Interval Enclosures of Upper Bounds of Roundoff Errors using Semidefinite Programming

A longstanding problem related to floating-point implementation of numerical programs is to provide efficient yet precise analysis of output errors. We present a framework to compute lower bounds of absolute roundoff errors for numerical programs implementing polynomial functions with box constrained input variables. Our study relies on semidefinite programming (SDP) relaxations and is complementary of over-approximation frameworks, consisting of obtaining upper bounds for the absolute roundoff error. Our method is based on a new hierarchy of convergent robust SDP approximations for certain classes of polynomial optimization problems. Each problem in this hierarchy can be exactly solved via SDP. By using this hierarchy, one can provide a monotone non-decreasing sequence of lower bounds converging to the absolute roundoff error of a program implementing a polynomial function. We investigate the efficiency and precision of our method on non-trivial polynomial programs coming from space control, optimization and computational biology.