The solution of an integral equation arising in a covariance factorization problem is obtained by a Newton-Raphson iteration that is almost always globally convergent. Interpretations of the iterates are given, and the result is shown to specialize to known algorithms when the covariance is stationary with a rational Fourier transform.

We study optimal quadrature formulas for convex functions in several variables. In particular, we answer the following two questions: Are adaptive methods better than nonadaptive ones? And: Are randomized (or Monte Carlo) methods better than deterministic methods?

It has been suggested in the literature that different quasi-uniform node sets on a sphere lead to quadrature formulas of highly variable quality. We analyze here the nature of these variations, and describe an easy-to-implement leastsquares remedy for previously problematic cases. Quadrature accuracies are then compared for different node sets ranging from fully random to those based on Gaussi...

A generalized N-point Birkhoff–Young quadrature of interpolatory type, with the Chebyshev weight, for numerical integration of analytic functions is considered. The nodes of such a quadrature are characterized by an orthogonality relation. Some special cases of this quadrature formula are derived. 2011 Elsevier Inc. All rights reserved.

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...

The zeros of semi-orthogonal functions with respect to a probability measure µ supported on the unit circle can be applied to obtain Szeg˝ o quadrature formulas. The discrete measures generated by these formulas weakly converge to the orthogonality measure µ. In this paper we construct families of semi-orthogonal functions with interlacing zeros, and give a representation of the support of µ in...

A flexible treatment of Gaussian quadrature formulas based on rational functions is given to evaluate the integral ∫ I f(x)W (x)dx, when f is meromorphic in a neighborhood V of the interval I and W (x) is an ill-scaled weight function. Some numerical tests illustrate the power of this approach in comparison with Gautschi’s method.