Showing papers on "Numerical analysis published in 1971"

Numerical solution of 2-dimensional scattering problems using a transmission-line matrix

Abstract: A numerical method using impulse analysis of a transmission-line matrix is introduced and used to obtain wave-impedance values in a waveguide. The method is demonstrated by applying it to the scattering caused by a waveguide bifurcation.

Projection method for solving a singular system of linear equations and its applications

Abstract: The iterative method for solving system of linear equations, due to Kaczmarz [2], is investigated. It is shown that the method works well for both singular and non-singular systems and it determines the affine space formed by the solutions if they exist. The method also provides an iterative procedure for computing a generalized inverse of a matrix.

Derivative free analogues of the Levenberg-Marquardt and Gauss algorithms for nonlinear least squares approximation

Abstract: In this paper we give two derivative-free computational algorithms for nonlinear least squares approximation. The algorithms are finite difference analogues of the Levenberg-Marquardt and Gauss methods. Local convergence theorems for the algorithms are proven. In the special case when the residuals are zero at the minimum, we show that certain computationally simple choices of the parameters lead to quadratic convergence. Numerical examples are included.

A General Computer Program to Determine the Perturbation of Alternating Electric Currents in a Two-Dimensional Model of a Region of Uniform Conductivity with an Embedded Inhomogeneity

Abstract: Summary A computer program to calculate the perturbation of alternating electric currents in a two-dimensional Earth model with a conductivity inhomogeneity is presented. The program provides for an inhomogeneity of arbitrary shape surrounded by a region of different conductivity. The equations and boundary conditions are solved by a numerical method for both E-polarization and H-polarization. The computer program allows for the solution over a grid of variable mesh dimensions and for a general model which consists of several conductivities. The program is given in detail and an example for a particular model is illustrated.

Scattering from a perfectly conducting surface with a sinusoidal height profile: TE polarization

Abstract: A method for the analysis of scattering from periodic structures based on the numerical solution of the integral equations is further developed. Using periodicity (Floquet's theorem), the range of the integral equations is reduced to a single period where the kernels are the Green's functions for periodic arrays. The numerical solution of the integral equations is obtained using the method of moments. Efficient numerical methods for the computation of the periodic Green's functions which allows their rapid evaluation with good accuracy are reported. A new treatment of the singularities which includes the effect of the surface curvature is given. Numerical results for the transverse electric scattering from a conducting surface with a sinusoidal height profile are presented, and several interesting physical phenomena are explored including Brewster angle effects and diffraction grating anomalies.

A stabilization of the simplex method

Abstract: This paper considers the effect of round-off errors on the computations carried out in the simplex method of linear programming. Standard implementations are shown to be subject to computational instabilities. An alternative implementation of the simplex method based upon L U decompositions of the basic matrices is presented, and its computational stability is indicated by a round-off error analysis. Some computational results are given.

Dispersion characteristics of slot lines

Abstract: A new numerical method is developed for the analysis of dispersion characteristics of slot lines. Galerkin's method is applied in the Fourier-transform domain to derive a determinantal equation for the propagation constants. It is shown that accurate numerical results can be obtained with even a 2 × 2 matrix.

Deflation techniques for the calculation of further solutions of a nonlinear system

Abstract: This paper defines several classes of methods which can be used to find additional solutions of a nonlinear system of equations. A theory which embraces these classes is presented and the theory is extended to the multiple root problem. The techniques developed can also be used in avoiding previously found extreme points when performing function minimization. Results of computer experiments are presented.

Finite element method applied to bilot's consolidation theory

Abstract: A VARIATIONAL PRINCIPLE EQUIVALENT TO THE GOVERNING EQUATIONS IN BIOT'S CONSOLIDATION THEORY IS DERIVED. ON THE BASIS OF THE VARIATIONAL PRINCIPLE, A NUMERICAL METHOD EFFECTIVE FOR NONHOMOGENEOUS, ANISOTROPIC SOILS IN ONE-, TWO-, OR THREE-DIMENSIONS IS DEVELOPED. TWO EXAMPLES ARE GIVEN TO ILLUSTRATE THE VALIDITY AND PRACTICALITY OF THE METHOD. /AUTHOR/

The searchlight problem with isotropic scattering

Abstract: A solution to Chandrasekhar's searchlight problem with isotropic scattering is presented for semi-infinite and finite geometries. The scattering and transmission functions for the Fourier components of intensity are given by expressions analogous to those of the plane-parallel case, except that the physical domain of the variable ζ, which replaces the usual direction cosine, is now the circular region |ζ 1 2 | ≤ 1 2 in the complex plane, and also that the corresponding H-, X-, and Y-functions now depend on a scalar wave number. Nonlinear integral equations for H, X and Y are derived, and numerical methods for computing these functions are given by means of the kernel approximation method.

Solution of Tung's axial dispersion equation by numerical techniques

Abstract: Numerical methods for the solution of Tung's axial dispersion equation have been developed and comprehensively evaluated. These methods are general and can be applied where the instrumental spreading function is unsymmetrical and nonuniform. Computation times required are comparable to those of the method of Chang and Huang being about 10 sec per case on the CDC6400 computer. Memory requirements are minimal and this should permit their use with minicomputers for data acquisition and processing.

