Program Verification in the presence of complex numbers, functions with branch cuts etc

Abstract

In considering the reliability of numerical programs,it is normal to “limit our study to the semanticsdealing with numerical precision” (Martel, 2005). On the otherhand, there is a great deal of work on the reliability ofprograms that essentially ignores the numerics. The thesisof this paper is that there is a class of problems that fallbetween these two, which could be described as “does the lowlevelarithmetic implement the high-level mathematics”. Manyof these problems arise because mathematics, particularlythe mathematics of the complex numbers, is more difficultthan expected: for example the complex function log is notcontinuous, writing down a program to compute an inversefunction is more complicated than just solving an equation,and many algebraic simplification rules are not universallyvalid.The good news is that these problems are theoreticallycapable of being solved, and are practically close to being solved,but not yet solved, in several real-world examples. However,there is still a long way to g