on the subject. Periodic conferences, as well as special meetings, are held on the subject. There is an increasing amount of software support for interval computations, and more resources concerning interval computations are becoming available through the Internet. In this paper, boldface will denote intervals, lower case will denote scalar quantities, and upper case will denote vectors and matrices. Brackets “[·]” will delimit intervals while parentheses “(·)” will delimit vectors and matrices. Underscores will denote lower bounds of intervals and overscores will denote upper bounds of intervals. Corresponding lower case letters will denote components of vectors. The set of real intervals will be denoted by IR. Interval vectors will also be called boxes.