Showing papers in "Bit Numerical Mathematics in 1962"
TL;DR: For difference equations with constant coefficients necessary and sufficient algebraic stability conditions are given for the stability definitions used by G. Forsythe and W. D. Richtmyer as discussed by the authors.
Abstract: For difference equations with constant coefficients necessary and sufficient algebraic stability conditions are given for the stability definitions used by G. Forsythe and W. Wasow (A) and P. D. Lax and R. D. Richtmyer (B). The application of these conditions for difference equations with variable coefficients is considered and it is shown that the stability condition of definitionA is not sufficient for stability. The same is true with respect to the definitionB if the difference equations are not parabolic and do not approximate first order systems. Therefore another stability definition is proposed and a number of properties are discussed.
150 citations
TL;DR: In this article, a generalization of the successive overrelaxation method for the solution of complex linear systems is studied, and an optimum complex over-Relaxation factor is determined and some numerical experiments verify the theory.
Abstract: A very natural generalization of the successive overrelaxation method for the solution of complex linear systems is studied. An optimum complex overrelaxation factor is determined and some numerical experiments verify the theory.
38 citations
TL;DR: The application of the Richardson second order iterative method to positive definite, symmetric linear equations is investigated and absolute and statistical bounds for the round-off error are derived.
Abstract: The application of the Richardson second order iterative method to positive definite, symmetric linear equations is investigated. Absolute and statistical bounds for the round-off error are derived. The statistical theory agrees well with numerical experiments, until the accumulated round-off error becomes of the order of magnitude of the error in the computed solution. After this point the statistical dependence between the local round-off errors makes the observed variances larger than the theoretical variances.
17 citations
TL;DR: To what extent this can be done is discussed in the paper and suitable methods are also suggested.
Abstract: A vast amount of tedious labour is nowadays spent at detecting and correcting errors in text. Such errors are normal occurrence in text written by humans, but also often in text produced from a transmission system. The birth of the computers provides possibilities for leaving a large part of that labour to machines. To what extent this can be done is discussed in the paper and suitable methods are also suggested. The discussion is partly conducted in terms of information theory.
15 citations
TL;DR: It has been found that a numerical approximation of a ship-designer's spline results in a smooth interpolating function and an algorithm which computes the function coefficients is presented in Algol.
Abstract: It has been found that a numerical approximation of a ship-designer's spline results in a smooth interpolating function. An algorithm which computes the function coefficients is presented in Algol. Methods to vary the stiffnes of the spline are included. A test of fairness is described and finally a more elaborate algorithm which leads to a closer approximation is outlined.
14 citations
IBM1
TL;DR: A method has been described for classifying a collection of items characterized by qualitative attributes, using mainly two concepts, typicality and similarity, which has been programmed for the IBM 1620 computer.
Abstract: A method has been described for classifying a collection of items characterized by qualitative attributes. The method uses mainly two concepts, typicality and similarity. The first of these has been derived by applying information-theoretical tools to the data. Typicality is a relation of one item to the whole collection. Similarity is a one-to-one relation between two items and depends on the number of common properties shared by them. The method has been programmed for the IBM 1620 computer.
13 citations
TL;DR: In this article, a complete system of ALGOLGOL procedures for arithmetic operations on complex numbers is presented, which enable arithmetic operations to be carried out upon complex numbers, such as ln, exp, sin, etc.
Abstract: This paper contains a complete system of ALGOL procedures which enable arithmetic operations to be carried out upon complex numbers. Further procedures for carrying out the evaluation of certain elementary functions (e.g. ln, exp, sin, ...) of a complex variable are given. Application of these procedures is then illustrated by their use in the computation of the confluent hypergeometric function and the Weber parabolic cylinder function. Procedures relating to the application of the e-algorithm to series of complex terms are described. Two integrated series of procedures, relating to Stieltjes typeS-fractions and to corresponding continued fractions respectively, are given. Complete programmes, which illustrate the use of these procedures, may be used for the computation of the incompleteβ-function, the incompleteΓ-function (of arguments of large and small modulus) and the Weber function.
13 citations
TL;DR: In this paper, a method for finding the best rational approximation p(x)/q(x) in the least-squares sense of a function known at discrete points of an interval was developed.
Abstract: A method is developed for finding the “best” rational approximationp(x)/q(x) in the least-squares sense of a functionf(x) known at discrete points of an interval. Especially for functions with a steep tangent the method yields far better results than polynomial approximation. The scheme can easily be extended to functions known at all points of an interval.
11 citations
TL;DR: In this paper, rational expressions for functions with exponential asymptotic behavior are approximated by rational expressions, where the numerators are obtained from the nodes of the functions, and the denominators are powers of polynomials, whose coefficients are obtained by expansion in orthogonal polynoms.
Abstract: Numerically given functions with exponential asymptotic behavior are approximated by rational expressions. The numerators are obtained from the nodes of the functions, and the denominators are powers of polynomials, whose coefficients are obtained by expansion in orthogonal polynomials. Different weights are discussed for uniform absolute or relative errors.
