Abstract: The development of algorithms for Radionuclide Computed Tomography (RCT) is complicated by the presence of attenuation of gamma-rays inside the body. Some of the existing RCT reconstruction algorithms apply approximation formulas to the projection data for attenuation correction, while others take attenuation into account through some iterative procedures. The drawbacks of these algorithms are that the approximation formulas commonly used are generally inadequate and the iterative procedures are usually very time-consuming. The method for attenuation correction in RCT, which we propose, applied a simple, effective two-step procedure to the uncorrected image. In this procedure the filtered back-projection algorithm is used for its fast speed. A simple mathematical basis and description of the procedure together with some illustrative computer results are given in this paper.

Abstract: This paper describes a method for the numerical solution of linear systems of equations. The method is a two-parameter generalization of the Successive Over- relaxation (SOR) method such that when the two parameters involved are equal it co- incides with the SOR method. Finally, a numerical example is given to show the su- periority of the new method. 1. Introduction. For the numerical solution of linear systems, numerous direct as well as indirect methods exist. Among the indirect or iterative methods the Succes- sive Overrelaxation (SOR) and related methods play a very important role and are the most popular ones. These methods are fully covered in the excellent books by Varga (1), by Wachspress (2) and in the most recent one by Young (3). The purpose of this paper is to present a two-parameter generalization of the SOR method and also the first basic results concerning this method which has been called Accelerated Overrelaxation (AOR) method. As will be seen, the well-known methods of Jacobi, of Gauss-Seidel, of Simultaneous Overrelaxation and of Successive Overrelaxation can be derived, as special cases, from the AOR method. Finally a characteristic numerical example, which we give in a special case, shows the superiority of the AOR method. 2. Derivation of the AOR Method. We consider a system of N linear equations with N unknowns written in matrix form

Abstract: The location problem is to find a point M whose sum of weighted distances from m vertices in p-dimensional Euclidean space is a minimum. The best-known algorithm for solving the location problem is an iterative scheme devised by Weiszfeld in 1937. The procedure will not converge if some nonoptimal vertex is an iterate, however. This paper solves the problem of vertex iterates and presents a general proof permitting a variable step length within certain bounds. This property is used, in particular, to show the convergence of a modified gradient Newton-Raphson type of procedure.

Abstract: An iteration based upon the Tchebychev polynomials in the complex plane can be used to solve large sparse nonsymmetric linear systems whose eigenvalues lie in the right half plane. The iteration depends upon two parameters which can be chosen from knowledge of the convex hull of the spectrum of the linear operator. This paper deals with a procedure based upon the power method for dynamically estimating the convex hull of the spectrum. The stability of the procedure is discussed in terms of the field of values of the operator. Results show the adaptive procedure to be an effective method of determining parameters. The Tchebychev iteration compares favorably with several competing iterative methods.

Abstract: The solution of parabolic control problems is characterized by a system of two equations parabolic with respect to opposite orientations. In this paper a fast iterative method for solving such problems is proposed.

Abstract: In this paper we consider the local rates of convergence of Newton-iterative methods for the solution of systems of nonlinear equations. We show that under certain conditions on the inner, linear i...

Abstract: A method for the optimal design of water distribution networks is presented. It defines the least-cost solution for a closed network, using standard diameters on the market. It can take into account the branches whose diameter value are assigned (existing); the range of velocities in the pipes and of the hydraulic heads can be imposed. And, lastly, the desired solution is obtained with acceptable computer time. The basic hydraulic laws and the other constraints are introduced into the objective function to be minimized. Thus, all constraint equations are eliminated, and the problem is to determine the minimum function (even if complex). The solution is found using an iterative method which considers that the most economical distribution system is always an open network. The method has been applied to a complex network (many loops), and has introduced acceptable computer times.

Abstract: Multi-grid methods are characterized by the simultaneous use of additional auxiliary grids corresponding to coarser step widths. Contrary to usual iterative methods the speed of convergence is very fast and does not tend to one if the step size approaches zero. The computational amount of one iteration is proportional toN, the number of grid points. Thus, a solution with accuracy ɛ requires 0 (|log ɛ|N) operations. In this paper we apply a multi-grid method to Helmholtz's equation (Dirichlet boundary data) in a general region and to a differential equation with variable coefficients subject to arbitrary boundary conditions.

Abstract: A regular iteration algorithm is constructed for the case of a nonlinear generalized thermal conductivity equation for determination of the nonstationary thermal flux. The algorithm is based on the method of conjugate gradients.

Abstract: In this paper we study linear stationary iterative methods with nonnegative iteration matrices for solving singular and consistent systems of linear equationsAx=b. The iteration matrices for the schemes are obtained via regular and weak regular splittings of the coefficients matrixA. In certain cases when only some necessary, but not sufficient, conditions for the convergence of the iterations schemes exist, we consider a transformation on the iteration matrices and obtain new iterative schemes which ensure convergence to a solution toAx=b. This transformation is parameter-dependent, and in the case where all the eigenvalues of the iteration matrix are real, we show how to choose this parameter so that the asymptotic convergence rate of the new schemes is optimal. Finally, some applications to the problem of computing the stationary distribution vector for a finite homogeneous ergodic Markov chain are discussed.

