Efficient and accurate computation of upper bounds of approximation errors

Sylvain Chevillard (1) , John Harrison (2) , Mioara Maria Joldes (3) , Christoph Lauter (2)

1 CARAMEL - Cryptology, Arithmetic: Hardware and Software
2 INTEL
3 ARENAIRE - Computer arithmetic

For purposes of actual evaluation, mathematical functions f are commonly replaced by approximation polynomials p. Examples include floating-point implementations of elementary functions, quadrature or more theoretical proof work involving transcendental functions. Replacing f by p induces a relative error epsilon = p/f - 1. In order to ensure the validity of the use of p instead of f, the maximum error, i.e. the supremum norm of epsilon must be safely bounded above. Numerical algorithms for supremum norms are efficient but cannot offer the required safety. Previous validated approaches often require tedious manual intervention. If they are automated, they have several drawbacks, such as the lack of quality guarantees. In this article a novel, automated supremum norm algorithm with a priori quality is proposed. It focuses on the validation step and paves the way for formally certified supremum norms. Key elements are the use of intermediate approximation polynomials with bounded approximation error and a non-negativity test based on a sum-of-squares expression of polynomials. The new algorithm was implemented in the Sollya tool. The article includes experimental results on real-life examples.

Mots clés: Sum-of-squares; Validation; Certification; Formal proof; Supremum norm; Approximation error; Taylor Models

Domaines: Arithmétique des ordinateurs

Sylvain Chevillard, John Harrison, Mioara Maria Joldes, Christoph Lauter. Efficient and accurate computation of upper bounds of approximation errors. Theoretical Computer Science, 2011, 412 (16), pp.1523-1543. ⟨10.1016/j.tcs.2010.11.052⟩. ⟨ensl-00445343v2⟩