Showing papers in "Siam Review in 1970"
394 citations
361 citations
250 citations
TL;DR: In this article, the authors provide a survey of direct methods for solving finite difference equations with rectilinear domains. But the authors do not discuss whether the methods are easily adaptable to more general regions, and to general elliptic partial differential equations.
Abstract: where G is a rectangle, Au = 82u/8x2 + 82u/8y2, and v, w are known functions. For computational purposes, this partial differential equation is frequently replaced by a finite difference analogue. These discrete models for (1) consist of linear systems of equations of very large dimension, and it is widely recognized that the usual direct methods (e.g., Gaussian elimination) are unsatisfactory for such systems [18, ?? 21.2-21.3]. Theoretical investigation has, therefore, been primarily directed toward the development of effective iterative methods for the solution of these problems [64], [66]. In recent years, however, direct methods that take advantage of the special block structure of these linear equations have appeared. For the rectangular regions under consideration, these methods can be considerably faster than iterative methods. The purpose of this survey paper is to provide brief summaries and a list of references for methods which can be used to directly solve the finite difference equations. Some of these methods can be applied to problems in more general domains. However, the extensions generally include only simple rectilinear regions, such as L-shaped or T-shaped domains. This is basically due to the fact that the direct methods require a great degree of regularity in the block structure of the matrix equation. In our discussion, we will indicate whether the methods are easily adaptable to more general regions, and to more general elliptic partial differential equations.
218 citations
TL;DR: A survey of the main methods for numerical evaluation of multiple integrals can be found in this article, where the Monte Carlo method and its generalizations are discussed, as well as number-theoretical methods, based essentially on the ideas of Diophantine approximation and equidistribution modulo 1; functional analysis approach, in which the quadrature error is regarded as a linear functional and one attempts to minimize its norm.
Abstract: This paper is an expository survey of the main methods that have been developed for numerical evaluation of multiple integrals. Among the approaches discussed are: the Monte Carlo method and its generalizations; number-theoretical methods, based essentially on the ideas of Diophantine approximation and equidistribution modulo 1; the functional analysis approach, in which the quadrature error is regarded as a linear functional and one attempts to minimize its norm; and the classical approach of designing formulas to be exact for polynomials of high degree while using as few values of the integrand as possible. Most of the research in this field is quite recent.
218 citations
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TL;DR: In this paper, a method for solving the m-center problem by solving a finite series of minimum set covering problems is presented. But the problem is not solved in this paper.
Abstract: An m-center set of a graph is any set of m points, belonging either to the edges or vertices, that minimizes the maximum distance from a vertex to its nearest m-center. This paper presents a method for solving the m-center problem by solving a finite series of minimum set covering problems.
191 citations
TL;DR: In this paper, it was shown that the duality of geometric programs is not affected by a "duality gap" and that the maximum of the dual geometric program is equal to the minimum of the primal geometric program.
Abstract: A geometric program concerns minimizing a function subject to constraint functions, all functions being of posynomial form. In this paper the posynomial functions are condensed to monomial form by the use of the inequality reducing a weighted arithmetic mean to a weighted geometric mean. The geometric mean is a monomial and by a logarithmic transformation it becomes a linear function. This observation shows that the condensed program is equivalent to a linear program. Moreover by suitable choice of the weights it is found that the minimum of the condensed program is the same as the minimum of the original programs. This fact together with the duality theorem of linear programming proves that the maximum of the dual geometric program is equal to the minimum of the primal geometric program. With this result as a basis a new approach to the duality properties of geometric programs is carried through. In particular it is shown that a “duality gap” cannot occur in geometric programming.
109 citations
TL;DR: In this article, the authors introduce and relate the more successful numerical methods for calculating the greatest value of a given real function (F(x_1,x_2, \ldots,x_n ) ).
Abstract: This paper is intended to introduce and relate the more successful numerical methods for calculating the greatest value of a given real function $F(x_1 ,x_2 , \ldots ,x_n )$. It also indicates some directions for research.
103 citations
TL;DR: Wei and Kendall as mentioned in this paper showed that the Wei-Kendall method provides a natural measure of how evenly the participants are matched, and they also showed that a parameter appearing in the Wei Kendall method can be used to measure how evenly players are matched.
