Abstract: Taylor-series estimation gives a least-sum-squared-error solution to a set of simultaneous linearized algebraic equations. This method is useful in solving multimeasurement mixed-mode position-location problems typical of many navigational applications. While convergence is not proved, examples show that most problems do converge to the correct solution from reasonable initial guesses. The method also provides the statistical spread of the solution errors.

Abstract: The theoretical basis for the integrated finite difference method (IFDM) is presented to describe a powerful numerical technique for solving problems of groundwater flow in porous media. The method combines the advantages of an integral formulation with the simplicity of finite difference gradients and is very convenient for handling multidimensional heterogeneous systems composed of isotropic materials. Three illustrative problems are solved to demonstrate that two- and three-dimensional problems are handled with equal ease. Comparison of IFDM with the well-known finite element method (FEM) indicates that both are conceptually similar and differ mainly in the procedure adopted for measuring spatial gradients. The IFDM includes a simple criterion for local stability and an efficient explicit-implicit iterative scheme for marching in the time domain. If such a scheme can be incorporated in a new version of FEM, it should be possible to develop an improved numerical technique that combines the inherent advantages of both methods.

Abstract: A rapidly converging iterative method is presented to solve the many‐electron Schrodinger equation within a Hilbert space confined to functions with at most two electrons outside an internal space defined by the orbitals of a reference function. The wavefunction is given in terms of external two‐electron clusters represented by coefficients and density matrices referring directly to the basis functions. All matrix elements are obtained from generalized Coulomb and exchange operators. Only one operator per correlated electron pair is required for each iteration cycle.

Abstract: [ l ] D. A. Gandolfo, C. L. Grasse, and G. D. O’clock, Jr., “Surface acoustic wave components in electronic warfare systems,’’ Proc. of International Specialist Seminar on Component Performance and Systems Applications of Surface Acoustic Wave Devices, Sept. 1973, Aviemore Scotland, pp. 231-42. [ 2 ] K. R. Laker, A. J . Budreau, and P. H. Carr, “Interconnecting SAW filters for low loss frequency multiplexers and frequency synthesizers,” Proc. of I974 Ultrasonics Symposium, Nov. 1974, Milwaukee, Wisconsin, pp. 161-163.

Abstract: This paper presents and develops an algebraic structure for determining the k shortest paths from a given node to all other nodes of a network. Three new methods for calculating such k shortest path information are examined and compared. These methods are based on a fairly strong analogy which exists between the solution of such network problems and traditional techniques for solving linear equations. On the basis of both theoretical and computational evidence, one of the three methods is seen to offer an extremely effective procedure for finding the k shortest paths from a given node in a network.

Abstract: Publisher Summary This chapter explains the problems of perturbations and approximations of generalized inverses of linear operators. The approximation theory of generalized inverses of linear operators has many subtle points involving several modes of convergence, analytic and computational tractability, and techniques that are not merely extensions of those used in the matrix case. Often the study of approximations for a given mathematical object leads to a deeper understanding of the properties of that object, as suggested by Bertrand Russell. Resolution of the difficulties arising in the approximations leads to sharper insight into the properties of the exact object. The approximation methods include projectional and iterative methods and collectively-compact operator approximations. The chapter discusses these methods.

Abstract: The formal refinement methods of least-squares adjustment or difference-map analysis give atomic positions in protein structures with standard deviations which are large compared with the standard deviations of accepted molecular dimensions. This paper describes a method of adjusting the Cartesian coordinates to obtain a properly weighted fit to both the positions from the refinement and the molecular parameters. The equations which have to be solved have many unknowns but few coefficients, and an effective iterative method can be used. The results of applications of the method to insulin are summarized.

Abstract: Two algorithms for least-squares estimation of parameters of a Hammerstein model are compared. Numerical examples demonstrate that the iterative method of Narendra and Gallman produces significantly smaller parameter covariance and slightly smaller rms error than the noniterative method of Chang and Luus, as expected from an analysis of the parameter estimators. In addition, the iterative algorithm is faster for high-order systems.

Abstract: Gallager's exponent function E_{\circ}, (\rho,p) plays a crucial role in the derivation of bounds for coding error probabilities. An iterative algorithm for computing the maximum of E_{\circ} (\rho,p) over the set of input probability distributions is presented. The algorithm is similar to that of Arimoto and Blahut for computing channel capacity. It is shown that the approximation error is at most inversely proportional to the number of iterations. A similar iterative algorithm for computing the source code reliability-rate function also is presented.

