Showing papers in "Siam Review in 1968"
7,199 citations
TL;DR: The proper justification of Monte Carlo integration must be based not on the randomness of the procedure, which is spurious, but on equidistribution properties of the sets of points at which the integrand values are computed as discussed by the authors.
Abstract: The proper justification of the normal practice of Monte Carlo integration must be based not on the randomness of the procedure, which is spurious, but on equidistribution properties of the sets of points at which the integrand values are computed. Besides the discrepancy, which it is proposed to call henceforth extreme discrepancy, another concept, that of mean square discrepancy, can be regarded as a measure of the lack of equidistribution of a sequence of points in a multidimensional cube. Determinate upper bounds can be obtained, in terms of either discrepancy, for the absolute value of the error in the computation of the integral. There exist sequences of points yielding, for sufficiently smooth functions, errors of a much smaller order of magnitude than that which is claimed by the Monte Carlo method. In the case of two dimensions, sequences with optimum properties can be generated with the help of Fibonacci numbers. The previous arguments do not apply to domains of integration which cannot be reduced to multidimensional intervals. Difficult questions arising in this connection still await an answer.
96 citations
TL;DR: In this article, it was shown that if A is an integral matrix having linearly independent rows, then the extreme points of the set of nonnegative solutions to Ax = b are integral for all integral b if and only if the determinant of every basis matrix is plus or minus 1.
Abstract: : It is shown that if A is an integral matrix having linearly independent rows, then the extreme points of the set of nonnegative solutions to Ax = b are integral for all integral b if and only if the determinant of every basis matrix is plus or minus 1. This provides a short proof of the Hoffman- Kruskal theorem characterizing unimodular matrices, i.e., matrices in which the determinant of each nonsingular submatrix is plus or minus 1. Their theorem is that if A is integral, then A is unimodular if and only if the extreme points of the set of nonnegative solutions to Ax = or b are integral for all integral b.
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TL;DR: In this paper, the convex conical hull of the rows of an m X n matrix A is defined as a convex convex hull for which the nonnegative orthant is present.
Abstract: Collatz [2, Chap. 3, ?23] treats square matrices of monotone kind and shows that for such matrices the above implication is equivalent to: A-' exists and A1 2 0.1 Matrices of monotone kind have useful applications in numerical analysis [2, Chap. 3], [7]. It is the purpose of this note to generalize Collatz's result to rectangular matrices, and also to show that, for the general rectangular case, a matrix of monotone kind can be further characterized as one for which the convex conical hull of the rows contains the nonnegative orthant. (For an m X n matrix A, the convex conical hull of the rows of A is defined as
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TL;DR: Reduction of nonstandard problems in mathematical control theory by transformation into equivalent ordinary differential form as mentioned in this paper is a common technique in control theory, and has been successfully applied to control theory.
Abstract: Reduction of nonstandard problems in mathematical control theory by transformation into equivalent ordinary differential form
TL;DR: In this article, the existence of best approximations and their dependence on the function being approximated are studied for continuous functions on a family of continuous functions, where the Chebyshev problem is to find an element g* E CG for which 1i f -g 11 attains its infimum p(f) over elements g of G. Such an element G* is called a best approximation to f on X.
Abstract: hagil = sup {Ig(x) I:x E X}. Let G be an approximating family, a nonempty subset of B(X). Let C(X) be the family of continuous functions on X. The Chebyshev problem is, for given f E C(X), to find an element g* E CG for which 1i f -g 11 attains its infimum p(f) over elements g of G. Such an element g* is called a best approximation to f on X. In this note we study the existence of best approximations and their dependence on the function f being approximated.