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Based upon the concept of Galerkin's approximate method for solving eigenvalue problems, a new scheme of numerical treatment is proposed for a class of nonconservative (circulatory) elastic stability problems. This is accomplishe by considering, together with the original system, a second system which is obtained by introducing an adjoint to the circulatory force field. The resulting problem is shown to be self-adjoint, with eigenfunctions which possess the property of reducing the original problem to a simple integral equation. This integral equation may be solved by quadrature and an estimation of error is also possible. The proposed method is especially suitable for direct evaluation on a digital computer and does not involve tedious integrations of functions encountered in the commonly adopted application of the Galerkin method.

The report describes the application of an iterative method for solving a set of simultaneous non-linear equations which minimizes the probability of error in a digital communication system with intersymbol interference. The form of the optimum linear receiver for such a system as well as the equations to be solved are described in detail in the introduction. The equations are approximated by a numerical quadrature formula. In the following section, an iterative method for the solution of the approximating equations is described. The equations are put into a suitable form for applying the method; a starting value is given. The third section consists of an APL computer program which uses the method and a program description. In the last section, the technique is illustrated with four examples with varying degrees of intersymbol interference. The probability of error for the optimum linear receiver is plotted as a function of signal to noise ratio of the 'desired' signal. The appendix contains a brief discussion of the error incurred in using the quadrature formula. (Author)

Quadrature formulas for double integrals are described. Examples are used to given an idea of what should be proven about quadrature formulas. Proofs are simple and cover formulas of third- and fifth-degree polygonal accuracy. Several problems are provided and should be worked to gain some idean of why quadrature formulas work, and how they can be created or altered to suit ones' need. (Author)

The nonexistence of 10-point 7th-degree quadrature formulas over fully symmetric regions is proven. Two 13-point, 7th-degree quadrature formulas over such regions are presented. First can be a 12-point quadrature formula if a condition is satisfied. The second can be also if another condition is satisfied. Five quadrature formulas of 3rd and 5th degree are extended to fully symmetric regions. (Author)

It is shown that the problem of computing astronomical refraction for any value of the zenith angle may be reduced to a simple, non-singular, numerical quadrature when the proper choice is made for the independent variable of integration. The angle between the radius vector and the light ray is such a choice. The implementation of the quadrature method is discussed in its general form and illustrated by means of an application to a piecewise polytropic atmosphere. The flexibility, simplicity and computational efficiency of the method is evident. (Author), Sponsored in part by Grant NSF-GP-26495.

Symmetric interpolation polynomials are defined for N-dimensional simplexes with the aid of a symmetric coordinate notation. These polynomials are used to produce symmetric interpolatory quadrature formulae of arbitrary degree of precision over simplexes of arbitrary dimensionality. Tabulated values of weight coefficients are given for triangles and tetrahedra. (Author)

The report summarizes the activities supported in full or part under the contract. The activities are broken into three major areas: The Special Year in Numerical Analysis, The Mathematical Software Symposium, and Research. (Author)

The purpose of the paper is to develop and analyze error estimates of order type for the numerical quadrature of integrands f which are integrable but may have singularities at the two-ends of the interval of integration. The case where some low order derivative of f has singularities is also discussed. The analysis is carried out for a class of integrands f called weakly singular functions. Convolution integrals are also discussed. (Author), Prepared in cooperation with Arizona State Univ., Tempe and California Univ., Berkeley. Lawrence Radiation Lab.

Optimal quadrature formulas are constructed for a class of integral functions having a limited derivative of order r > 2 and with higher order of approximation. The number of nodes is limited such that only certain nodes will vary. (Author), Trans. of unidentified mono. pub. by Institut Tochnoi Mekhaniki i Vychislitel'noi Tekhniki AN SSSR (Institute of Precision Mechanics and Computer Technology, Acad. of Sci, USSR), Moscow, 1969 p1-38.

Recent papers of P. Davis and P. Rabinowitz and of P. Rabinowitz study the problem of 'ignoring the singularity' in numerical quadrature of singular integrands. Among the other things they show that for certain quadratures if the singularity occurs at an end point of the interval and if the integrand is monotone in a neighborhood of this singularity, then the quadrature does indeed approximate the integral. The purpose of this paper is to show that the assumption of monotonicity of the integrand can be replaced by the assumption that the integrand can be dominated by a monotone, integrable function. The paper also contains error estimates for approximate quadratures of singular integrands. (Author)

The report summarizes the recent work in the application of the LIE-series method to the solution of ordinary and partial differential equations. The power series method which is a special case of the Lie series method of chapter III is described in chapter II. Chapter III deals with the numerical evaluation of the Lie series perturbation formula. In chapter IV we prove Grobner's integral equation which leads to short proofs of the formulas of chapter III and to various generalizations of the method. A survey of these is presented at the end of this summary. Chapter V generalizes the concept of Runge-Kutta to methods with multiple nodes, which is possible with the use of the Lie differential operator D. Chapter VI deals with the step-size control and chapter VII shows the application of generalized Lie series to the calculation of switch-on transients occurring in the telegraphic equation.

Approximate rules for evaluating linear functionals are often obtained by requiring that the rule shall give exact value for a certain linear class of functions. The parameters of the rule appear hence as the solution of a system of equations. This can generally not be solved exactly but only 'numerically'. Sometimes large errors occur in the parameters defining the rule, but the resultant error in the computed value of the functional is small. In the present paper, efficient methods of computing a strict bound for this error are developed in the case when the parameters of the rule are determined from a linear system of equations. (Author)

The construction of best quadrature formulae is generalized in two ways. Firstly, integrals are discussed having an arbitrary preassigned weight function. Secondly, the most general boundary conditions are determined. (Author)

A critical examination is made of two common approaches to the automatic evaluation of definite integrals. It is shown that on the grounds of efficiency and reliability it is generally preferable to use families of common point high precision formulae directly over the whole interval of integration than to apply low precision formulae to an adaptively subdivided range. Many examples are given to illustrate the conclusions. (Author)

The absence on many computers of subroutines for performing multiple quadrature creates a need for improvization by each programmer. A simple method which can be employed, but is often overlooked, is the use of subroutines for single quadrature to accomplish multiple quadrature. This report describes the method and illustrates its use through a simple program for evaluating the circular coverage function used in weapons systems analysis. (Author)

Contents: Concerning the question of approximation of characteristics of statistical structure; Statistical structure of the precipitation field; Certain characteristics of the three-dimensional macrostructure of air temperature; Three-dimensional objective analysis of the field of air temperature; The effect of instrument inertia on the accuracy of meteorological measurements; Optimum formulas for numerical quadratures for random functions of a special type; One method of restoration of the geopotential field of the 500 millibar surface; Statistical structure of averaged values of meteorological elements; Correlation functions for a case of exponential structure (two-dimensional averaging); Correlation functions for a case of power structure; The problem of the accuracy of analysis of synoptic maps.

SCOPUS: ar.j, info:eu-repo/semantics/published