Abstract: Chemical engineers have taken a new look at the critical-point mathematics for multicomponent hydrocarbon mixtures and have found an alternative way to evaluate the two nonlinear functions of the intensive variables in the critical phase; the procedure greatly facilitates the computations involved, allowing applications to very large systems (with up to 43 components) and to systems with high-density (liquid-liquid) critical points. The calculation procedure is robust and converges (in all the cases studied so far) in three to five iterations.

Abstract: In this paper we shall present two new algorithms for solution of the diserete-time algebraic Riccati equation. These algorithms are related to Potter's and to Laub's methods, but are based on the solution of a generalized rather than an ordinary eigenvalue problem. The key feature of the new algorithms is that the system transition matrix need not be inverted. Thus, the numerical problems associated with an ill-conditioned transition matrix do not arise and, moreover, the algorithm is directly applicable to problems with a singular transition matrix. Such problems arise commonly in practice when a continuous-time system with time delays is sampled.

Abstract: Some inverse problems are characterized by a model consisting of a piecewise continuous function and a set of discrete parameters. For linear problems of this general type, which we call mixed, we show that when the number of data d is greater than the number of parameters p, it is always possible to construct a set of a least d - p equations that are independent of the values of the discrete part of the model. These equations, which we call the annulled data set, can be used to estimate the continuous part of the model. The discrete part of the model can be estimated from a second set of p equations that relate the discrete and continuous parts of the model. The linearization of the nonlinear travel time functional that enter in the hypocenter location problem leads to a mixed inverse problem. The splitting procedure is natural to this problem if the hypocenters are estimated initially by conventional nonlinear least squares by using travel times calculated from some initial estimate of the velocity model. The annulled data are a set of linear combinations of the residuals that are unbiased by that initial location, and as a result, they can be used directly to estimate a perturbation to the velocity model by a Backus-Gilbert procedure. This makes an iterative algorithm possible that consists of a conventional hypocenter location followed by estimating a perturbation of the velocity model from the annulled data set. The uniqueness of the final velocity model is assessed via the linear resolution analysis of Backus and Gilbert (1968, 1970). We also construct a set of Frechet derivatives that relate perturbations of each hypocenter component to perturbations of the velocity model. These kernels are used to assess the possible error of the hypocenters due to inadequate knowledge of the velocity structure by an application of the generalized prediction approach of Backus (1970a). Good results are obtained when the procedure is applied to a simple synthetic data set.

Abstract: We shall in this paper consider the problem of computing a generalized solution of a given linear system of equations. The matrix will be partitioned by blocks of rows or blocks of columns. The generalized inverses of the blocks are then used as data to Jacobi- and SOR-types of iterative schemes. It is shown that the methods based on partitioning by rows converge towards the minimum norm solution of a consistent linear system. The column methods converge towards a least squares solution of a given system. For the case with two blocks explicit expressions for the optimal values of the iteration parameters are obtained. Finally an application is given to the linear system that arises from reconstruction of a two-dimensional object by its one-dimensional projections.

Abstract: We present an iterative approach which uses the Schwinger variational principle to solve the Lippmann-Schwinger equation for electron-molecule scattering. This method combines the use of discrete basis functions to describe the effects of the noncentral molecular potential with an iterative procedure which provides systematic convergence of the scattering solutions. Results for electron-H2 scattering in the static-exchange approximation show that the method converges rapidly and gives very accurate results.

Abstract: A new formulation is proposed for obtaining the SCF wave functions. It is based on an exponential transformation of spin–orbitals which obviates the use of Lagrangian multipliers. A general method is developed for determining explicit expressions for the successive derivatives of the energy with respect to the new variables. The total energy and the wave function are obtained by an iterative procedure, the convergence of which is shown to be quadratic. The method itself provides information as to the Hartree–Fock stability — or instability — of the SCF solution. The method of exponential transformation of molecular orbitals is applicable to closed‐shell systems, as well as to a large variety of open‐shell systems. As an illustration of the procedure the results of ab initio calculations for ammonia, methane, formaldehyde, and aziridine are given.

Abstract: For solving the Euclidean distance Weber problem Weiszfeld proposed an iterative method. This method can also be applied to generalized Weber problems in Banach spaces. Examples for generalized Weber problems are: minimal surfaces with obstacles, Fermat's principle in geometrical optics and brachistochrones with obstacles.

Abstract: McDonald's (1970) generalization of Tucker's (1958) inter-battery factor analysis model to multiple batteries is considered. The identification of parameters is examined and an iterative algorithm for obtaining maximum-likelihood estimates is provided. Consideration of the relationship between inter-battery factor analysis and canonical correlation analysis in the case of two batteries suggests a generalization of canonical coefficients to the situation where there are several batteries. Examples of the application of the procedure are given.

Abstract: An image-restoration method applying the iterative method to solve simultaneous linear equations is described. The advantages of this method are that the memory capacity to be used is minimal, the computation time is very short, and the man-machine interaction in the course of processing is easily effected. Owing to these advantages, this method seems to be superior from a practical viewpoint to other recently proposed linear-algebraic approaches for image restoration. The mathematical basis of this iterative image-restoration method is described and the suitability of this method is presented. The characteristics of this method are clarified through analysis in frequency space. Nonlinear constraints can also be introduced in this method, which restrain occurrence of erratic results caused by noise amplification. Experimental results using a minicomputer-base digital image-processing system demonstrate that the method is very effective and applicable in practice.

