Showing papers in "International Journal of Computer Mathematics in 1968"
TL;DR: In this article, three approaches to the quantitative definition of information are presented: information-based, information-aware and information-neutral approaches to quantifying information in the context of information retrieval.
Abstract: (1968). Three approaches to the quantitative definition of information. International Journal of Computer Mathematics: Vol. 2, No. 1-4, pp. 157-168.
2,661 citations
IBM1
TL;DR: This paper is intended for computer programmers and other scientists who may be interested in linguistics and has no previous knowledge of linguistics, but some experience with logical or mathematical reasoning.
Abstract: This paper is intended for computer programmers and other scientists who may be interested in linguistics. We assume that the reader has no previous knowledge of linguistics, but some experience with logical or mathematical reasoning. *
29 citations
TL;DR: In this article, a method for converting Fredholm integral equations into systems of ordinary differential equations that can be readily solved by modern analog and digital computers is given, and a method is given for converting the integral equation into a system of differential equations.
Abstract: : A method is given for converting Fredholm integral equations into systems of ordinary differential equations that can be readily solved by modern analog and digital computers. (Author)
26 citations
TL;DR: The problem of computing N Chebyshev coefficients is considered when , and two methods are discussed.
Abstract: The problem of computing N Chebyshev coefficients is considered when . Two methods are discussed. The first method is related to the Fast Fourier Transform (FFT) and required a total number of operations proportional to N log2 N. The second method, although not as efficient as efficient as the FFT exemplified interesting properties of the discrete Chebyshev polynomials.
12 citations
TL;DR: The method of operation of some sparse matrix routines written for use on the Atlas Computer is briefly described and their application to some frame problems is discussed.
Abstract: Many matrix formulations of structural analyses involve matrices having a large proportion of zero elements. By storing only the non-zero elements of all matrices, machine store and time may be utilised effectively, making the simplest matrix formulations feasible as a method of computer solution even for fairly large frame problems. The method of operation of some sparse matrix routines written for use on the Atlas Computer is briefly described and their application to some frame problems is discussed. The term “searching factor” is introduced as a guide to the efficiency of sparse matrix routines.
11 citations
TL;DR: In this paper, the authors present the causes of instabilities which arise during the numerical solution of ordinary differential equations using the numerical integration routines presently available, one actually approximates the differential equation by a difference equation if the difference equation is of higher order than the original differential equation.
Abstract: This paper presents the causes of instabilities which arise during the numerical solution of ordinary differential equations Using the numerical integration routines presently available, one actually approximates the differential equation by a difference equation If the difference equation is of higher order than the original differential equation, the approximate solution contains extraneous solutions which are not at all related to the true solution It is the behavior of these extraneous solutions that one is concerned with in a stability analysis
11 citations
TL;DR: In this paper, a comparison of the errors committed when integrating the functions I x l a and (1-x)" over the interval I= [-I, 11] with Gaussian rules of various orders given in Tables 1 and 2 reveals significant differences in the error patterns.
Abstract: A comparison of the errors committed when integrating the functions I x l a and (1-x)" over the interval I= [-I, 11 with Gaussian rules of various orders given in Tables 1 and 2 reveals significant differences in the error patterns. First of all, the errors in the latter case are much smaller than those in the former. This is true not only when comparing errors arising from using the same rule, but more strikingly, a rule such as GI, applied to I x 1' is less accurate than G, applied to (1 -x)" for values of ac as large as 2+. Furthermore, the errors in the former case oscillate in sign and do not even decrease monotonically in absolute value as the order of the integration rule increases. In the latter case, there is a clear pattern of convergence from the outset, with all errors of the same sign and monotonically decreasing with increasing order of the integration rule. How can one explain this? The standard error term for the m-point Gaussian integration rule contains the 2mth derivative of the integrand and is certainly not applicable here. Similarly, the method of Davis and Rabinowitz [3] is out since it requires analyticity of the integrand. The methods using approximation theory [4] or the Peano kernel [9] involve the maximum value of the derivative of highest order which is bounded in I and hence cannot distinguish between I x l a and (1 -x)". Hence the question remains as to why the results are so much better in one case than in the other. A possible explanation, together with a method for estimating the error in Gaussian integration of functions with branch points at the end points of the interval of integration, is given by the use of expansions in series of Chebyshev polynomials. A slightly different approach was taken by Chawla and Jain [2] who got related asymptotic error estimates in the Gaussian integration of functions with singularities on the real axis.
9 citations
TL;DR: The problem of minimizing the number of zero elements that become non-zero during the computation when a sparse symmetric matrix is reduced to a triple diagonal form, either by Givens' or Householder's method, is discussed.
Abstract: The problem of minimizing the number of zero elements that become non-zero during the computation when a sparse symmetric matrix is reduced to a triple diagonal form, either by Givens' or Householder's method, is discussed. Algorithms for minimizing the growth of such non-zero elements are given.
6 citations
IBM1
5 citations
TL;DR: A version of the Logic Theory Machine of Newell, Shaw and Simon was written in LISP 1.5 and provided an experimental model for form-map, a method of information storage and retrieval particularly suited for list structures.
