December 1962 - Vol. 5 No. 12

It is unfortunate that almost all of the presently used algebraic languages do not provide the capability of linear algebra. Operations such as the inner product of vectors, the product of two matrices, and the multiplication of a matrix by a scaler must inevitably be written out in detail in terms of the individual components. The reasons usually given for avoiding linear algebra in these languages are (1) the difficulties which would arise in scanning linear algebraic expressions, and (2) the uncertainty involved as to the amount of temporary storage needed during the evaluation of linear algebraic expressions when the program is executed. The purpose of this paper is to show how these two types of difficulties can be overcome. Although suggestions have been made for even further increasing the general capability of ALGOL such as including the ability to form a matrix from a collection of vectors, we shall be content here to consider the ordinary operations of linear algebra. Even if this much becomes available in algebraic languages, considerable progress will have been made. The following remarks constitute a suggestion for the addition to ALGOL of linear algebraic expressions.

Currently a great deal of time and effort is being spent on the development of bigger and better compiler languages, multiprogram executive systems, etc. Since the implementation of of new methods and procedures is not instantaneous, but rather occurs by an evolutionary process, we should be concerned also with the problem of maintaining, improving and incorporating new ideas into existing systems. It is with this somewhat neglected area that the author is interested. A method employing a decision matrix is presented for the handling of a standard systems programming problem, that of providing a data input routine.

Scientific users of small-scale variable-word-length computers, such as the IBM 1401, may frequently have the occasion to use fixed-word-length arrays. For instance, it is common practice to store matrices row-wise in linear arrays. A linear array whose elements are addresses is a common storage allocation scheme for handling pushdown lists.

The suggested method of manipulating alphabetic characters in FORTRAN which appeared in this section of the August, 1962, issue of the Communications of the ACM has been incorporated into the IBM 1620 FORTRAN II system.

This paper points out a variety of ways computer systems used in business and industry can be involved in legal entanglements and suggests that computer specialists have a responsibility to call for assistance in forestalling or minimizing those entanglements during the planning stage. Techniques are suggested for making legal clearance effective with the least burden on the new technology and for achieving a favorable legal climate for it generally. Computer specialists also are alerted to potential opportunities to interpret to lawyers the technical aspects of computer systems involved in legal situations.

The common techniques for solving two-point boundary value problems can be classified as either "shooting" or "finite difference" methods. Central to a shooting method is the ability to integrate the differential equations as an initial value problem with guesses for the unknown initial values. This ability is not required with a finite difference method, for the unknowns are considered to be the values of the true solution at a number of interior mesh points. Each method has its advantages and disadvantages. One serious shortcoming of shooting becomes apparent when, as happens altogether too often, the differential equations are so unstable that they "blow up" before the initial value problem can be completely integrated. This can occur even in the face of extremely accurate guesses for the initial values. Hence, shooting seems to offer no hope for some problems. A finite difference method does have a chance for it tends to keep a firm hold on the entire solution at once. The purpose of this note is to point out a compromising procedure which endows shooting-type methods with this particular advantage of finite difference methods. For such problems, then, all hope need not be abandoned for shooting methods. This is desirable because shooting methods are generally faster than finite difference methods. The organization is as follows: I. The two-point boundary value problem is stated in quite general form. II. A particular shooting method is described which is designed to solve the problem in this form. III. The two-point boundary value problem is then restated in such a way that: (a) the restatement still falls within the general form, and (b) the shooting method now has a better chance of success when the equations are unstable.

The first computers operating in post-war Germany had been constructed within Germany. These were developed at the Max Plank Institute in Göttingen, with two additional university machines under construction at Darmstadt and Munich [1]. These machines did not attain any importance, however, in the German economy.