Abstract: A systematic and iterative method of computing the capacity of arbitrary discrete memoryless channels is presented. The algorithm is very simple and involves only logarithms and exponentials in addition to elementary arithmetical operations. It has also the property of monotonic convergence to the capacity. In general, the approximation error is at least inversely proportional to the number of iterations; in certain circumstances, it is exponentially decreasing. Finally, a few inequalities that give upper and lower bounds on the capacity are derived.

Abstract: The numerical solution of problems of elastic stability through the use of the iteration method of Newton is examined. It is found that if the equations of equilibrium are completed by a simple auxiliary equation, problems governed by a snapping condition can, in principle, always be calculated as long as the problem at hand is properly formulated. The effectiveness of the proposed procedure is demonstrated by means of an elementary example.

Abstract: We study an iterative technique for the numerical solution of strongly elliptic equations of divergence form in two dimensions with Dirichlet boundary conditions on a rectangle. The technique is based on the repeated solution by a fast direct method of a discrete Helmholtz equation on a uniform rectangular mesh. The problem is suitably scaled before iteration, and Chebyshev acceleration is applied to improve convergence. We show that convergence can be exceedingly rapid and independent of mesh size for smooth coefficients. Extensions to other boundary conditions, other equations, and irregular mesh spacings are discussed, and the performance of the technique is illustrated with numerical examples.

Abstract: In Newton load-flow solutions the mathematical decoupling of busbar-voltage angle and magnitude calculations has several computational and conceptual attractions. A hybrid Newton formulation exploiting this principle has been developed and well tested. For moderately-accurate solutions the method has advantages over the formal Newton approach in terms of computer storage and speed, particularly in adjusted solutions, and is at least as reliably convergent.

Abstract: A new iterative method is described for the solution of the generalised nonlinear least squares problem: where the model may be nonlinear in its parameters and in the independent variable(s) and all variables are subject to error. The method is described for the case of two arbitrarily related variables; does not require the analytic calculation of derivatives; leads to exceptionally close satisfaction of the least squares conditions; and exhibits especially rapid convergence arising from the use of somewhat unconventional numerical approximations for partial derivatives. Examples are given which compare the results of the method of those of other existing techniques.

Abstract: With the increasing availability of high-speed multiplication units in large computers it is attractive to develop an iterative procedure to compute division and square root, using multiplication as the primary operation. In this paper, we present three new methods of performing square rooting rapidly which utilize multiplication and no division. Each algorithm is considered for convergence rate, efficiency, and implementation. The most typical and efficient one of the already-known algorithms which utilizes multiplication, here called the N algorithm, is introduced for the purpose of comparison with the new algorithms. The effect and importance of the initial approximation is considered. (One of the algorithms, here called the G algorithm, is described in detail with the emphasis on its high efficiency.)

Abstract: The paper presents a number of alternative procedures for the solution of elasto-plastic problems by the matrix displacement or finite element method. In particular, the iterative initial strain and stress approaches are fully developed. Attention is also focused on the tangent stiffness method in association with an efficient modification technique. Convergence criteria are set up for the initial strain and stress methods. It is proved that the latter will only diverge, if an actual break-down of the structure occurs. In order to speed up the eventually slow convergence of the initial stress method a procedure is suggested which has proved very useful in conjunction with the initial strain approach. The theoretically deduced convergence properties of the iterative methods are verified by the numerical results obtained for some illustrative elasto-plastic problems.

Abstract: The iterative methods by Ben-Israel and others for computing the Moore-Penrose inverse of a matrix are examined. Ill conditioned test matrices are inverted by the methods and some difficulties are found out. The iterative methods do not seem superior to direct ones.

Abstract: A well-known source of sparse matrix problems is the systems of linear algebraic equations which arise when we solve elliptic boundary value problems by finite difference methods, A great deal of effort has been devoted to the design and study of iterative methods for the solution of such linear systems (Varga (1962A), Wachspress (1966A) and Young (1971B)). Among direct methods, i.e. methods which give an exact solution of the finite difference equations in absence of round-off errors, Gaussian elimination and its variants are undoubtedly the best known. In this paper we will concentrate on recently developed direct methods other than Gaussian elimination. The best known of these are due to Hockney and Buneman (Buzbee, Golub and Nielson (1970A), Dorr (1970A), Golub (1971A) and Hockney (1965A), (1970A)). Originally the methods of Buneman and Hockney were used only for Poisson’s equation on rectangular domains. The two methods are of comparable speed and very fast. According to Hockney (1970A) his method produces an accurate solution of the standard five-point difference approximation of Poisson’s equation on a 128 × 128 mesh in a time corresponding to that of 3 steps of a successive over-relaxation method for a problem of the same size. The amount of storage needed is about the same as that required for an iterative method and these direct methods therefore compare very favorably with standard band or block Gaussian elimination methods even in that respect.

Abstract: Using an unbiased objective function and a fast algorithm which employs an iterative least-squares technique, consistent estimates are obtained for the parameters of a linear difference equation which describes a system having an output corrupted by correlated noise. The method is applied to a simulated process and compared with the generalised least-squares method.

