Context: Petri net slicing is a technique to reduce the size of a Petri net to ease the analysis or understanding of the original Petri net. Objective: Presenting two new Petri net slicing algorithms to isolate those places and transitions of a Petri net (the slice) that may contribute tokens to one or more places given (the slicing criterion). Method: The two algorithms proposed are formalized. The maximality of the first algorithm and the minimality of the second algorithm are formally proven. Both algorithms, together with three other state-of-the-art algorithms, have been implemented and integrated into a single tool so that we have been able to carry out a fair empirical evaluation. Results: Besides the two new Petri net slicing algorithms, a public, free, and open-source implementation of five algorithms is reported. The results of an empirical evaluation of the new algorithms and the slices they produce are also presented. Conclusions: The first algorithm collects all places and transitions that may contribute tokens (in any computation) to the slicing criterion, while the second algorithm collects the places and transitions needed to fire the shortest transition sequence that contributes tokens to some place in the slicing criterion. Therefore, the net computed by the first algorithm can reproduce any computation that contributes tokens to any place of interest. In contrast, the second algorithm loses this possibility, but it often produces a much more reduced subnet (which still can reproduce some computations that contribute tokens to some places of interest). The first algorithm is proven maximal, and the second one is proven minimal.

Abstract Earlier authors have shown experimentally that the relative effect of rounding errors in the evaluation of polynomial approximations to certain special functions may be significantly reduced by first extracting an exponential factor. The critical importance of evaluating this exponential by an accurate library function is demonstrated, and the two principal methods of representing the polynomial part are shown to be equally accurate provided the recommended evaluation schemes are employed.

An automatic integration algorithm, based on the Clenshaw-Curtis method, is described which can adapt either the formula in use or the interval subdivision at any stage. Numerical tests demonstrate its competitiveness with both whole-interval formulae and adaptive algorithms for well and badly behaved integrands.

Most numerical methods for producing approximate Chebyshev series solutions to ordinary differential equations lead to a system of algebraic equations for the coefficients, while the truncation error (or a first order approximation to it for non-linear equations) can be formulated as an infinite series. By solving the algebraic system with additional right-hand sides and by extrapolating the size of the exact coefficients from the computed ones, the first few terms in the error expansion result, giving an accurate error estimate varying with the independent variable unless the series is slowly convergent. The technique is illustrated by numerical examples.