Abstract: played (i.e., there are no ties). The outcome of the tournament may be represented by an n x n tournament matrix T where lij = 1 if player i defeats player j (symbolically, i -+ j) and zero otherwise. An n x n matrix P with nonnegative entries is a generalized tournament matrix if P + ptr = J I, where J denotes the matrix of l's and I the identity matrix. We interpret the entry Pij as the a priori probability that player i defeats player j when they meet (we assume Pii = 0 since no one plays against himself). Suppose the association organizing the tournament wishes to devise an equitable system of handicaps so as to neutralize the advantage strong players have over weak players. In ? 3 we discuss two general methods for doing this making use of various properties of generalized tournament matrices developed in ? 2. In the first method, each player bets a certain amount of his own money on each game he plays; in the second method the association awards a certain amount to the winner of each game. The bets and awards for each game are to depend on the two players involved. The association is to determine the amounts of the bets or awards in advance so that every player has the same expected net gain. There will be many fair systems in general and we consider the existence of systems satisfying various additional conditions. There are many ways in which a generalized tournament matrix P might arise; Pij could represent the frequency with which team i has beaten team j in some atheletic competition or Pi, might represent the frequency with which certain consumers prefer item i to itemj in a paired comparison test. One important problem is to determine some reasonable ranking of the n objects and, if possible, to obtain some quantitative measure of their relative merit. Several ranking methods have been proposed (see David [5]). The most obvious method is simply to rank the participants according to the number of games they have won. One well-known method is due to Zermelo, Bradley, Terry and Ford and another to Wei and Kendall. We obtain a new rationale for their methods as special cases of our betting and awards systems. Furthermore, in ? 3.3 we show that a parameter appearing in the Wei-Kendall method provides a natural measure of how evenly the participants are matched. Landau [12] has shown when there exists an ordinary tournament matrix with prescribed row sums; in ? 3.4 we obtain an analogous result for generalized tournament matrices. Some of the results in this paper were announced at the IFIP '68 Congress [161; some of these results were also obtained independently by Daniels [41.
92 citations
TL;DR: A survey of the current widely used methods for generating rational or polynomial approximations to continuous functions is given in this article, with emphasis on functions of one variable and on the Chebyshev and $L_1 $-norms.
Abstract: This paper surveys current widely used methods for generating rational or polynomial approximations to continuous functions. Analytic methods and numerical algorithms are discussed with emphasis on functions of one variable and on the Chebyshev and $L_1 $-norms. A previously unpublished algorithm, the method of artificial poles for handling nearly degenerate cases of rational Chebyshev approximation, is also presented.
59 citations
TL;DR: In this article, a new formalism, termed the resolution space, is presented within which the theory of causal systems may be unified and extended, which allows the various aspects of network and system theory which are dependent on the time parameter to be studied in operator theoretic context without the detailed structure of a function space.
Abstract: A new formalism, termed the resolution space, is presented within which the theory of causal systems may be unified and extended. The resulting formalism, which is defined as a Hilbert space together with a resolution of the identity, readily includes the commonly encountered function and sequence space causality concepts yet is sufficiently straightforward to allow the various aspects of network and system theory which are dependent on the time parameter to be studied in an operator theoretic context without the detailed structure of a function space. Specific results include additive and multiplicative decomposition theorems for causal operators which naturally extend the “realizable part” and “spectral” decompositions of classical system theory and an integral representation theorem for linear operators on a resolution space. The general theory is illustrated with a number of examples concerning passive “networks”, those including an operator theoretic approach to the passive synthesis problem over an arbitrary resolution space.
TL;DR: In this paper, the Moore-Penrose inverse of A is defined as At = A*TA*, where Te A*AA*{1}. Proofs for At = AtAAt = At(AAt)A*TA*A(AtA*A*At) = AtAtAAt.
Abstract: (2) AAtPR(A) = v (3) AtPR(A) At = 0 is the Moore-Penrose inverse of A [12]. An equivalent definition is given in [10]. As is well known, every A has one and only one At. 2. Results. For a given matrix B we denote by B{ 1 } the set B{ 1 } {X:BXB = B} . THEOREM. For any matrix A, (4) At = A*TA*, where Te A*AA*{1}. Proof. At = AtAAt = At(AAt)*A(AtA)*At = At(At)*A*AA*(At)*At = At(At)*A*AA*TA*AA*(At)*At = At(AAt)*AA*TA*A(AtA)*At = (AtA)A*TA*(AAt) = A*TA*. All these equalities follow directly from (1), (2) and (3). COROLLARY 1. Let A A(l A12) A2 1 A22 where A11 is a square nonsingular matrix such that rank Al = rank A. Then
TL;DR: In this paper, a discussion of regular differential operators and their inverses or generalized inverse is given, which is the analogue for differential operators of the Moore-Penrose generalized inverse for matrices.