9 citations
TL;DR: In this paper, a new method of generating permutations in lexicographical order is developed, which is superior to the known method of generation by addition, and a simple method for generating combinations is also described.
Abstract: In this paper a new method of generating permutations in lexicographical order is developed. Analysis shows that the method is superior to the known method of generation by addition. A simple method of generating combinations is also described.
7 citations
TL;DR: It is shown that simple serial-access memories such as magnetic tapes, when used together with a fair amount of ferrite-core storage, constitute a most efficient tool for the kind of data-handling involved.
Abstract: The computation of parts requirements or the timing of these requirements constitute important parts of the data processing in connection with production control. This kind of data processing is much more complicated than the simple updating of files. It is therefore often regarded as a problem for which random access memories are necessary. It is shown in this paper, however, that simple serial-access memories such as magnetic tapes, when used together with a fair amount of ferrite-core storage, constitute a most efficient tool for the kind of data-handling involved. The main theme of the paper, however, is to show how matrix algebra can be used as a very convenient tool for treating problems in production control both for theoretical discussions and practical data processing. In order to make the paper readable also for people who are not familiar with matrices, this concept is introduced on the basis of simple elements of the problem of computation of parts requirements. Different machine procedures for the matrix handling are discussed.
TL;DR: For comparison of performance and prices of certain computers, artificial test problems were designed by a Swedish governmental committee and (theoretically) solved by the manufacturers.
Abstract: For comparison of performance and prices of certain computers, artificial test problems were designed by a Swedish governmental committee and (theoretically) solved by the manufacturers. Tested were Bull Gamma 30—RCA 301, IBM 1401, ICT 1301, and Saab D21. Winner was the Swedish Saab D21, loser IBM 1401, with wide margins, but the committee selected IBM 1401, mainly—it is assumed—because IBM 1401 was considered more thoroughly tested; the project (for 20 computers) was considered extremely important, and the committee apparently did not want to accept any risks for delay.—The decision of the committee has been criticized.
TL;DR: In this article, a simple algorithm is described for inverting the operatorDxDy (Dx and Dy here and subsequently denote partial differentiation with respect to tox andy respectively) which occurs in the iterative solution of the equation.
Abstract: A simple algorithm is described for inverting the operatorDxDy (Dx andDy here and subsequently denote partial differentiation with respect tox andy respectively) which occurs in the iterative solution of the equationDxDyf (x, y)=g (x, y, f, Dxf, Dx2f,DxDyf, Dy2f) when boundary values off(x,y) are given along the sides of the rectangle in thexy-plane whose corners are at the points (a,b); (a+(n+1)k,b); (a+(n+1)k,b+(n+1)h); (a,b+(n+1)h).
TL;DR: In this article, the search limits on divisorsq of 2p − 1, even when no divisor up to this limit was found, were published, and this list, therefore, should avoid time consuming double work.
Abstract: On the urgent request of several coenthusiasts around the globe in the field of Factorization of Mersenne Numbers, the author publishes here for the first time the search limits on divisorsq of 2p − 1, even when no divisor up to this limit was found. This list, therefore, should avoid time consuming double work.
TL;DR: Some points of view concerning the need felt by many computer-people for more intensive education and research activities in administrative data processing and the responsibility of different institutions to certify these needs are contained.
Abstract: This paper contains some points of view concerning the need felt by many computer-people for more intensive education and research activities in administrative data processing. After a short out-line of the main problems in administrative data processing we will try to indicate some topics where research is most needed. Then follows a survey of the need of knowledge specified for different categories of persons working in this field. Finally, we will draw some conclusions regarding the responsibility of different institutions to certify these needs.
TL;DR: Some critical remarks are made in connection with the use of symbols in ALGOL in order to improve the quality of the text.
Abstract: Some critical remarks are made in connection with the use of symbols in ALGOL. The underlined words and the for-statements are especially considered.
TL;DR: This is a report on the Swedish experiment with the MEDSAC method used for diagnostic purposes in the outpatient departments of two general hospitals and in health screening.
Abstract: This is a report on the Swedish experiment with the MEDSAC method used for diagnostic purposes in the outpatient departments of two general hospitals and in health screening.
TL;DR: In the history of the art of measurement, computer type instruments have been in existence for a very short time as discussed by the authors and they can still be considered by-products of the computer technology, but there is no doubt that they will in a future have the same importance for the human welfare and the evolution of sciences as electronic computers have to-day.
Abstract: In the history of the art of measurement, computer type instruments have been in existence for a very short time. They can still be considered by-products of the computer technology, but there is no doubt that computer type instruments will in a future have the same importance for the human welfare and the evolution of sciences as electronic computers have to-day. In this article two examples are described, related to paper industry and nuclear physics research.
TL;DR: A procedure for constructing a planning and scheduling network, where each activity to be scheduled is represented uniquely by a line, where no events need be defined manually.
Abstract: A procedure for constructing a planning and scheduling network is given, where each activity to be scheduled is represented uniquely by a line. No events need be defined manually. Input is the set of activities with a list of preceding (but not necessarily immediately preceding) activities for each of them. Some comparisons with events network of Pert type are made.