April 1961 - Vol. 4 No. 4

April 1961 issue cover image

Features

Research and Advances

Eigenvalues of a symmetric 3 × 3 matrix

Recently, in order to find the principal moments of inertia of a large number of rigid bodies, it was necessary to compute the eigenvalues of many real, symmetric 3 × 3 matrices. The available eigenvalue subroutines seemed rather heavy weapons to turn upon this little problem, so an explicit solution was developed. The resulting expressions are remarkably simple and neat, hence this note.
Research and Advances

Bessel functions of integral order and complex argument

The FORTRAN II source language [1, 2] places rather severe restrictions on the form a subscript may take, primarily because of the manner in which indices are incremented in iterative loops. In the process of constructing a compiler for a medium-sized (8008-word memory) computer which will accept the FORTRAN II source language, it became clear that the “recursive address calculation” scheme, as used in the FORTRAN compilers to minimize object-program running time, was probably not the best one to use. This system, described in some detail by Samelson and Bauer [3], requires that the subscript expression be a linear function of the subscripting variable. The alternative, which requires complete evaluation of the “storage mapping function”, is usually rejected because of the time required for the object program to perform the necessary address calculation.
Research and Advances

Programmed error correction on a decimal computer

In a previous paper [1], B. Dimsdale and I reported on the use of programmed Hamming codes for error correction on a 7090. The paper generated much interest, but several readers remarked that they could not use the technique on their decimal machines since they could not manipulate the bit structure of the characters. This paper presents a modification of Hamming's technique to adapt it to such machines. It generalizes to any number base for which addition is built into the machine, and so could be used for alphabetic error correction on a machine where such operations as (A + B, literally) or (R+2, literally) or (H + $, literally) are unequivocally defined. The technique will be described for single strings of characters, but may be generalized to parallel techniques if parallel, no-carry addition is available on the machine in question.
Research and Advances

Further survey of punched card codes

The valuable “Survey of Punched Card Codes” prepared by Smith and Williams (Comm. ACM 3, Dec. 1960, 638) unfortunately omits the card codes of European equipment, other than IBM. These are presented in the table on page 181. This information has been extracted from a Ferranti publication, “Collected Information on Punched Card Codes” (List CS 266) and has been set out in much the same way as the table by Smith and Williams.
Research and Advances

Some numerical experiments using Newton’s method for nonlinear parabolic and elliptic boundary-value problems

Using a generalization of Newton's method, a non-linear parabolic equation of the form ut - uxx = g(u), and a non-linear elliptic equation uxx + uyy = eu, are solved numerically. Comparison of these results with results obtained using the Picard iteration procedure show that in many cases the quasilinearization method offers substantial advantages in both time and accuracy.
Research and Advances

Laviathan studies

The Leviathan studies are investigations into how people operate in large social organizations. Examples of such groups are a large military command, a governmental agency like the U. S. Bureau of Internal Revenue, or an industrial combination like an international oil corporation.
Research and Advances

Advanced computers

The objective is to investigate significant developments in advanced computers and to explore the relationship between programming and new concepts of computer organization. Two specific studies are in progress: Data Sequencing and Vertical Data Processing.

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