Showing papers in "Journal of The Society for Industrial and Applied Mathematics in 1964"
TL;DR: An algebraic proof of the existence of equilibrium points for two-person non-zero-sum games is given in this paper, leading to an efficient scheme for computing an equilibrium point, which is valid for any ordered field.
Abstract: An algebraic proof is given of the existence of equilibrium points for bimatrix (or two-person, non-zero-sum) games. The proof is constructive, leading to an efficient scheme for computing an equilibrium point. In a nondegenerate case, the number of equilibrium points is finite and odd. The proof is valid for any ordered field.
1,087 citations
TL;DR: In this paper, an exact recursive procedure for numerical inversion of an arbitrary positive definite Toeplitz matrix of finite order was derived, which takes full advantage of the strong restrictions placed on its elements by (1.1), (1 2 ), and (1 3 ).
Abstract: possesses properties (1.1), (1.2), and (1.3). In order to find the joint probability density function of (yo, yi , , yn), or of any n + 1 successive variates, it is necessary to invert the matrix Tn . In this paper, we derive an exact recursive procedure for the numerical inversion of an arbitrary positive definite Toeplitz matrix of finite order, which takes full advantage of the strong restrictions placed on its elements by (1.1), (1.2), and (1.3). The number of multiplications required for the inversion of an nth order Toeplitz matrix, using this procedure, is proportional to n2, rather than to n', as in the case of methods which are suitable for arbitrary Hermitian matrices. To the author's knowledge, this inversion algorithm is the first to be specifically designed to take advantage of the peculiar simplicity of the general Toeplitz matrix. In addition, the closing section of the paper is devoted to a statement of an algorithm for the inversion of lnon-Hernmitian matrices of the form (1.1).
563 citations
416 citations
300 citations
TL;DR: In this article, the authors describe a method for computing the extreme equilibrium points of a bimatrix game which are finite in number, i.e., determining all equilibrium points in the game.
Abstract: The purpose of this note is to describe a method for computing the extreme equilibrium points of a bimatrix game which are finite in number. This in essence determines all equilibrium points of the game, The present method seems to be simpler than that of Vorob’ev’s method [7] and Kuhn’s simplification thereof [3]. A computer program has been written to implement the present method.
104 citations
100 citations
TL;DR: In this paper, the Peaceman-Rachford method was used to approximate the solution of a matrix equation with a linear operator on a Hilbert space H. Two theorems were proved showing that the solution u can be approximated arbitrarily closely by an iterative method which is analogous to the PEARACHF method for solving matrix equations.
Abstract: 1. Consider an equation Au = f, where A is a linear operator on a Hilbert space H. Suppose one can write A = Al + A2, where the operators Al and A2 satisfy certain conditions given below. Two theorems are proved showing that the solution u can be approximated arbitrarily closely by an iterative method which is analogous to the Peaceman-Rachford method for solving matrix equations. (See [1], [2]. The operators Al and A2 play the role of H and V in the Peaceman-Rachford method.) In particular the conditions are satisfied when A is the unbounded operator
92 citations
90 citations
TL;DR: Rodrigues's formula can be applied also to (1.1) and 1.3) as discussed by the authors, but here the situation is slightly more involved in that the integrals with respect to σ^2 are of fractional order and their inversion requires the knowledge of differentiation and integration.
Abstract: Rodrigues’s formula can be applied also to (1.1) and (1.3) but here the situation is slightly more involved in that the integrals with respect to σ^2 are of fractional order and their inversion requires the knowledge of differentiation and integration of fractional order. In spite of this complication the method has its merits and seems more direct than that employed in [1] and [3]. Moreover, once differentiation and integration of fractional order are used, it seems appropriate to allow a derivative of fractional order with respect to σ^-1 to appear so that the ultraspherical polynomial in (1.3) may be replaced by an (associated) Legendre function. This will be done in the present paper.
83 citations
TL;DR: Algorithms for min-cost synthesis of a communication network which is able to handle simultaneous flows of all time periods are given.
