History
Interval arithmetic is not a completely new phenomenon in mathematics; it has appeared several times under different names in the course of history. For example Archimedes calculated lower and upper bounds 223/71 < π < 22/7 in the 3rd century BC. Actual calculation with intervals has neither been as popular as other numerical techniques, nor been completely forgotten.
Rules for calculating with intervals and other subsets of the real numbers were published in a 1931 work by Rosalind Cicely Young, a doctoral candidate at the University of Cambridge. Arithmetic work on range numbers to improve reliability of digital systems were then published in a 1951 textbook on linear algebra by Paul Dwyer (University of Michigan); intervals were used to measure rounding errors associated with floating-point numbers.
The birth of modern interval arithmetic was marked by the appearance of the book Interval Analysis by Ramon E. Moore in 1966. He had the idea in Spring 1958, and a year later he published an article about computer interval arithmetic. Its merit was that starting with a simple principle, it provided a general method for automated error analysis, not just errors resulting from rounding.
Independently in 1956, Mieczyslaw Warmus suggested formulae for calculations with intervals, though Moore found the first non-trivial applications.
In the following twenty years, German groups of researchers carried out pioneering work around Götz Alefeld and Ulrich Kulisch at the University of Karlsruhe and later also at the Bergische University of Wuppertal. For example, Karl Nickel explored more effective implementations, while improved containment procedures for the solution set of systems of equations were due to Arnold Neumaier among others. In the 1960s Eldon R. Hansen dealt with interval extensions for linear equations and then provided crucial contributions to global optimisation. Classical methods in this often are have the problem of determining the largest (or smallest) global value, but could only find a local optimum and could not find better values; Helmut Ratschek and Jon George Rokne developed branch and bound methods, which till then had only applied to integer values, by using intervals to provide applications for continuous values.
In 1988, Rudolf Lohner developed Fortran-based software for reliable solutions for initial value problems using ordinary differential equations.
The journal Reliable Computing (originally Interval Computations) has been published since the 1990s, dedicated to the reliability of computer-aided computations. As lead editor, R. Baker Kearfott, in addition to his work on global optimisation, has contributed significantly to the unification of notation and terminology used in interval arithmetic (Web: Kearfott).
In recent years work has concentrated in particular on the estimation of preimages of parameterised functions and to robust control theory by the COPRIN working group of INRIA in Sophia Antipolis in France (Web: INRIA).
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