On the Automatic Computation of Error Bounds for Solutions of Nonlinear Equations

A large portion of software is used for numerical calculations in mathematics, physics and engineering applications. Among the things that make verification in this domain difficult is the quantification of numerical errors, such as roundoff errors and errors due to the approximate numerical method. Much of numerical software uses self-stabilizing iterative algorithms, for example, to find solutions of nonlinear equations. To support such algorithms, we present a runtime verification technique that checks, given a nonlinear equation and a tentative solution, whether this value is indeed a solution to within a specified precision. Our technique combines runtime verification approaches with information about the analytical equation being solved. It is independent of the algorithm used for finding the solution and is therefore applicable to a wide range of problems. We have implemented our technique for the Scala programming language using our affine arithmetic library and the macro facility of Scala 2.10.