Software (in Matlab) is developed for computing variable-precision recurrence coefficients for orthogonal polynomials with respect to the weight functions 1+sin(1/t), 1 + cos(1/t), e−1/t on [0, 1], as well as e−1/t−t on [0,∞] and e−1/t 2 −t on [−∞,∞]. Numerical examples are given involving Gauss quadrature relative to these weight functions.

The existence and uniqueness of the Gaussian interval quadrature formula with respect to the Hermite weight function on R is proved. Similar results have been recently obtained for the Jacobi weight on [−1, 1] and for the generalized Laguerre weight on [0,+∞). Numerical construction of the Gauss–Hermite interval quadrature rule is also investigated, and a suitable algorithm is proposed. A few n...

Gaussian formulas are among the most often used quadrature formulas in practice. In this survey, an overview is given on stopping functionals for Gaussian formulas which are of the same type as quadrature formulas, i.e., linear combinations of function evaluations. In particular, methods based on extended formulas like the important Gauss-Kronrod and Patterson schemes, and methods which are bas...

In the paper we consider new estimators of expected values E w(X) of functions of a random variable X. The new estimators are based on Gauss quadrature, a numerical method frequently used to approximate integrals over finite intervals. We apply the new estimators in Partial Cross Validation, a numerical method for finding optimal smoothing parameters in nonparametric curve estimation. We show t...

Accurate numericalintegrationof singularfunctions usually requireseither adaptivity or product integration. Both interfere with fast summation techniques and thus hamper large-scale computations. This paper presents a method for computing highly accurate quadrature formulas for singular functions which combine well with fast summation methods. Given the singularity and the N nodes, we rst const...