Verified computed Peano constants and applications in numerical quadrature

Abstract

Peano kernels are critical for the error estimates of numerical approximation of linear functionals. In the literature, considerable efforts have been devoted to the numerical computation or estimation of Peano constants, however, their discussions have mainly focused on the Peano kernels of some specific higher orders related to the degrees of the underlying error functionals. Limiting to such higher orders either requires certain smoothness of the approximated functions, which is not always fulfilled, or is not always the optimal choice for error estimates, even if the approximated functions are sufficiently smooth. Practical considerations in computing Peano constants for the full range of orders are given in this paper. Numerical examples are also given to demonstrate that much better error bounds can be derived while Peano constants of lower orders are considered. (2000): 41-04, 41A44, 41A55, 41A80, 65-04, 65A05, 65D20, 65D30, 65G20, 65G30. AMS subject classification

Convergence analysis and adaptive strategy for the certified quadrature over a set defined by inequalities

“…As real numbers are identified to degenerated intervals, the interval arithmetic actually generalizes the real arithmetic, and mixed operations like 1 + [1, 2] = [2,3] are interpreted using (4).…”

“…Finally, the framework is instantiated with Taylor model based quadrature methods: Although other methods allow computing certified quadrature (e.g. [3][4][5]), Taylor models are efficient and have good asymptotical convergence properties. Experiments on standard benchmarks from the literature [6][7][8][9] are presented in Section 6 to support this convergence analysis.…”