Seymour Haber

Parameters for Integrating Periodic Functions of Several Variables

A number-theoretical method for numerical integration of periodic functions of several variables was developed some years ago. This paper presents lists of numerical parameters to be used in implementing that method. The parameters define quadrature formulas for functions of 2, 3,..,8 variables; error bounds for those formulas are also tabulated. The derivation of the parameters and error bounds is described.

Stochastic quadrature formulas

A class of formulas for the numerical evaluation of multiple integrals is described, which combines features of the Monte-Carlo and the classical methods. For certain classes of functions-defined by smoothness conditions-these formulas provide the fastest possible rate of convergence to the integral. Asymptotic error estimates are derived, and a method is described for obtaining good a posteriori error bounds when using these formulas. Equal-coefficients formulas of this class, of degrees up to 3, are constructed.

Numerical indefinite integration of functions with singularities

We derive an indefinite quadrature formula, based on a theorem of Ganelius, for H p functions, for p > 1, over the interval (-1,1). The main factor in the error of our indefinite quadrature formula is O(e -π √ N/q ), with 2N nodes and 1/p + 1 - = 1. The convergence rate of our formula is better than that of the Stenger-type formulas by a factor of √2 in the constant of the exponential. We conjecture that our formula has the best possible value for that constant. The results of numerical examples show that our indefinite quadrature formula is better than Habers indefinite quadrature formula for H p -functions.

On a Theorem of Piatetsky-Shapiro and Approximation of Multiple Integrals

Let f be a function of s real variables which is of period 1 in each variable, and let the integral I of f over the unit cube in s-space be approximated by

Adaptive integration and improper integrals

Let R be the class of all functions that are properly Riemann-integrable on (0, 1], and let IR be the class of all functions that are properly Riemann-integrable on [a, 1] for all a > 0 and for which