Certified and fast computation of supremum norms of approximation errors

Sylvain Chevillard (1) , Mioara Maria Joldes (1) , Christoph Lauter (1)

1 ARENAIRE - Computer arithmetic

In many numerical programs there is a need for a high-quality floating-point approximation of useful functions f, such as exp, sin, erf. In the actual implementation, the function is replaced by a polynomial p, leading to an approximation error (absolute or relative) epsilon = p-f or epsilon = p/f-1. The tight yet certain bounding of this error is an important step towards safe implementations. The main difficulty of this problem is due to the fact that this approximation error is very small and the difference p-f is highly cancellating. In consequence, previous approaches for computing the supremum norm in this degenerate case, have proven to be either unsafe, not sufficiently tight or too tedious in manual work. We present a safe and fast algorithm that computes a tight lower and upper bound for the supremum norms of approximation errors. The algorithm is based on a combination of several techniques, including enhanced interval arithmetic, automatic differentiation and isolation of the roots of a polynomial. We have implemented our algorithm and timings on several examples are given.

Mots clés: supremum norm; approximation error; certified computation; elementary function; interval arithmetic; automatic differentiation; roots isolation technique

Domaines: Arithmétique des ordinateurs

Sylvain Chevillard, Mioara Maria Joldes, Christoph Lauter. Certified and fast computation of supremum norms of approximation errors. 19th IEEE Symposium on Computer Arithmetic (ARITH 19), Jun 2009, Portland, United States. pp.169 -- 176, ⟨10.1109/ARITH.2009.18⟩. ⟨ensl-00334545v2⟩