Abstract: This paper contains a discussion of regular differential operators and their inverses or generalized inverses. Several examples of generalized inverses are given. The examples are selected to illustrate various properties of the generalized inverse and also to illustrate various noninvertible cases for differential operators. The generalized inverse used is the analogue for differential operators of the Moore–Penrose generalized inverse for matrices.
TL;DR: Asymptotic behavior of polynomial root locus in presence/absence of time lag is discussed in this paper, along with procedures for constructing root-locus diagrams.
Abstract: Asymptotic behavior of polynomial root locus in presence/absence of time lag, outlining procedures for constructing root locus diagrams
TL;DR: A survey of methods used in the solution of such inverse problems in a number of areas of physics can be found in this article, including particle scattering, electromagnetic and acoustic scattering and diffraction, quantum optical correlation theory and plasma physics.
Abstract: In many areas of physics one calculates quantities of experimental or applicational interest from the solutions of differential equations which contain fundamental functions such as potential energies, or dispersive indices of refraction. The reconstruction of these basic functions, or of some of their properties (e.g., moments) from the more or less directly measurable properties of the solutions, constitutes the general class of inverse problems. The paper surveys the methods used in the solution of such inverse problems in a number of areas of physics. These include particle scattering, electromagnetic and acoustic scattering and diffraction, quantum optical correlation theory, and plasma physics.
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TL;DR: In this paper, a graph G = {N, E} is taken to be a finite set of nodes N together with a set of distinct edges E which are unordered pairs of distinct nodes, and a matching M of the graph is a subset of the edges E with the property that no two edges of M are incident at a node.
Abstract: Introduction. A graph G = {N, E} is taken to be a finite set of nodes N together with a set of distinct edges E which are unordered pairs of distinct nodes. A matching M of the graph is a subset of the edges E with the property that no two edges of M are incident at a node. A matching M is perfect if every node is incident to an edge of M. For any finite set X, let IXI denote its cardinality. If S c N let G(N S) be the subgraph of G consisting of nodes N S (nodes of N but not S) and all edges of E which only join nodes of N S. For any subgraph H of G, let o(H) be the number of connected components of H each of which consists of an odd number of nodes. If, for some S c N, o(G(N S)) > ISI, we say that the set S is an imperfectable set of nodes of G.
TL;DR: In this article, it was shown that the Moore-Penrose generalized inverse of a matrix is sufficient and necessary for a linear system with singular coefficient matrix to be convergent. But this is not the case for linear systems with nonsingular coefficient matrix.
Abstract: It is well known that the necessary and sufficient condition for a matrix to be convergent is that all of its eigenvalues in magnitude be less than unity. For linear systems with nonsingular coefficient matrix the convergence of the iterations is equivalent to the convergence of the so-called iteration matrix associated with the scheme. However, this is not the case for a linear system with singular coefficient matrix. In a paper by H. B. Keller [2] a condition on the iteration matrix is determined when the system is singular. Further, this condition is shown to be necessary and sufficient for the convergence of the iterates to a solution if it has one. In [2], however, the author considers a singular linear system with a coefficient matrix of order n. The popular concept of the Moore-Penrose generalized inverse of a matrix [3], [4] now makes it possible to extend the results of the ? 2 of [2] to include systems with rectangular coefficient matrix. The purpose of this note is to give a kind of decomposition of a coefficient matrix which enables us to carry out the aforementioned extension to rectangular systems. The development here parallels that of [2] for a square matrix. Consider a general system of equations
TL;DR: Several familiar special functions possess a hidden permutation symmetry which accounts for some of their transformation properties, e.g., the transformation of the elliptic integral of the integral of K(k) as discussed by the authors.
Abstract: Several familiar special functions possess a hidden permutation symmetry which accounts for some of their transformation properties. For example the transformation of the elliptic integral $K(k)$ i...