Research and Advances

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.

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.

A network of directed line segments free of circular elements is assumed. The lines are identified by their terminal nodes and the nodes are assumed to be numbered by a non-topological system. Given a list of these lines in numeric order, a simple technique can be used to create at high speed a list in topological order.

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.

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.