  This paper presents a fast and powerful method for the computation of
eigenvalue bounds for Hessian matrices $\nabla^2 \varphi(x) $ of nonlinear
functions $\varphi: U \subseteq R^n\rightarrow R$ on hyperrectangles $B \subset
U$. The method is based on a recently proposed procedure for an efficient
computation of spectral bounds using extended codelists. Both the previous
approach and the one presented here substantially differ from established
methods in that they do deliberately not use any interval matrices and thus
result in a favorable numerical complexity of order $O(n)\,N(\varphi)$, where
$N(\varphi)$ denotes the number of operations needed to evaluate $\varphi$ at a
point in its domain. We improve the previous method by exploiting sparsity,
which naturally arises in the underlying codelists.