Newton's method under mild differentiability conditions with error analysis

Abstract: The concept of majorizing sequences introduced by Rheinboldt (SIAM J.N.A. 1968) is used to prove convergence for Newton's method for operator equations of the formT f=? when the operator satisfied the condition that the Frechet derivative is Holder continuous. A detailed analysis of computational errors is given for Newton's method applied to operators with Holder continuous derivatives. This analysis is shown to reduce the analysis of Lancaster (Num. Math. 1968) when the operator has a continuous second derivative. The above analysis is applied to an example of a second order differential equation.

Linear and Regular Celestial Mechanics: Perturbed Two-body Motion Numerical Methods Canonical Theory

Abstract: Book on linear and ordinary celestial mechanics covering perturbed two body motion, numerical methods, canonical theory and initial value problems

An Improved Numerical Technique for Solving Multi-Dimensional Miscible Displacement Equations

Abstract: Analog difference equations of high order accuracy describing stable miscible displacement are presented. The high order difference scheme eliminates almost all the numerical smearing, which is the result of the normally used approximations, and leaves only the effect of the physical dispersion in the solution. In a one-dimensional system, the technique involves an addition of a negative dispersion term to the continuity equation. The negative dispersion term which is called the numerical dispersion coefficient, depends upon the flow velocity, the time-step size, and the block size. The procedure is extended to multi-dimensional systems. As a check, comparisons of the computed results with analytical solutions in one and 2- dimensional systems are made. The error function solution in a one-dimensional miscible system and a 5-spot fractional flow curve computed from the potentiometric model are considered as the analytical solutions in the 2 cases.

On the solution of the optimal linear control problems under conflict of interest

Abstract: In the control of multi-input plants, the optimal choice of the feedback control inputs depends upon the choice of disturbance inputs by the competitor, enemy, or nature. Under that viewpoint, the formulation of the control problem under conflict of interest is translated to a differential game. This study considers a two-person conflicting situation, described by linear plant dynamics, while the performance indices are quadratic functionals. A theorem and an iterative numerical technique, based on Newton's method, are developed, for the actual computation of the closed-loop solution in the stationary case of the nonzero and zero-sum differential game. Explicit solutions are also presented for the finite terminal time problem arising in the zero-sum linear differential game, and a simple sufficient condition for the existence of such solutions is included. Two examples are solved to illustrate the procedures described.

Numerical design of transonic airfoils

Abstract: Publisher Summary This chapter discusses numerical design of transonic airfoils. It describes an inverse method of computing plane transonic flows past air foils that are not only free of shocks but also have adverse pressure gradients so moderate that no separation of the turbulent boundary layer should take place. Up-to-date existence and uniqueness theorems combine with the experimental evidence to assure that these flows are physically realistic and will occur in practice. The chapter presents the partial differential equations governing steady two-dimensional flow of an inviscid compressible fluid by numerical analysis of characteristic initial value problems for the analytic continuation of the solution into the complex domain. The finite difference scheme presented in the chapter was originally introduced to describe the detached shock wave in front of a blunt body but is actually better suited to the inverse problem of shaping air foils so as to achieve shock-free transonic flow. It is related to Bergman's integral operator method and does exploit simplifications associated with the linearity of the equations of motion in the hodograph plane.

Numerical Analysis of Free Surface Seepage Problems

Abstract: The finite element numerical technique is used to effect a solution to linear seepage problems. Particular reference is made to the time dependent movement of a phreatic surface. The flow domain is represented by subdomains composed of isoparametric elements, which permit a close approximation to curved boundaries. The ease of application to seepage through inhomogeneous media is demonstrated during the development of the finite element discretization technique. It is shown that relatively few of the sophisticated isoparametric elements yield solutions to particular flow situations which compare favorably with other numerical and experimental methods for both steady state and time dependent problems. Then generality of approach is shown by particular examples, such as flow towards wells and through an earth dam which amplify the versatility of the particular numerical technique used.

Function generation by approximation employing interative interpolation

Abstract: A method of solving a function for values of a variable when values of an independent variable are given, the method being especially valuable for use in or with a computer, an advantage being minimization of storage requirements. The method is an extension of the branch of mathematics known as numerical analysis, and specifically of the division of that branch known as approximation. An expression is developed for a locus of points which approach points on the given function, i.e., a polynomial expression having a high degree of convergence. The method includes finding solutions to the terms of the polynomial expression by reiterated interpolation. Only a relatively small number of factors need be stored. The method can be employed to calculate values to predetermined accuracy, and is suitable for many functions although it is especially well suited for many transcendental functions. Embodiments of apparatus suitable for performing the method are also disclosed. The apparatus includes elements of electronic data processing such as shift registers, adders, and the like to perform the interpolation involving addition, subtraction and division by 2. The algorithm developed as a manifestation of this method is describable as an add-shift algorithm.

The theory of top-loaded antennas: Integral equations for the currents

Abstract: The general problem involving the distribution of current and the driving-point impedance of a top-loaded antenna is formulated. Consideration is given to an idealized structure having one or more cylindrical top-loading elements. For the practically important case involving conductors of small radius, the axial currents are obtained as the solution to a pair of coupled integral equations. Approximate solutions obtained by numerical methods for the inverted L -, T -, and four-element top-loaded antennas are compared with measured driving-point impedances. Satisfactory agreement indicates the possibility of utilizing the theory in the analysis of practical configurations involving many top-loading elements.