Abstract: A communication network is a set of nodes connected by arcs. Every arc has associated with it a nonnegative number called the branch capacity which indicates the maximum amount of flow that can pass through the arc. A communication network must have large enough branch capacities such that all message requirements (which can be regarded as flows of different commodities) can reach their destinations simultaneously. In general, these requirements vary with time. The present paper gives algorithms for min-cost synthesis of a communication network which is able to handle simultaneous flows of all time periods.
79 citations
TL;DR: Inequalities for theory of finite difference equations and more general discontinuous functional equations were shown in this article for finite difference and functional equations. But they were not shown for the theory of functional functions.
Abstract: Inequalities for theory of finite difference equations and more general discontinuous functional equations
TL;DR: The problem of finding an equivalence relation E, which “best approximates” a given symmetric relation R in the sense of minimizing the number of elements of E, is solved for a class of relations R representing two-level and three-level “hierarchical” structures.
Abstract: The problem of finding an equivalence relation E, which “best approximates” a given symmetric relation R in the sense of minimizing the number of elements of $( E - R) \cup ( R - E)$, is solved for a class of relations R representing two-level and three-level “hierarchical” structures.
TL;DR: In this paper, Frank and Wolfe showed that sp(z) > 0 for all z E C. If M is positive semi-definite (not necessarily syonmetric) and conditions (2) and (3) are consistent, then so are conditions (1)-(3).
Abstract: For the moment, ill need have no special properties other than being real and of size N X N. Clearly, C is a polyhedral convex set in real N-space, RN. We observe that sp(z) > 0 for all z E C. A result of AMl. Frank and P. Wolfe [5, appendix i, p. 108] implies that when C is nonempty, the infiinum of sp(z) over C is attained there. Having observed this fact, we proceed to our variant of Dorn's Theorem. THEOREM. If M is positive semi-definite (not necessarily syonmetric) and conditions (2) and (3) are consistent, then so are conditions (1)-(3). Proof. If (2) and (3) are consistent, the set C is nonempty, and the infimum of Sp(z) over C is attained there for some z( E C. Showing that
TL;DR: In this article, it was shown that for all sufficiently large integers s there are precisely 2s and 2s + 2 such zeros of fp and gq respectively within the circle I z I = (s+ 2)ir/{ k 1 1 1 ) for k = 0.
Abstract: arise regularly in a variety of physical problems with circular or cylindrical geometry, and one is usually interested in either the nature of the z-zeros of these functions for fixed v and k or that of their v-zeros with z and k given It has been known for some time that for real v and positive k the z-zeros of fL(k, z) are all real and simple [1] In this paper we show that these same results hold for gq(k, z) also Furthermore, by means of contour integrations it is proved that for all sufficiently large integers s there are precisely 2s and 2s + 2 such zeros of fp and gq respectively within the circle I z I = (s + 2)ir/{ k 1 1 This result enables us to conclude that the asymptotic expansions of McMahon [2] for the larger real z-zeros actually give the sth and (s + 1 )st zeros of fL and gv respectively (v 5 0) and not, as stated by McMahon, the sth in both cases The v-zeros are also considered in this paper When z is real1 and k > 0 it is established that both f, and gq have a countable number of v-zeros of which only at most a finite number are real, the remainder being purely imaginary All of these zeros are shown to be simple except for that occurring at the origin (v = 0) which can appear only as a zero of the second order Asymptotic expressions are derived for the large imaginary v-zeros The case of negative or complex k is quite another matter For instance, f312(k, z) and g112(lc, z) both have two imaginary zeros for k < 0 Although one expects that Hurwitz-type results [3] will be true in the more general situations, these will not be considered here Now f(1(k, z) and g,(k, z) are both entire functions of the complex variable v In addition, f,(k, z) is also regular analytic for all z, while gq(k, z) is in general regular everywhere in the finite z-plane except at the origin where it has a pole of the second order If v = 0 this pole does not occur, and go(k, z) is an entire function of z Both f, and g, are even functions of z and v Moreover since f, (/lk, kz) =-f, (k, z) and g, ( /lk, kz) = g, (k, z), there is a one-one correspondence
TL;DR: In this article, it was shown that the conditions for convergence for linear homogeneous equations, which are provided by the Lax-Richtmnyer theory, are sufficient for the more general inhomogeneous equations.