Abstract: Publisher Summary This chapter discusses the methods for sparse linear least squares problems. It reviews direct and iterative methods for solving sparse least squares problems. These problems generally arise in the same contexts as sparse linear equations. Among the applications are geodesy, photogrammetry, statistical computations, and structural analysis. As the subject is rapidly developing and so far not enough is known about the algorithms, it has generally not been possible to make final assertions about the relative efficiency of different algorithms. Direct methods have the general advantages over iterative methods that subsequent right hand sides can be treated efficiently and it is easier to obtain elements of related inverse matrices. With direct methods, it is also possible to use the technique of iterative refinement. The most straightforward method to solve the least squares problem is to form the normal equations and then compute the Cholesky factorization. For dense problems, iterative refinement is a cheap and simple way to improve the accuracy of a computed solution. It also has the advantage that it gives useful information about the condition of the problem.

Abstract: A general method of characteristics for solving the multigroup transport equations is developed. This is combined with an adaptive difference scheme, called the modified diamond scheme, and is then applied to the finite difference form of the equation. This formulation is obtained from the discrete ordinates equation, which in turn derives from the multigroup equation, both on the basis of consistency arguments. In this connection two forms of the multigroup equation are used, and the diffusion and other important limits also have a bearing on the final difference equation. The new approaches resolve a number of theoretical and practical difficulties with S/sub n/-type transport calculations, in particular in curved and multidimensional geometries. They lead to a firmer basis for discrete ordinates quadrature sets and to better control, mesh cell by mesh cell, over flux extrapolation, including methods to smooth out unwanted flux oscillations. The total effect is a more consistent treatment of the transport equation together with improved accuracy, fewer breakdowns, and more speed in the calculations, while keeping close to the physics of the problem and retaining the basic simplicity of the difference approach.

Abstract: This paper applies the asymptotic stability theory for ordinary differential equations to Gavurin's continuous analogue of several well-known nonlinear iterative methods. In particular, a general theory is developed which extends the OrtegaRheinboldt concept of consistency to include the widely used finite-difference approximations to the gradient as well as the finite-difference approximations to the Jacobian in Newton's method. The theory is also shown to be applicable to the LevenbergMarquardt and finite-difference Levenberg-Marquardt methods.

Abstract: Conditions necessary for the convergence of the van Cittert iterative method of deconvolution are studied. Conditions that can be expressed in the function domain are derived, some of which are readily apparent restrictions on the shape of the impulse response. The position of the response function along the abscissa is considered, since the shifting of the function can influence convergence. Only real, piecewise-analytic responses with pointwise Fourier transforms and pointwise Fourier inversion are considered.

Abstract: A numerical technique was developed to solve the three-dimensional potential distribution about a point source of current located in or on the surface of a half-space containing arbitrary two-dimensional conductivity distribution. Finite difference equations are obtained for Poisson's equations, making point as well as area discretization, of the subsurface. Potential distribution at all points in the set defining the half-space are simultaneously obtained for multiple point sources of current injection. The solution is obtained with direct, explicit, matrix inversion techniques. An empirical mixed boundary condition is used at the ''infinitely distant'' edges of the lower half-space. Accurate solutions using area discretization method were obtained with significantly less attendant computational costs than with the relaxation, finite-element or network solution techniques, for models of comparable dimensions.

Abstract: This paper considers the parameter identification of nonlinear systems using Hammerstein model and in the presence of correlated output noise. Existing identification methods are all iterative. The proposed method, called MSLS, is a noniterative four-stage least square solution procedure. Therefore, it is computationally simpler. The estimates so obtained are statistically consistent. Two examples are included to demonstrate the utility of this method.

Abstract: Optimal impulsive control of systems arising from linear compartment models for drug distribution in the human body is considered. A system of linear, time-invariant, homogeneous differential equations is given along with a set of continuous constraints on state and control. The object is to develop a constructive algorithm for the computation of the optimal control relative to a convex cost functional. It is first shown that under suitable hypotheses, satisfying the continuous constraints is equivalent to satisfying the constraints at a finite set of abstractly definedcritical points. Once these critical points have been determined, the solution of the optimal control problem is found as the solution of a finite-dimensional convex programming problem. The set of critical points can often be determineda priori solely from the qualitative behavior of the solutions of the system. A class of such problems, generalizing the so-calledplateau effect, is considered in detail. It is shown that the solution achieving the plateau effect is indeed optimal in certain cases. In a subsequent paper, an iterative algorithm will be given for the solution of these problems when the critical points cannot all be determineda priori.

