Interval Enclosures of Upper Bounds of Roundoff Errors using Semidefinite Programming

TL;DR

This paper introduces a semidefinite programming-based framework to compute lower bounds on maximum roundoff errors in polynomial numerical programs, complementing existing upper bound methods for more precise error enclosures.

Contribution

It proposes a novel hierarchy of SDP relaxations that provide convergent lower bounds on roundoff errors, enhancing the accuracy of error analysis in floating-point computations.

Findings

Efficient computation of lower bounds on roundoff errors for polynomial programs.

The hierarchy converges to the true maximum roundoff error, providing tight bounds.

Demonstrated effectiveness on real-world applications from space control, optimization, and biology.

Abstract

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 on largest absolute roundoff errors, for a particular rounding model. This method applies to numerical programs implementing polynomial functions with box constrained input variables. Our study is based on three different hierarchies, relying respectively on generalized eigenvalue problems, elementary computations and semidefinite programming (SDP) relaxations. This is complementary of over-approximation frameworks, consisting of obtaining upper bounds on the largest absolute roundoff error. Combining the results of both frameworks allows to get enclosures for upper bounds on roundoff errors. The under-approximation framework provided by the third hierarchy is based on a new sequence of convergent robust SDP approximations for certain classes of polynomial optimization problems. Each problem in this hierarchy can be solved exactly 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, applying for a particular rounding model. We investigate the efficiency and precision of our method on non-trivial polynomial programs coming from space control, optimization and computational biology.