Abstract: On the basis of an existence theorem for solutions of nonlinear systems, a method is given for finding rigorous error bounds for computed eigenvalues and eigenvectors of real matrices. It does not require the usual assumption that the true eigenvectors span the whole space. Further, a priori error estimates for eigenpairs corrected by an iterative method are given. Finally the results are illustrated with numerical examples.

Abstract: An iterative method for solving simultaneous linear equations for image restoration has an inherent problem of convergence. The introduction of the procedure called “reblur” solves this convergence problem. This reblurring procedure also serves to suppress noise amplification. The characteristics of the iterative image-restoration method modified by the reblurring procedure are discussed through an analysis in frequency space. Two-dimensional simulations using this method indicate that a noisy image degraded by linear motion can be well restored without noticeable noise amplification.

Abstract: This paper is concerned with the acceleration, by Chebyshev acceleration or conjugate gradient acceleration, of basic iterative methods for solving systems of linear algebraic equations. It is shown that under certain conditions these acceleration procedures are equivalent to similar procedures applied to the “double method” corresponding to two applications of the original basic iterative method. This result is applied to show the equivalence of certain acceleration procedures applied to the Jacobi methods for “red/black” systems, and similar procedures applied to the “reduced system,” which is obtained from the original system by eliminating some of the unknowns. The result is also used to study the behavior of the generalized conjugate gradient procedure of Concus and Golub and of Widlund, for solving linear systems where the matrices are positive real rather than symmetric and positive definite.

Abstract: An identification scheme is developed for the determination of several parameters of a modified "Windkessel" model of the systemic arterial system for an individual patient undergoing cardiac catheterization. The scheme utilizes a modification of the Prony method [10], [11] as a "starter method" to determine good nominal values for the model parameters being varied. These values then serve as input to a well-known iterative nonlinear least-squares identification method (Marquardt method [14]) which then converges rapidly to frmal values of the parameters. Solution of the model equations with these parameter values yields the best fit in a least-squares sense of model-generated and observed aortic and brachial artery pressures. This two stage or sequential Prony-Marquardt technique represents an extension of our previous work associated with the analysis of multiexponential decay curves [18], and is applied here to the identification of parameters associated with the humam arterial system. When coupled with a method of determining the contractile mechanics of the left ventricle (eg., the ventricular elastance concept [l]-[5]), this identification scheme permits a functional characterization of the hemodynamic properties of the left ventricle and its systemic load, for an individual subject.

Abstract: Given a generalized complementarity problem (i.e., complementarity problem over a cone), Habetler and Price introduced an iterative method to solve it under the conditions that the cone is solid and the function is continuous and strongly copositive on the cone. In this paper, we provide an easier iterative method to solve this problem provided that the function is Lipschitz continuous and strongly monotone on the (maybe nonsolid) cone. A separate consideration is given to polyhedral cones.

Abstract: We consider a linear system Ax = b of n simultaneous equations, where A is a strictly diagonally dominant matrix. We get bounds for the spectral radius of the matrix L rsl,which is accociated with the Accelerated Overrelaxation iterative method (AOR). Sufficient conditions for the convergence of that method will be given, which improve the results of Theorem 3, Section 4 of [2], applied to this type of matrices.

Abstract: This paper proposes a method for finding optimal, or near optimal, solutions for problems involving m objective functions, where there is an overall criterion which is a weighted sum of the m objective functions, but where the weights are, initially, unknown. The process is an interactive one, beginning with a set within which the actual weighting vector is known to lie, and progressively cutting down the size of the set until an acceptable solution is found. A by-product of the procedure is an iterative method for finding the generators of the polyhedral cones, within which the weighting vector must lie, at each stage.

Abstract: A generalization of the classical quasi-static method is described. ''Amplitude'' and ''shape'' functions are treated on the same footing by applying an iterative Newton method to the integral form of the multigroup diffusion equations. Provision is made for one amplitude function per neutron group. It is shown, further, how this model encompasses Ott and Meneley's improved quasi-static algorithm.

Abstract: We present in this paper a method for the numerical solution of the steady and unsteady Navier-Stokes equations for incompressible viscous fluids. This method is based on the following techniques: • A mixed finite element approximation acting on a pressure-velocity formulation of the problem, • A time discretization by finite differences for the unsteady problem, • An iterative method using — via a convenient nonlinear least square formulation — a conjugate gradient algorithm with scaling; the scaling makes a fundamental use of an efficient Stokes solver associated to the above mixed finite element approximation.

Abstract: Spatial equilibrium problems are frequently formulated as large scale quadratic programming problems or linear complementarity problems. We show that these problems can be reduced to two or more smaller problems with Generalized Benders Decomposition. The procedure then becomes iterative with the repetitive solution of the smaller problems. In practice, the iterative procedure has converged rapidly.

Abstract: The letter presents an efficient technique for determination of voltage and Current waveforms when a microwave network containing one or more nonlinear elements is excited by a sinusoidal source. This algorithm is a generalisation of an approach previously reported by Gupta and Lomax and features a ready application to microwave networks represented by a large number of equivalent circuit elements, either lumped or distributed.