Abstract: Machine contouring must not introduce information which is not present in the data. The one-dimensional spline fit has well defined smoothness properties. These are duplicated for two-dimensional interpolation in this paper, by solving the corresponding differential equation. Finite difference equations are deduced from a principle of minimum total curvature, and an iterative method of solution is outlined. Observations do not have to lie on a regular grid. Gravity and aeromagnetic surveys provide examples which compare favorably with the work of draftsmen.

Abstract: The concept of a regular splitting of a real nonsingular matrix, introduced by R. Varga, is used in iterative methods for solving linear systems. The purpose of this paper is to extend this concept to rectangular linear systems \[ ( * )\qquad Ax = b, \] in two directions. Firstly, this is accomplished by replacing $A^{ - 1} $ by $A^\dag $, the Moore–Penrose inverse of A, and, secondly, by considering matrices that leave a cone invariant.Let $A \in R^{m \times n} $. The splitting $A = M - N$ is called proper if $R(A) = R(M)$ and $N(A) = N(M)$. Such is the case when A and M are nonsingular. Consider the iteration \[ ( * * )\qquad x^{i + 1} = M^ \dag Nx^i + M^ \dag b. \] It is shown that for a proper splitting, $\rho (M^ \dag N)$ (the spectral radius of $M^ \dag N$) is less than l, if and only if the iteration (**) converges to $A^ \dag b$, the best least squares approximate solution of the system (*). This approach has the advantage of avoiding the normal system $A^T Ax = A^T b$ in solving (*). Necessary an...

Abstract: In this paper some of the numerical problems associated with computing the generalized inverse of a matrix are discussed and illustrated by a detailed analysis of an iteration of Ben-Israel and Cohen.

Abstract: A probabilistic, multidimensional version of Coombs' unfolding model is obtained by assuming that the projections of each stimulus and each individual on each axis are normally distributed. Exact equations are developed for the single dimensional case and an approximate one for the multidimensional case. Both types of equations are expressed solely in terms of univariate normal distribution functions and are therefore easy to evaluate. A Monte Carlo experiment, involving 9 stimuli and 3 subjects in a 2 dimensional space, was run to determine the degree of accuracy of the multidimensional equation and the feasibility of using iterative methods to obtain maximum likelihood estimates of the stimulus and subject coordinates. The results reported here are gratifying in both respects.

Abstract: A work-in-progress report is presented dealing with several methods of iterative 3-dimensional reconstruction that are more accurate than the usual methods, and the behavior of a number of algorithms upon varying the number of iterations, the number of projections, the counts per projection, and the field of view of the detector.

Abstract: In many of the experimental situations to which the analysis-of-covariance model applies, the values of the covariates are known prior to the actual experiment so that they can be used in allocating the experimental units to the treatments. Due to the large number of possible allocations, the computation of an allocation that is D-optimal for inferences on the treatment means will generally not be practical. Good allocations can be constructed by a multistage procedure that allocates one or more units at each stage. These allocations can be made even better by applying an iterative algorithm that induces small changes in the design at each iteration. The techniques were applied to an example taken from the chemical industry and produced highly efficient allocations. Similar design techniques can be used in experimental situations where the experimental units become available and must be allocated in stages.

Abstract: The nonlinear minimization problem that results from recursive digital filter design with phase constraints is simplified somewhat by working in the time domain. This paper describes techniques that utilize the time samples of the desired response as target values for an iterative minimization. Initial values for the α and β (feedforward and feedback) coefficients can be obtained by one of several reliable methods and fed into iterative routines that lead to a locally optimal solution for the coefficients. The initial guess procedures, stemming from regressionlike equations, only require the solution of a set of linear equations. In addition, the iteration procedures described in this paper lead to recursive filter designs requiring little computer time. Examples are presented to illustrate a range of applications.

Abstract: A differential equation formulation of the multichannel coupled equations of Kouri and Levin describing rearrangement scattering is given. The different channels are linked together in these equations through the presence of the channel coupling array $W$. It is shown that the earlier results of Hahn, derived using the reduction method, follow directly from the multichannel coupled equations by making specific choices of the array $W$. These particular $W'\mathrm{s}$ guarantee that the iterated kernels of the coupled equations are connected in those channels which are explicitly considered, so that standard numerical techniques can be applied to their solution.

Abstract: The eigenvalue problem Ax = XBx, where A and B are large and sparse symmetric matrices, is considered. An iterative algorithm for computing the smallest eigenvalue and its corresponding eigenvector, based on the successive overrelaxation splitting of the matrices, is developed, and its global convergence is proved. An ex- pression for the optimal overrelaxation factor is found in the case where A and B are two-cyclic (property A). Further, it is shown that this SOR algorithm is the first order approximation to the coordinate relaxation algorithm, which implies that the same overrelaxation can be applied to this latter algorithm. Several numerical tests are reported. It is found that the SOR method is more effective than coordinate relaxation. If the separation of the eigenvalues is not too bad, the SOR algorithm has a fast rate of convergence, while, for problems with more severe clustering, the c-g or Lanczos algorithms should be preferred.