Abstract: Because of its simplicity, Richardson''s non-stationary iterative scheme is a potentially powerful method for the solution of (linear) operator equations However, its general application has more or less been blocked by (a) the problem of constructing polynomials, which deviate least from zero on the spectrum of the given operator, and which are required for the determination of the iteration parameters of the non-stationary method, and (b) the instability of this scheme with respect to rounding error effects Recently, these difficulties were examined in two Russian papers In the first, Lebedev [1969] constructed polynomials which deviate least from zero on a set of subintervals of the real axis which contains the spectrum of the given operator In the second, Lebedev and Finogenov [1971] gave an ordering for the iteration parameters of the non-stationary Richardson scheme which makes it a stable numerical process Translation of these two papers appear as Appendices 1 and 2, respectively, in this report The body of the report represents an examination of the properties of Richardson''s non-stationary scheme and the pertinence of the two mentioned papers along with the results of numerical experimentation testing the actual implementation of the procedures given in them

Abstract: Most of the known results concerning convergence of iterative methods for solving linear systems involve either positive definiteness or monotonicity. In this paper a new concept, called K-semipositivity, is introduced, which provides a link between convergence theory, monotonicity, and positive definiteness. By using this concept, together with partial orderings in Euclidean n-space, several new convergence theorems are proved. Application to Jacobi's methods and the theory of regular splittings shows the usefulness of these new results.

Abstract: An iterative method for dual linear programming problems is described, and its rate of convergence is established.

Abstract: A new characterization of row-monotone matrices is given and is related to the Moore-Penrose generalized inverse. The M-matrix concept is extended to rectangular matrices with full column rank. A structure theorem is provided for all matrices A with full column rank for which the generalized inverse A+ > 0. These results are then used to investigate convergent splittings of rectangular matrices in relation to iterative techniques for computing best least squares solutions to rectangular systems of linear equations.

Abstract: A coupled pair of harmonic equations is solved by the application of Chebyshev acceleration to the Jacobi, Gauss-Seidel, and related iterative methods, where the Jacobi iteration matrix has purely imaginary (or zero) eigenvalues. Comparison is made with a block SOR method used to solve the same problem. Introduction. In (4), we proposed a general block SOR method for solving the biharmonic equation as a coupled set of finite-difference equations. Here, we consider related methods and compare them to the SOR method. The methods considered here are Chebyshev accelerated Jacobi and Gauss-Seidel as well as others, the cyclic Chebyshev semi-iteration method (5), and the unsymmetric modified SOR method (20), (21). It is shown, by comparing spectral radii, that the SOR method of (4) is at least as fast as any of the above methods. The analysis applies to a block cyclic matrix of index 2, or one that can be written in the form (2.1). The interesting feature of the analysis is that the Jacobi matrix has purely imaginary (or zero) eigenvalues, whereas most previous work assumes real eigenvalues, or some complex. To fix notation, consider the iterative process

Abstract: A linear iterative method for evaluating the Moore-Penrose generalized inverse of a matrix is described and conditions for optimizing its rate of convergence are given

Abstract: A rigorous iteration method for the solution of nonlinear equations in Banach spaces, due to Isaac Newton and Kantorovic, is reported. We consider its application to three nonlinear problems of atomic and nuclear physics, in finite-dimensional spaces: the ordinary, the constrained and a generalized Hartree-Fock problem. We have proved the existence of a solution of the Hartree-Fock equations and its uniqueness in a definite small region of the functional space, for the nucleus16O. This method converges faster than the usual Hartree algorithm and displays some further advantages.

Abstract: Spectral factorization is a powerful tool in deriving the steady-state solution of Kalman filtering equations. It is an algebraic, nortrecursive method, thus economical in terms of computing cost, when compared with the conventional iterative algorithm. In this paper the technique is extended to time-varying systems having periodic coefficient matrices for both discrete and continuous systems. The tracking of low-thrust spacecraft from an Earth-based station is used as an example and a sensitivity study is performed using a computer program incorporating the algorithm.

Abstract: A new algorithm for finding the roots of polynomials is presented. The method is based upon an accurate first approximation to a root which is then used to initiate an iterative solution. The method is reliable, rapid, has a high order of convergence, and excellent convergence in the large.

Abstract: An iterative procedure applying matrix methods to accomplish an efficient algorithm for automatic computer reduction of wind-tunnel force-balance data has been developed. Balance equations are expressed in a matrix form that is convenient for storing balance sensitivities and interaction coefficient values for online or offline batch data reduction. The convergence of the iterative values to a unique solution of this system of equations is investigated, and it is shown that for balances which satisfy the criteria discussed, this type of solution does occur. Methods for making sensitivity adjustments and initial load effect considerations in wind-tunnel applications are also discussed, and the logic for determining the convergence accuracy limits for the iterative solution is given. This more efficient data reduction program is compared with the technique presently in use at the NASA Langley Research Center, and computational times on the order of one-third or less are demonstrated by use of this new program.