Abstract: The Prony method is extended to handle the nonsymmetric algebraic eigenvalue problem and improved to search automatically for the number of dominant eigenvalues. A simple iterative algorithm is given to compute the associated eigenvectors. Resolution studies using the QR method are made in order to determine the accuracy of the matrix approximation. Numerical results are given for both simple well defined resonators and more complex advanced designs containing multiple propagation geometries and misaligned mirrors.

Abstract: An iterative algorithm is presented: it builds a continuous stress and displacement solution starting from the solution of a classical displacement finite element analysis. The modified solution satisfies the virtual work principle, and is much better than the starting solution. The algorithm is very easily included in existing programs. A number of examples shows the efficiency of the method.

Abstract: The relaxation field for solutions of mixed electrolytes of any type is calculated. the calculation is based on the well-known treatment due to Fuoss-Onsager with the same distance parameter for all the ions in solution. A general conductance-continuity equation has been established and an improved iterative method of calculation, using Laplace transforms, is proposed. The relaxation-field results are derived to the second iteration in the perturbation method of integration.

Abstract: We describe in this report the numerical analysis of a particular class of nonlinear Dirichlet problems. We consider an equivalent variational inequality formulation on which the problems of existence, uniqueness and approximation are easier to discuss. We prove in particular the convergence of an approximation by piecewise linear finite elements. Finally, we describe and compare several iterative methods for solving the approximate problems and particularly some new algorithms of augmented lagrangian type, which contain as special case some well-known alternating direction methods. Numerical results are presented.

Abstract: A least squares method is presented for computing approximate solutions of indefinite partial differential equations of the mixed type such as those that arise in connection with transonic flutter analysis. The method retains the advantages of finite difference schemes namely simplicity and sparsity of the resulting matrix system. However, it offers some great advantages over finite difference schemes. First, the method is insensitive to the value of the forcing frequency, i.e., the resulting matrix system is always symmetric and positive definite. As a result, iterative methods may be successfully employed to solve the matrix system, thus taking full advantage of the sparsity. Furthermore, the method is insensitive to the type of the partial differential equation, i.e., the computational algorithm is the same in elliptic and hyperbolic regions. In this work the method is formulated and numerical results for model problems are presented. Some theoretical aspects of least squares approximations are also discussed.

Abstract: An iterative method has been used to produce rain-rate estimates for an attenuating frequency radar. An infinite number of higher-order estimates are shown to converge in the limit to the Hitschfeld-Bordan solution under certain conditions. An error analysis was performed by a model accounting for the randomness of the radar return power, the k-Z, Z-R relations, and offsets in the radar calibration constant. Since the behavior of the estimates strongly depends on system errors, the choice of the best estimate requires a knowledge of the variance and range of offsets in meteorological data and the calibration constant. As the errors increase, the use of the lower orders avoids significant overpredictions. In order to obtain reliable rain-rate predictions in the presence of realistic errors, an antenna pointing angles away from the horizontal at frequencies in the lower end of the X-band may be used. Such antenna configurations insure low attenuation.

Abstract: A simple computational test for existence of a solution to a nonlinear system of equations and convergence of iterative methods is given for n-cubes. The test is eventually satisfied by any convergent Newton-type sequence.

Abstract: In this paper we discuss an iterative method for the calculation of the boundary values of the conformal mapping of a simply connected regionG onto a regionH with smooth boundary. The method is based on a certain Riemann-Hilbert-problem. It turns out that this problem is the linearised version of a singular integral equation of the second kind. Hence the method is a Newton-method. Whenever the boundaries ofG andH are sufficiently smooth, convergence is locally quadratic. In caseG is the unit circle, the solution of the linearised problem can explicitly be represented in terms of integral-transformations. From this one derives a quadratically convergent Newton-like numerical method which avoids the numerical solution of systems of linear equations and therefore is in comparison with other methods, which are based on integral equations, rather economical in terms of computer time and storage requirements.

Abstract: We consider the multi-facility location problem of placing m new facilities optimally among n demand points (or existing facilities) so that the sum of all weighted lp distance pairs (facility to facility and facility to demand point) is minimized. A method involving the solution of differential equations by numerical integration is presented. This method is computationally comparable to many existing heuristic and iterative methods. It avoids the frequent convergence difficulties associated with many iterative methods for p > 1 and has no difficulties in dealing with Targe “clusters” of facilities of p = 1.

Abstract: A method is presented for obtaining temperature and pressure perturbations within convective clouds using detailed in-cloud motion data as input. Initial testing of the iterative method indicates that it converges to a solution consistent with the input motion field. Potential applications of the method are discussed.

Abstract: Monotone iterative methods have been successfully used to generate improvable two-sided point-wise bounds on solutions of nonlinear boundary value problems for both ordinary and partial differential equations. While such procedures take a simple form when the nonlinearities are independent of gradient terms [6, 9], the extension of such techniques to fully nonlinear problems has been quite formidable. In the case of scalar ordinary differential equations of the type