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The article presents an algorithm for the numerical integration over a finite interval automatically. The algorithm's purpose is to automatically calculate the integral over a finite interval with relative error not exceeding a specified value. It utilizes a basic integration algorithm applied under the control of algorithms which invoke adaptive or nonadaptive subdivision of the range of integration. The subdivision processes will only normally be needed on extremely difficult integrals as the basic algorithm is sufficiently powerful.

The article presents the procedures for computing, smoothing and interpolating natural splines. Several mathematical formulae with mathematical and numerical analyses are cited. Problems indicating the application of smoothing and interpolating are discussed. The purpose of the procedure is to determine the coefficients in the representation of a natural spline of degrees in terms of a local basis. The procedure is based on algorithms. The parameters maxit should be a positive integer specifying the maximum number of iterations desired in computing the problem. Illustrations in applying such mathematical formulae are offered.

The article presents the procedures for computing the finite Fourier transform of a complex data vector which is based on the Cooley-Tukey algorithm. An overview of the complextransform and realtransform's procedures are offered. Either the complex Fourier transform or its inverse are the procedure computed by complextransform, while realtransform computes either the Fourier coefficients of a sequence of real data points or evaluates a fourier series. The topic also includes orthogonal transformation, permutations, and spectral analysis.

The FORMAC (FORmula MAnipulation Compiler) programming system provides a powerful tool for performing mathematical analysis. It is an extension of FORTRAN IV which permits the use of the computer to perform the tedious algebraic computations that arise in many different fields. Among the areas in which it has been successfully used are: differentiation of complicated expressions, expansion of truncated power series, solution of simultaneous equations with literal coefficients, nonlinear maximum likelihood estimation, tensor analysis, and generation of the coefficients of equations in Keplerian motion. These types of analysis—which arose in the solution of specific practical problems in physics, engineering, astronomy, statistics and astronautics—are discussed in the paper. In addition to its usage for specific problem solutions, FORMAC can also be used to automate the analysis phase in certain production programming. Several such applications are presented. [ABSTRACT FROM AUTHOR]

A large class of problems leading to digital computer processing can be formulated in terms of the numerical solution of systems of ordinary differential equations. Powerful methods are in existence for the solution of such systems. A good general purpose routine for the solution of such systems furnishes a powerful tool for processing many problems. This is true from the point of view of ease of programming, ease of debugging, and minimization of computer time. A number of examples are discussed in detail. [ABSTRACT FROM AUTHOR]

A method which finds simultaneously all the zeros of a polynomial, developed by H. Rutishauser, has been tested on a number of polynomials with real coefficients. This slowly converging method (the Quotient-Difference (Q-D) algorithm) provides starting values for a Newton or a Bairstow algorithm for more rapid convergence. Necessary and sufficient conditions for the existence of the Q-D scheme ore not completely known; however, failure may occur when zeros have equal, or nearly equal magnitudes. Success was achieved, in most of the cases tried, with the failures usually traceable to the equal magnitude difficulty. In some cases, computer roundoff may result in errors which spoil the scheme. Even if the Q-D algorithm does not give all the zeros, it will usually find a majority of them. [ABSTRACT FROM AUTHOR]

The article presents mathematical procedures using Legendre functions for cases involving arguments larger than one. The integer Legendre 1 procedure is said to generate the associated Legendre functions of the first kind. Integer Legendre 2 generates associated Legendre functions of the second kind to d significant digits. The integer Legendre 3 generates the Legendre functions of the second kind to d significant digits and stores in the array Q. Legendre 1 procedure evaluates the Legendre functions to d significant digits. Other procedures include Legendre 2, conical and toroidal.

The article discusses matrix transposition in place. Several algorithms have been used for solving the problem of transposing a rectangular matrix in place. A significantly speedier algorithm is described, based on a number of theoretical analysis, and experimentally compared with existing algorithms. The theory calls for the storing of a matrix in a vector in one of two ways, either rowwise or columnwise. The matrix is transposed by its conversion from one mode of storage to the other through permutation.

The article discusses the mathematical concept called arrow to precedence network transformation. Many of the application programs in the area of critical path scheduling and resource allocation are written for the precedence networking convention. The method of transforming arrow convention networks into precedence convention is required, because only few of these programs admit networks defined by arrow convention directly. The algorithm generates the required transformation by creating a list of followers for each non-dummy arrow. The logic used in the transformation can be used to create a list of predecessors if they are desirable.

This article presents procedures for solving of equations related to algorithm. Procedure determines the least-cost flow over an upper and lower bound capacitated flow network. Each directed network arc a is defined by nodes I[a] and J[a], has upper and lower flow bounds hi{a] and lo[a], and cost per unit of flow cost[a]. Costs and flow bounds may be any positive or negative integers. An upper flow bound must be greater than or equal to its corresponding lower flow bound for a feasible solution to exist. There may be any number of parallel arcs connecting any two nodes.

The article discusses four combinatorial algorithms. The different algorithms, by successive calls, produce a sequence of all combinatorial configurations belonging to the appropriate type, such as PERMU for permutations, COMBI for combinations of natural numbers, COMPOMIN for the composition of an integer, and COMPOMAX, which is the same as COMPOMIN but with maximum for each term. The four algorithms uses a single operation for producing a new configuration from the old one, in which PERMU is transformed by a single transposition of two adjacent elements, COMBI by replacing a single element x by a y, and COMPOMIN by changing the values of two adjacent terms.

The article discusses the mathematical concept called Transitivity Sets. Several mathematical formulae with mathematical and numerical analyses are cited. Problems indicating the application of Transitivity Sets are discussed. The algorithm can cause an infinite integer. It can manifest whenever the corrected point called the centroid of the remaining complex points is given, and every point on the line segment has functional values lower than the functional values at each of the remaining complex points.

The article comments on algorithms that offer symbolic expansion of algebraic expressions and prime number generator. The first algorithm expands arbitrarily parenthesized expressions into monomials and has direct distribution without intermediate expansion of lower level expressions. It has also been used as part of algebra programs in theoretical physics. The second algorithm has the intention of giving only the number of primes less than or equal to m. The algorithm has been tested for a large range of values including m =5, 10, 501 and 2000.

The article focuses on arithmetic over a finite field. An algorithm for the rational operations of arithmetic over a finite field of elements is presented. The detailed procedure of the algorithmic operation is described. For small values, it is suggested that multiplication tables and addition be generated by the algorithm. The method and its generalization to a multi-step process is also described. A list of primitive irreducible polynomials is provided, as well as other useful information on different functions.

The author comments regarding the method for performing Boolean OR, AND and NOT operations using arithmetic and conditional transfer operations presented by M. Morris Mano. He states that the operations are useful in many cases but they are seriously limited by the OR operation in many ways. He notes that in a Fortran program, not all Boolean functions that contain an OR operation can be placed in the parenthesis of an IF statement. He also reveals that because of the number of necessary conditional transfers, a function which includes all three operations is more difficult to program.