Evaluating crystallographic likelihood functions using numerical quadratures

Intensity-based likelihood functions in crystallographic applications have the potential to enhance the quality of structures derived from marginal diffraction data. Their usage, however, is complicated by the ability to efficiently compute these target functions. Here, a numerical quadrature is developed that allows the rapid evaluation of intensity-based likelihood functions in crystallographic applications. By using a sequence of change-of-variable transformations, including a nonlinear domain-compression operation, an accurate, robust and efficient quadrature is constructed. The approach is flexible and can incorporate different noise models with relative ease.

Fractional Ostrowski Type Inequalities via Generalized Mittag–Leffler Function

If we study the theory of fractional differential equations then we notice the Mittag–Leffler function is very helpful in this theory. On the contrary, Ostrowski inequality is also very useful in numerical computations and error analysis of numerical quadrature rules. In this paper, Ostrowski inequalities with the help of generalized Mittag–Leffler function are established. In addition, bounds of fractional Hadamard inequalities are given as straightforward consequences of these inequalities.

A note on the accuracy of adaptive Gauss–Hermite quadrature

Summary Numerical quadrature methods are needed for many models in order to approximate integrals in the likelihood function. In this note, we correct the error rate given by Liu & Pierce (1994) for integrals approximated with adaptive Gauss–Hermite quadrature and show that the approximation is less accurate than previously thought. We discuss the relationship between the error rates of adaptive Gauss–Hermite quadrature and Laplace approximation, and provide a theoretical explanation of simulation results obtained in previous studies regarding the accuracy of adaptive Gauss–Hermite quadrature.