Abstract: Iterative methods in real Hilbert spaces.- Iterative methods in complex Hilbert spaces.- Successive approximation methods.- Gradient methods.

Abstract: In this paper, we shall give some results on the approximation and on the numerical solution of some non linear elliptical problems. It is also shown that the iterative method used to solve the approximate problems is also useful for solving other non linear problems arising in mechanics and physics.

Abstract: The problem of determining a linear feedback control that depends only on the instantaneous value of the system output is solved for a class of linear stochastic; systems with quadratic cost criterion. For the finite time (steady-state) version it is shown that a two-point boundary-value problem (two non-linear algebraic equations) must be solved to implement the optimal control. The iterative methods to solve the above equations are briefly reviewed. A number of examples solved on a digital computer are included in the paper.

Abstract: We study stationary iterative methods of maximal order for calculating zeros of operator equations. These methods use the values of the operator and its first s Frechet derivatives at n previous iteration points. We introduce a sufficient condition for an iterative method to have maximal order in a certain class of admissible methods. We prove the maximality of the interpolatory method $I_{n,s} $ in the scalar case (see Traub [11, p. 60 and ff.]). For the m-dimensional case, $2 \leqq m \leqq + \infty $, we prove that interpolatory iteration is maximal for $n = 0$ in the class of iterations using values of the first s derivatives at n previous points.

Abstract: An optical second degree iterative method for accelerating a basic linear stationary iterative method of first degree with zeal eigenvalues has been studied by Young [18], [19] and Young and Kincaid [20]. The derivation of this method is now established for the case when the eigenvalues are contained in an elliptical region in the complex plane. By extending results developed by Kublanowskaya (Faddeev–Faddeeva [1]) and N iethammer [8], this second-degree method is obtained through an application of conformal mapping and summation techniques.

Abstract: This paper is concerned with a class of iterative processes of the formu k+1 =Tu k (k = 0, 1, ⋯) for solving nonlinear operator equationsu = Tu orFu = 0. By studying the relationship between a linear functional inequalityϕ(Ah) β(h) + γ(h) ⩽ ϕ(h) and estimates for the iteration operatorT a general semilocal convergence theorem is obtained. The theorem contains as special cases theorems for various iterative methods. Numerical examples illustrate the accuracy of the error estimates for the approximationu k .

Abstract: A study of four techniques is examined to determine which method has the greater overall utility. An iterative least-squares error minimization method provides the greatest accuracy, while a linear algebraic method is accurate only for two-exponential data with small error. Consequently, a three-method sequential identification scheme is devised utilizing a graphical method, a transform method, and an iterative method. This scheme is successfully applied to physiological data containing three and four exponents.

Abstract: An iterative algorithm, simple enough to be executed on a desk top automatic computer, is given for computing the inverse of the function x = erfc(y) for small values of x. In the present note, a simple method is proposed for computing values of y for which the function

Abstract: A method is developed for computing an exact nonorthogonal analysis of variance using cell means. This is accomplished without forming or using computer storage for X 0, or X′0 X 0, or an orthogonal transformation of X 0, where X 0 is the N × p nonorthogonal design matrix. The method is a convergent iterative method which utilizes balanced analysis-of-variance estimates and residuals iteratively in solving the relevant normal equations and conducting tests of hypotheses. A monotonicity property of the method is derived to minimize iteration for nonsignificant factors or interactions in hypothesis testing.

Abstract: An iterative algorithm is presented for finding the optimal feedback controller for a linear jump parameter System with state-dependent transition probabilities. Conditions guaranteeing convergence are provided, and an example illustrates the application of the algorithm.

Abstract: This paper fills a gap in the theory of multipoint iteration function exemplified by the question: How does the secant method converge to a multiple zero? To answer this question a theory for the asymptotic behavior of a certain type of homogeneous, nonlinear difference, equation is developed, and it is shown that two broad classes of iterative methods can fit this theory. The results of numerical investigations based on the theory suggest that Muller’s method applied to a zero of multiplicity greater than 2 will inevitably produce complex iterates.

Abstract: An iterative method of computing the capacity of a discrete memoryless channel, whose channel matrix has m row vectors and is of rank t, has been proposed independently by Arimoto (1972) and Blahut (1972) . The amount of computation involved depends upon the size of the channel matrix used. It is shown that it is sufficient to use a set of t linearly independent row vectors as channel matrix in the computation of the capacity of a discrete memoryless channel. For the case m > t, a criterion for selecting a set of t linearly independent row vectors as channel matrix is presented.

Abstract: The present method is based on a decomposition of the velocity potential into two new functions. These two functions, forming a new system, can be stably integrated in opposite radial directions. A far-field boundary condition in free air is established at a finite radius by means of the asymptotic Guderley far-field expansion. Porous wind tunnel walls also can be handled by this method. A pronounced feature of this iterative method, considering only one new function at a time, is a rather high rate of convergence. The agreement with experimental data and other comparable methods is found to be very good.