Research and Advances

It is often desired to solve eigenvalue problems of the type (A - &lgr;1)C = 0 or (A - &lgr;B)C = 0 repeatedly for similar values of the matrix elements Aij, where A and B are Hermitean or real symmetric matrices. Among the various methods to find all eigenvalues and eigenvectors, Jacobi's method of two-dimensional rotations [1] has been very popular for its numerical stability, although it is comparatively time-consuming. The purpose of this note is to show how existing subroutines can be used to reduce substantially the computing time, if approximate eigenvectors are known from the previous solution of a similar problem.

This paper describes a new algorithm for the selection of the pivot row in matrix inversion when using the product form of the inverse. This algorithm has been developed for linear programming codes; however, it would be valuable for the inversion of any non-dense matrix. The procedures described in this paper have been thoroughly tested and have been in operation on the Esso Research and Engineering IBM 7090 computer for nine months. Substantial computer cost savings have been realized because of this procedure.

If one has a set of observables (z1, ··· , zm) which are bound in a relation with certain parameters (a1, ··· , an) by an equation &zgr;(z1, ··· , a1, ···) = 0, one frequently has the problem of determining a set of values of the ai which minimizes the sum of squares of differences between observed and calculated values of a distinguished observable, say zm. If the solution of the above equation for zm, zm = &eegr;(z1, ··· ; a1, ···) gives rise to a function &eegr; which is nonlinear in the ai, then one may rely on a version of Gaussian regression [1, 2] for an iteration scheme that converges to a minimizing set of values. It is shown here that this same minimization technique may be used for the solution of simultaneous (not necessarily linear) equations.

The programming of a proof procedure is discussed in connection with trial runs and possible improvements.

Automatic computation methods were first developed and applied to the problem of radiation therapy treatment planning by the Physics staff at Memorial Hospital and Sloan-Kettering Institute in 1954 and reported in 1955 [1]. The field of radiation from a single port was stored as a matrix in a library of punched cards, and a sorter and accounting machine were used to combine various fields for rotation, cycling and multi-port therapy. This system was in continuous routine use from then until 1961, when the equipment was replaced by a Bendix G15-D digital computer. Subsequent work by Sterling [2] followed essentially the same method of describing the radiation field as used by the Physics staff at Memorial Hospital [1], except that more powerful equipment has been used. An analytic expression for the dose distribution produced by rotation had been previously applied successfully in 1951 to treatment planning with high-energy X-rays [3].

In this note we propose a method of generating stationary noise with a prescribed auto-covariance function by digital methods. The need for such a technique often arises in testing the performance of data processing and engineering systems, where inputs corrupted with correlated noise (of a known form) are required. The technique is quite simple and produces strict-sense stationary noise which agrees approximately with R(&tgr;), the prescribed auto-covariance function (acf), over an interval [- T0, T0]. The method consists of approximating the spectral density by a periodic process with spectral lines, and then synthesizing the periodic noise with random phases and appropriate amplitudes. In order to simplify discussion of the statistical properties of the noise generated, the technique is first presented in terms of exact harmonic analysis. In practice, discrete harmonic analysis as presented in the third section is used.

Record linkage in the updating of files is accomplished in many establishments through the use of a preassigned number, such as payroll number, customer number, or social security number. In vital and health records, however, a unique number is generally not preassigned to an individual for purposes of reporting services received to the health department. In order to determine whether different physician reports refer to the same individual, name and other identification must be compared. This is a laborious operation which is subject to various errors because of name misspellings, changes of name upon marriage, and other problems. We are interested in the maintenance of a psychiatric case register in Maryland, where many of the reports from over a hundred psychiatric agencies refer to the same patient. These records must be linked in order to provide unduplicated counts of individuals under care and longitudinal records of psychiatric history. An earlier paper [1] describes our general procedures for register maintenance by use of a digital computer (Honeywell 800). Here we present in more detail our initial procedures for the person-matching process in order to elicit comments and suggestions from persons who have had experience in matching.

The logic of English grammar is being investigated by a process of approximative logical synthesis. Beginning with a kernel language which is an alphabetically spelled form of symbolic logic with English-like vocabulary, English-like extensions are constructed simulating selected features of English grammar. For each extended language, an algorithm is given which would permit a computer to make a left-to-right, word-by-word scan of an input sequence, to transform it by stages (as long as the sequence is found to be a well-formed formula of the extended language) into a sentence in the logical grammar of the kernel (or into a logical symbolization if desired). The algorithm thus constitutes simultaneously (1) a recognition grammar of the extended language, i.e., not just formation rules or productions, but a decision method for sentencehood, and (2) a portion of its transformation rules (defining a consequence relation). To date, languages have been constructed simulating the English determiners, (e.g., “every”, “any”, “no”, “a”) and the system of groupers (“either”, “both”, “if”) with precedence-ordered connectives (e.g., “and”, “and furthermore”, “or”, “or else”) by which English achieves the effect of parentheses. Work soon to be reported extends the analysis to relative, general and indefinite pronouns. It is expected that the algorithms will be embodied in computer programs within the year for purposes of comparison with results of human parsers.