Up-to-date Interval Arithmetic: From Closed Intervals to Connected Sets of Real Numbers

This paper unifies the representations of different kinds of computer arithmetic. It is motivated by ideas developed in the book The End of Error by John Gustafson [5]. Here interval arithmetic just deals with connected sets of real numbers. These can be closed, open, half-open, bounded or unbounded. The first chapter gives a brief informal review of computer arithmetic from early floating-point arithmetic to the IEEE 754 floating-point arithmetic standard, to conventional interval arithmetic for closed and bounded real intervals, to the proposed standard IEEE P1788 for interval arithmetic, to advanced computer arithmetic, and finally to the just recently defined and published unum and ubound arithmetic [5]. Then in chapter 2 the style switches from an informal to a pure and strict mathematical one. Different kinds of computer arithmetic follow an abstract mathematical pattern and are just special realizations of it. The basic mathematical concepts are condensed into an abstract axiomatic definition. A computer operation is defined via a monotone mapping of an arithmetic operation in a complete lattice onto a complete sublattice. Essential properties of floating-point arithmetic, of interval arithmetic for closed bounded and unbounded real intervals, and for advanced computer arithmetic can directly be derived from this abstract mathematical model. Then we consider unum and ubound arithmetic. To a great deal this can be seen as an extension of arithmetic for closed real intervals to open and halfopen real intervals. Essential properties of unum and ubound arithmetic are also derived from the abstract mathematical setting given in chapter 2. Computer executable formulas for the arithmetic operations of ubound arithmetic are derived on the base of pure floating-point arithmetic. These are much simpler, easier to implement and faster to execute than alternatives that would be obtained on the base of the IEEE 754 floating-point arithmetic standard which extends pure floating-point arithmetic by a number of exceptions. The axioms of computer arithmetic given in section 2 also can be used to define ubound arithmetic in higher dimensional spaces like complex numbers, vectors and matrices with real and interval components. As an example section 4 indicates how this can be done in case of matrices with ubound components. Execution of the resulting computer executable formulas once more requires an exact dot product. In comparison with conventional interval arithmetic The End of Error may be a too big step to easily get accepted by manufacturers and computer users. So in the last section we mention a reduced but still great step that might easier find its way into computers in the near future.