**The Synthesis of Algorithmic Systems**

Alan J. Perlis

Pages: 1-9

DOI: 10.1145/321371.321372

**Sieving Procedures on a Digital Computer**

Marvin C. Wunderlich

Pages: 10-19

DOI: 10.1145/321371.321373

Many sequences are most efficiently generated on a digital computer with a sieving procedure in which one represents in the main memory of the machine a set of elements known to contain the desired sequence and then systematically sieves out...

**Statistical Discrimination of the Synonymy/Antonymy Relationship Between Words**

P. A. W. Lewis, P. B. Baxendale, J. L. Bennett

Pages: 20-44

DOI: 10.1145/321371.321374

A basic hypothesis is stated about the contextual and co-occurrence properties of synonymous words. On the basis of this hypothesis, several statistics are derived for use in discriminating between pairs of words which are synonymous and pairs...

**A Computer Program for the Nonnumerical Testing and Reduction of Sets of Algebraic Partial Differential Equations**

Carl H. Brans

Pages: 45-62

DOI: 10.1145/321371.321375

A program is described which enables a digital computer to perform the formal algebraic manipulations and differentiations required to test a set of algebraic partial differential equations for consistency, and, if the set is consistent, to...

**Computable Error Bounds for Direct Solution of Linear Equations**

Bruce A. Chartres, James C. Geuder

Pages: 63-71

DOI: 10.1145/321371.321376

An error analysis of direct methods (i.e., Gaussian elimination or triangular factorization) of solving simultaneous linear algebraic equations is performed in the backward mode, in which the computational errors are expressed as perturbations...

**A Modification of Davidon's Minimization Method to Accept Difference Approximations of Derivatives**

G. W. Stewart, III

Pages: 72-83

DOI: 10.1145/321371.321377

A modification of Davidon's method for the unconstrained minimization of a function of several variables is proposed in which the gradient vector is approximated by differences. The step sizes for the differencing are calculated from information...

**A Multistep Generalization of Runge-Kutta Methods With Four or Five Stages**

John C. Butcher

Pages: 84-99

DOI: 10.1145/321371.321378

To obtain high-order integration methods for ordinary differential equatic which combine to some extent the advantage of Runge-Kutta methods on one hand and line multistep methods on the other, the use of “modified multistep” or...

**Fourier Analysis of Uniform Random Number Generators**

R. R. Coveyou, R. D. Macpherson

Pages: 100-119

DOI: 10.1145/321371.321379

A method of analysis of uniform random number generators is developed, applicable to almost all practical methods of generation. The method is that of Fourier analysis of the output sequences of such generators. With this tool it is possible to...

**Binomial-Weighted Orthogonal Polynomials**

Tzay Y. Young

Pages: 120-127

DOI: 10.1145/321371.321380

This paper discusses a set of polynomials, {&phgr;r(s)}, orthogonal over a discrete range, with binomial distribution, b(s; n,...

**Subresultants and Reduced Polynomial Remainder Sequences**

George E. Collins

Pages: 128-142

DOI: 10.1145/321371.321381

Let @@@@ be an integral domain, P(@@@@) the integral domain of polynomials over @@@@. Let P, Q ∈ P(@@@@) with m @@@@ deg (P) ≥ n = deg...

**Modified Multistep Methods Based on a Nonpolynomial Interpolant**

Brian Shaw

Pages: 143-154

DOI: 10.1145/321371.321382

A new class of multistep formulas with variable coefficients is derived for the numerical integration of the ordinary differential equation y′ = f(x, y) whose theoretical...

**Multistep Methods With Modified Predictors and Correctors**

J. J. Kohfeld, G. T. Thompson

Pages: 155-166

DOI: 10.1145/321371.321383

In April, 1964, Gragg and Stetter published a set of numerical methods for solving y′ = f(x, y) which rely on some very accurate correctors, using a “nonstep”...

**A Remark on Post Normal Systems**

Ann Yasuhara

Pages: 167-171

DOI: 10.1145/321371.321384

Let N(T) be the normal system of Post which corresponds to the Thue system, T, as in Martin Davis, Computability and Unsolvability (McGraw-Hill, New York, 1958), pp. 98-100. It...

**Stack automata and compiling**

Seymour Ginsburg, Sheila A. Greibach, Michael A. Harrison

Pages: 172-201

DOI: 10.1145/321371.321385

Compilation consists of two parts, recognition and translation. A mathematical model is presented which embodies salient features of many modern compiling techniques. The model, called the stack automaton, has the desirable feature of being...