Abstract: A version of the Logic Theory Machine of Newell, Shaw and Simon was written in LISP 1.5 on an IBM-7094. The program provides an experimental model for form-map, a method of information storage and retrieval particularly suited for list structures. The program was able to prove 17 out of 20 theorems from Principia Mathematica in 21.5 seconds.
5 citations
TL;DR: It is shown that any one-step method for the numerical solution of the initial value problem has asympotic error expansion, and an extrapolation theorem on accelerating the convergence of the numerical method is proved.
Abstract: Accelerating convergence of one step methods for numerical solution of ordinary differential equations, discussing various initial value problems
TL;DR: In this article, some results in differential approximation are presented. But they do not consider the problem of differential approximation in the context of computer games. International Journal of Computer Mathematics: Vol. 2, No. 1-4, pp. 231-245.
Abstract: (1968). Some results in differential approximation. International Journal of Computer Mathematics: Vol. 2, No. 1-4, pp. 231-245.
TL;DR: The problem of minimizing the number of zero elements that become non-zero during the computation, when a large sparse matrix is reduced to the Hessenberg (almost triangular) form by Gaussian similarity transformations, is discussed.
Abstract: The problem of minimizing the number of zero elements that become non-zero during the computation, when a large sparse matrix is reduced to the Hessenberg (almost triangular) form by Gaussian similarity transformations, is discussed. Algorithms for minimizing the growth of such non-zero elements are given.
TL;DR: In this article, an initial-value method for solving Fredholm integral equations on an infinite interval was proposed, which makes use of a generalized Ambarzumian integral equation in connection with an initial value problem to solve the Fredholm equation.
Abstract: : A new initial-value method is given for solving Fredholm integral equations on an infinite interval. Instead of solving the initial-value problem previously derived for the finite interval case and allowing the interval to tend to infinity, the method makes use of a generalized Ambarzumian integral equation in connection with an initial-value problem to solve the Fredholm equation. A numerical example from radiative transfer indicates that the results obtained by use of the method agree well with previously published results. (Author)
TL;DR: In this article, techniques for incremental plotting display of multiple regression lines, contours, and trajectories are described for incremental regression line, contour, and trajectory analysis, respectively.
Abstract: Techniques are described for incremental plotting display of multiple regression lines, contours, and trajectories.
TL;DR: An algorithm is suggested that takes advantage of the freedom of choice at the branching step in order to split the given problem into several smaller problems.
Abstract: After a statement of the general problem underlying Quine's methods of simplifying logical expressions, a few examples, and a survey of various approaches to the problem, attention is focussed on the question of how to branch when the straightforward simplication rules give no further progress. An algorithm is suggested that takes advantage of the freedom of choice at the branching step in order to split the given problem into several smaller problems. As the difficulty of the problem grows exponentially with its size, this results in a great saving of effort. Both the hand and computer versions of the algorithm are described since they differ appreciably. The pattern recognition example used to illustrate the paper is chosen as typical of the wide variety of practical questions in which the above general problem arises.
TL;DR: This semitutorial paper presents some new material by the author and some well-known material that may be applied to utility theory, automatic extracting, game playing, international relations, purchasing and selling, and the evaluation of personnel and computer programs.
Abstract: This semitutorial paper presents some new material by the author and some well-known material. A linear evaluation function is a function of the form CY where Y is a feature vector and where C is a coefficient (weight) vector. The author calls the following important problem the m,n-evaluation problem. Given a set { is preferred to Y 1} of m preferences in n-space, find a coefficient vector C such that, for as many preferences as possible, . This problem is important because a subject may be either unable or unwilling to divulge his evaluation function but may supply preferences, either implicitly by his behavior or explicitly. Finding an evaluation function may be applied to utility theory, automatic extracting, game playing, international relations, purchasing and selling, and the evaluation of personnel and computer programs The m, (n–1)-pattem problem consists of finding a hyperplane in (n–1)-space which approximately separates m pattern sample instances into two pattern classes. Once found, a linear ...
TL;DR: The paper describes transforma-tions of non-directed alpha-numeric networks: edges are separated and rearranged so as to form new networks, which lead to the formulation of algorithms, which have been programmed and solved on a ZUSEZ 22 digital computer.
Abstract: This paper describes the analysis of networks, using digital computers. If the nodes of the network are assigned alpha-numeric symbols, the network is said to be alpha-numerical. The paper describes transforma-tions of non-directed alpha-numeric networks: edges are separated and rearranged so as to form new networks. The results of these investiga-tions lead to the formulation of algorithms, which have been programmed and solved on a ZUSEZ 22 digital computer. Networks are stored as incidence matrices and transformed by matrix operations. Transforma-tions which are effected by separation and rejoining of at most two edges, are called elementary transformations. It will be shown that every struc-tural transformation can be represented as a logical consequence of elementary transformations. Our investigations permit the following ex-positions: Alpha-numerical networks can be interpreted as chemical structure formulae; transformations can be interpreted as chemical reactions.