Wilfred Kaplan

A test for copositive matrices

Abstract The paper presents necessary and sufficient conditions that a symmetric matrix be copositive or strictly copositive. The conditions are given in terms of the eigenvalues and eigenvectors of the principal submatrices of the given matrix.

A copositivity probe

Abstract The paper explores ways of determining whether a given symmetric matrix is copositive. In particular, a computational procedure is proposed for determining (if it exists) a representation of the matrix as a sum of a positive semidefinite matrix and a nonnegative matrix. The procedure is found to be successful in a significant number of cases.

Duality theorem for a generalized Fermat-Weber problem

The classical Fermat-Weber problem is to minimize the sum of the distances from a point in a plane tok given points in the plane. This problem was generalized by Witzgall ton-dimensional space and to allow for a general norm, not necessarily symmetric; he found a dual for this problem. The authors generalize this result further by proving a duality theorem which includes as special cases a great variety of choices of norms in the terms of the Fermat-Weber sum. The theorem is proved by applying a general duality theorem of Rockafellar. As applications, a dual is found for the multi-facility location problem and a nonlinear dual is obtained for a linear programming problem with a priori bounds for the variables. When the norms concerned are continuously differentiable, formulas are obtained for retrieving the solution for each primal problem from the solution of its dual.

PROCEDURES FOR MULTIPLE INTEGRATION

Abstract This paper presents a brief description of a numerical procedure for evaluating multiple integrals over the unit cube in N -dimensional space, for N = 2,…,8. A familiar smoothing process enhances the speed and accuracy. Examples are given.

Ordinary Differential Equations.

together with the initial condition y(t0) = y0 A numerical solution to this problem generates a sequence of values for the independent variable, t0, t1, . . . , and a corresponding sequence of values for the dependent variable, y0, y1, . . . , so that each yn approximates the solution at tn yn ≈ y(tn), n = 0, 1, . . . Modern numerical methods automatically determine the step sizes hn = tn+1 − tn so that the estimated error in the numerical solution is controlled by a specified tolerance. The Fundamental Theorem of Calculus gives us an important connection between differential equations and integrals.