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The paper reports new significant enhancement of the robustness and effectiveness of the Bayesian automatic adaptive quadrature over macroscopic integration ranges. The implementation of a classical m-panel rule (CC-32, Clenshaw-Curtis quadrature of algebraic degree of precision m = 32) is thought again. It involves new global and local decisions blocks which, on the one side, provide sharp diagnostics redirecting the advancement to the solution and, on the other side, take advantage of the progress in the available hardware to accelerate and to increase the accuracy of the computations. Where the decision power of CC-32 is exhausted, identification and precise characterization of the features of the integrand profile which prevent quick convergence are obtained by means of three-point Simpson rules spanned at triplets of successive CC-32 knots. This local complementary investigation tool provides scale insensitive diagnostics concerning the occurrence of integrand irregularities and prevents the activation of inappropriate decision blocks which would result in fake outputs. [ABSTRACT FROM AUTHOR]

Two topics concerning the use of Clenshaw-Curtis quadrature within the Bayesian automatic adaptive quadrature approach to the numerical solution of Riemann integrals are considered. First, it is found that the efficient floating point computation of the coefficients of the Chebyshev series expansion of the integrand is to be done within a mathematical structure consisting of the union of coefficient families ordered into complete binary trees. Second, the scrutiny of the decay rates of the involved even and odd rank Chebyshev expansion coefficients with the increase of their rank labels enables the definition of Bayesian decision paths for the advancement to the numerical output. [ABSTRACT FROM AUTHOR]

Two conceptual developments in the Bayesian automatic adaptive quadrature approach to the numerical solution of one-dimensional Riemann integrals are reported. First, it is shown that the numerical quadrature which avoids the overcomputing and minimizes the hidden floating point loss of precision asks for the consideration of three classes of integration domain lengths endowed with specific quadrature sums: microscopic (trapezoidal rule), mesoscopic (Simpson rule), and macroscopic (quadrature sums of high algebraic degrees of precision). Second, sensitive diagnostic tools for the Bayesian inference on macroscopic ranges, coming from the use of Clenshaw-Curtis quadrature, are derived. [ABSTRACT FROM AUTHOR]