Abstract: The linear, functional equation approach to the problem of the convergence of Pade approximants.- Construction of variational bounds for the N-body eigenstate problem by the method of Pade approximations.- Rational polynomial approximants in N variables.- Convergence of rows of the Pade table.- The use of Pade approximation in numerical integration.- Determination of shock waves by convergence acceleration.- Cyclic iterative method applied to transonic flow analyses.- A technique for accelerating iterative convergence in numerical integration, with application in transonic aerodynamics.- The rise of a bubble in a fluid.- Rational approximations to the solution of the blunt-body & related problems.- Wave front expansions and Pade' approximants for transient waves in linear dispersive media.- Application of methods for acceleration of convergence to the calculation of singularities of transonic flows.- The use of Pade fractions in the calculation of nozzle flows.- A bibliography on Pade approximation and some related matters.

Abstract: Publisher Summary This chapter focuses on optimal control of a system governed by the navier-stokes equations coupled with the heat equation; and discusses the problem of free convection, thermal diffusivity, kinematic viscosity, the problem of optimal control, iterative method for the construction of an element that satisfies the necessary condition for optimality, the method of perturbation (often called the method of artificial compressibility) a convergence theorem which proves the solution of the perturbed system corresponding to a optimal control converges, optimal control of the system of equations, formulation of an adjoint system, the original system of equations in which a system of Cauchy-Kowaleska type is obtained, some preliminary results for a fixed control, various theorems, and lemmas.

Abstract: Iterative methods for the solution of tridiagonal systems are considered, and a new iteration is presented, whose rate of convergence is comparable to that of the optimal two-cyclic Chebyshev iteration but which does not require the calculation of optimal parameters. The convergence rate depends only on the magnitude of the elements of the tridiagonal matrix and not on its dimension or spectrum. The theory also has a natural extension to block tridiagonal systems. Numerical experiments suggest that on a parallel computer this new algorithm is the best of the iterative algorithms considered.

Abstract: Three different algorithms for objective wind field analysis were tested on the same set of initial conditions: Dickerson-Sasaki's 'strong constraint' algorithm, a fixed-vorticity algorithm, and a newly proposed fixed-station-velocity algorithm. The three methods are compared with respect to the degree of minimization of wind divergence and the accuracy of wind data at a measured station. The first two techniques, though they reduce wind divergence, produce wind vectors substantially different from the observed values. The proposed iterative scheme is similar to Endlich's (1967) procedure for treating a macroscale wind field, and minimizes divergence while retaining the observed wind vectors.

Abstract: A procedure for numerical quadrature over a semi-infinite range is described which does not require storing weights or nodes. It is applicable to a wide variety of integrands including oscillatory functions and forms with integrable singularites. A computer program and numerical examples are presented.

Abstract: The problem of determining the points in a program at which variables are “live” (will be used again) is introduced and discussed. Two solutions, one which uses a simple iterative algorithm and one which uses an algorithm based on “Cocke–Allen interval” analysis, are presented and analyzed. These algorithms are compared on “self replicating“ families of reducible program flow graphs. The results are inconclusive in that the interval method requires fewer bit-vector steps on some graphs and more on others. If n is the number of nodes in a program flow graph and the number of edges is linearly proportional to n, then both algorithms require $O(n^2 )$ steps in the worst case.

Abstract: The problem of identifying constant and variable parameters in multi-input, multi-output, linear and nonlinear systems is considered, using the maximum likelihood approach. An iterative algorithm, leading to recursive identification and tracking of the unknown parameters and the noise covariance matrix, is developed. Agile tracking, and accurate and unbiased identified parameters are obtained. Necessary conditions for a globally, asymptotically stable identification process are provided; the conditions proved to be useful and efficient. Among different cases studied, the stability derivatives of an aircraft were identified and some of the results are shown as examples.

Abstract: This paper studies iterative methods for the global flow analsis of computer programs. We define a hierarchy of global flow problem classes, each solvable by an appropriate generalization of the "node listing" method of Kennedy. We show that each of these generalized methods is optimum, among all iterative algorithms, for solving problems within its class. We give lower bounds on the time required by iterative algorithms for each of the problem classes.

Abstract: There exist several algorithms for the optimal orthogonalization of the Direction Cosine Matrix used in navigation, control and simulation One of these recursive algorithms is shown to be derived from a dual solution to the optimal orthogonalization problem. The duality of the algorithm is demonstrated and its convergence properties are investigated Quadratic convergence is proven and the condition for convergence is determined and illustrated with numerical examples.