Interval Arithmetic Revisited

Summary This paper deals with interval arithmetic and interval mathematics. Interval mathematics has been developed to a high standard during the last few decades. It provides methods which deliver results with guarantees. However, the arithmetic available on existing processors makes these methods extremely slow. The paper reviews a number of basic methods and techniques of interval mathematics in order to derive and focus on those properties which by today’s knowledge could effectively be supported by the computer’s hardware, by basic software, and by the programming languages. The paper is not aiming for completeness. Unnecessary math- ematical details, formalisms and derivations are left aside whenever possi- ble. Particular emphasis is put on an efficient implementation of interval arithmetic on computers. Interval arithmetic is introduced as a shorthand notation and automatic calculus to add , subtract, multiply, divide, and otherwise deal with inequalities. Interval operations are also interpreted as special powerset or set operations. The inclusion isotony and the inclusion property are central and important consequences of this property. The basic techniques for enclosing the range of function values by centered forms or by subdivision are discussed . The Interval Newton Method is developed as an always (globally) convergent technique to enclose zeros of functions . Then extended interval arithmetic is introduced . It allows division by intervals that contain zero and is the basis for the development of the extended Interval Newton Method . This is the major tool for computing enclosures at all zeros of a function or of systems of functions in a given domain. It is also the basic ingredient for many other important applications like global optimization, subdivision in higher dimensional cases or for computing error bounds for the remainder term of definite integrals in more than one variable. We also sketch the techniques of differentiation arithmetic, sometimes called automatic differentiation , for the computation of enclosures of derivatives, of Taylor coefficients , of gradients, of Jacobian or Hessian matrices. The major final part of the paper is devoted to the question of how interval arithmetic can effectively be provided on computers. This is an essential prerequisite for its superior and fascinating properties to be more widely used in the scientific computing community. With more appropriate processors, rigorous methods based on interval arithmetic could be comparable in speed with today’s “approximate” methods. At processor speeds of gigaFLOPS there remains no alternative but to furnish future computers with the capability to control the accuracy of a computation at least to a certain extent.