# Interval Arithmetic

Chapter
Part of the Studies in Computational Intelligence book series (SCI, volume 766)

## Abstract

The first applications of interval arithmetic and interval methods mainly concerned the problem of assuring the correctness and reliability of numerical computations, in view of rounding errors resulting from the imperfect machine representation of real numbers. Because of this representation, two types of rounding errors can occur. The first one follows from the approximation of real input data by machine numbers, and the second is due to the approximation of intermediate results by machine numbers. As an example of the latter, consider quadratic equation $$x^2+2px-1=0$$. The roots of a quadratic equation are usually computed by using the following quadratic formula: $$x_{1,2}=-p\pm \sqrt{p^2+1}$$. In IEEE 754 floating-point (FP) arithmetic, which is the most-widely-used form of computer arithmetic, if $$1/p^2\leqslant {\mathbf {u}}$$, where $${\mathbf {u}}$$ is machine epsilon (half the distance between 1 and its successor), then $$p^2+1=p^2$$, and we obtain $$x_1=-p+p=0$$ in computer arithmetic (which is incorrect). Since $${\mathbf {u}}=2^{-53}$$ in binary64 floating-point representation, the binary64 floating-point arithmetic will produce the same incorrect result for each $$p\geqslant 10^8$$. What is worse, no warning will be displayed, so an inexperienced user will trust the result. In order to manage the problem, the root of the smaller magnitude should be computed via equation $$x_1=q/\sqrt{p^2+q}$$ ($$p>0$$). Then, $$x_1=5\cdot 10^{-9}$$ is produced, which is the correct value. Interval arithmetic in this case is not very helpful, since it yields range $$[0,1.4901\cdot 10^{-8}]$$ for $$x_1$$. This range takes rounding errors into account, of course, but it is not very informative. In Rump, Acta Numer 19:287–449, 2010, [234] it is argued that the wide range, such as the one obtained for $$x_1$$, brings no information about the sensitivity of the problem. In the general case, this is true since the wide range may be attributed to overestimation, which is inherent to interval computation. In this example, however, the wide range does indicate the sensitivity of the problem since the computation is relatively simple and the input data is crisp. Interval arithmetic “tells” us, in this case, that we should rearrange the computation or use higher-precision arithmetic. Several other examples of the pitfalls of floating-point processing can be found in Forsythe, Am Math Mon 77(9):931–956, 1970, [53], Moore, Reliability in computing: the role of interval methods in scientific computing, Academic Press, New York, 1988, [148], Revol and Théveny, IEEE Trans Comput 63(8):1915–1924, 2014, [200], Rump, Acta Numer 19:287–449, 2010, [234], for example. One of the best-known examples of the catastrophic consequences of rounding errors is the failure of the MIM-104 Patriot missile defense system during Operation Desert Storm at Dahram, Saudi Arabia, on Feb 25, 1991 (Software Problem Led to System Failure at Dhahran, Saudi Arabia. IMTEC-92-26: Published: Feb 4, 1992. Publicly Released: Feb 27, 1992, [284]). Interval arithmetic, if carefully implemented, can be useful in tracking rounding errors in numerical computing. However, like any other tool, it is not a panacea for all problems in scientific computing. If used inappropriately, the results might be useless (Rump, Acta Numer 19:287–449, 2010, [234]).