Abstract: In particular, they showed that for a consistent approximation stability and convergence are equivalent The results presented here have to do with the problem of approximating solutions of the inhomogeneous and quasi-linear equations which are obtained when terms of the form f(t) or g(t, u(t)) are added to equations such as the one above Finite difference schemes for specific types of inhomogeneous equations and quasi-linear equations have been studied many times (see, for example, [1], [2], [4], and [7]) An extensive bibliography is included in [2] In [7] Strang deals with linear inhomogeneous parabolic and hyperbolic partial differential equations; he shows that here, too, with appropriate definitions, stability and convergence are equivalent Fritz John [4] deals with parabolic equations and includes a section on quasi-linear equations John's paper is particularly remarkable in that he does not assume existence of solutions for the differential equations but proves existence using properties of the difference approximations Frequently a difference approximation for an inhomogeneous equation or a quasi-linear equation is a simple modification of an approximation for the associated homogeneous equation In this paper it will be shown that, it is often sufficient to consider the problem of convergence only for the homogeneous equation That is, the conditions for convergence for linear homogeneous equations, which are provided by the Lax-Richtmnyer theory, are sufficient for the more general equations Duhamel's method is used to show that a convergent approximation for a linear homogeneous equation
TL;DR: In this paper, it was shown that a sequence of polynomials PN(Z) can be determined numerically such that PN (Z) -> g(z) uniformly on G.
Abstract: It will be shown that a sequence of harmonic functions {I Uk} can be determined numerically such that Uk -->u uniformly on G U C as I zj} becomes dense in C and ,u -> 0. Since the Uk are determined by solving a certain linear programming problem which fits the boundary data F(z) with a well defined harmonic function, an a posteriori estimate of the error can be obtained. Some numerical results of the linear programming method are discussed in the final section. All of the results of this paper follow from the proof of the basic theorem of Runge and its extension by Walsh. THEOREM. If g(z) is an analytic function in G which is continuous in U = G U C, then there exists a sequence of polynomials PN(Z) such that PN(Z) -> g(z) uniformly on G.
TL;DR: In this paper, the problem of determining the structure of the interpolating polynomial in the case where p is independent of i has been studied, and it is shown that the solution depends upon the Bell polynomials which we now discuss.
Abstract: where Dt d/dt and aj,m is a Kronecker symbol. These conditions are used by Householder [5, pp. 193-195] to derive the formulas for p = 1, 2. The formula for p = 3 is given by Salzer [9]. The solution for n = 0 is given by Taylor's formula. Many authors have reported on the case where p depends on i. General prescriptions for a solution in this more general case may be found in Fort [2, pp. 85-88], Greville [3], Hermite [4], Krylov [6, pp. 45-49], Kuntzmann [7, pp. 167-169], and Spitzbart [12]; but these prescriptions do not determine the structure of the interpolating polynomial. By restricting ourselves to the case where p is independent of i, which is the most important case in practice, we can determine the structure. Salzer [10] discovered some of the properties of P ,, (t) by semiempirical means. We shall obtain, by a partial fraction expansion, a solution of surprising simplicity. [See (3.6), (3.7), or (3.8).] The solution depends upon the Bell polynomials which we now discuss.
TL;DR: In this article, it was shown that the truncation error can be made arbitrarily small by making the mesh size h small, and that such an error bound may be obtained for certain finite difference analogues to a class of smooth problems.
Abstract: 1. Introduction. A common method for obtaining an approximate solution to the various self-adjoint two point boundary value problems for the Sturm-Liouville equation is by finite differences. In defining any finite difference analogue, it is necessary to know that the truncation error (i.e., the difference between the exact solution and the solution of the approximating finite difference problem) can be made arbitrarily small by making the mesh size h small. In addition to knowing the order of convergence, it is desirable to have a priori error bounds of that order to assist one in the selection of h. It is the aim of this paper to show how such an error bound may be obtained for certain finite difference analogues to a class of smooth problems. The point of view taken here is to utilize the existence of a pointwise bound for any sufficiently smooth function y considered. In particular, it can be shown that