Philip Rabinowitz

Monomial cubature rules since “Stroud”: a compilation

A bibliography of references to cubature rules which have appeared since the publication of Strouds book (1971) is presented. The standard regions that are treated in this paper are the n-cube, the n-simplex, the n-sphere and the entire space.

Numerical integration based on approximating splines

Abstract Convergence results are proved for product integration rules based on approximating splines. These results are both for bounded and unbounded integrands. Pointwise and uniform convergence results are proved for sequences of Cauchy principal values of these approximating splines.

Advances in Orthonormalizing Computation

Publisher Summary The chapter presents a survey of least square approximation techniques in numerical analysis and some recent numerical results in this field, which were obtained at the National Bureau of Standards on an IBM 704 computer. The chapter describes aspects of this theory, which are of utility in numerical analysis and which display a spectrum of applications of the least square idea. The chapter discusses a wide variety of inner product spaces that appear in theoretical arguments. In the discrete case, inner products are the sums of ordinary products whereas in the continuous case, inner products are integrals. The chapter describes the methods of orthogonalization and least square methods for ordinary differential equations. The chapter also presents numerical experiments in the solution of boundary value problems by using the method of orthonormalized particular solutions. These experiments indicate the accuracy that can be achieved with the least square method in a wide variety of situations.

The Application of Integer Programming to the Computation of Fully Symmetric Integration Formulas in Two and Three Dimensions

A method is given for constructing fully symmetric integration rules over fully symmetric two- and three-dimensional regions of any degree d with a minimal number of evaluation points. This method is based on the solution of integer programming problems in which the constraints are the conditions for linear consistency of the system of nonlinear algebraic equations ensuring exactness of the rule for all polynomials of degree

Ignoring the Singularity in Approximate Integration

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Product Integration in the Presence of a Singularity

. The solution of the integer programming problem determines the number of generators of each type in the system and indicates how to solve it. In the Appendix, almost all known real fully symmetric integration rules for the regions

The exact degree of precision of generalized Gauss-Kronrod integration rules

C_n

Numerical integration in the presence of an interior singularity

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Abscissas and weights for Lobatto quadrature of high order

S_n

A Multiple Purpose Orthonormalizing Code and its Uses

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