Showing papers on "Iterative method published in 1973"

Iterative perturbation calculations of ground and excited state energies from multiconfigurational zeroth‐order wavefunctions

Abstract: A method is proposed to calculate the effect of configuration interaction by a Rayleigh‐Schrodinger perturbation expansion when starting from a multiconfigurational wavefunction. It is shown that a careless choice of H0 may lead to absurd transition energies between two states, at the first orders of the perturbation, even when the perturbation converges for both states. A barycentric defintion of H0 is proposed, which ensures the cancellation of common diagrams in the calculated transition energies. A practical iterative procedure is defined which allows a progressive improvement of the unperturbed wavefunction ψ0; the CI matrix restricted to a subspace S of strongly interacting determinants is diagonalized. The desired eigenvector ψ0 of this matrix is perturbed by the determinants which do not belong to S. The most important determinants in ψ1 are added to S, etc. The energy thus obtained after the second‐order correction is compared with the ordinary perturbation series where ψ0 is a single determinant...

A Family of Fourth Order Methods for Nonlinear Equations

Abstract: A family of fourth order iterative methods for finding simple zeros of nonlinear functions is displayed. The methods require evaluation of the function and its derivative at the starting point of e...

A Generalization of the Additive Correction Methods for the Iterative Solution of Matrix Equations

Abstract: A general formulation of the additive correction methods of Poussin [4] and Watts [7] is presented. The methods are applied to the solution of finite difference equations resulting from elliptic and parabolic partial differential equations. A new method is developed for anisotropic and heterogeneous problems. For such difficult problems the method presented here is comparable with Stone’s strongly implicit method, while all other methods tested require much greater computational effort. The correction approach discussed here can be easily applied with any iterative method.

An examination of an iterative method for the solution of the phase problem in optics and electron optics: I. Test calculations

Abstract: An evaluation is made of an iterative method for determining the amplitude and phase from the image intensity recorded in optical systems. The method, which requires two images recorded at different lens defocus values, is tested with simulated data subject to error arising from the photographic recording of the image. In the case of error-free data, the solution for the phase distribution appears to be indeterminate to within a constant. The results for photographic noise levels of up to 20% of the maximum image intensity reflect the small effect of error on the calculated phase distribution. The calculation of phase distributions for both symmetric and asymmetric amplitude-phase distributions shows that the use of two images, taken at defocus values differing by about 100 nm in electron optics and about 1 mm in optics (depending on the numerical aperture of the objective lens), may be used to determine the complex object wave-function in both dark-field and bright-field optics.

A descent numerical method for optimization problems with nondifferentiable cost functionals

Abstract: In this paper we consider the numerical solution of convex optimization problems with nondifferentiable cost functionals. We propose a new algorithm, the $\varepsilon $-subradient method, a large step, double iterative algorithm which converges rapidly under very general assumptions. We discuss the application of the algorithm in some problems of nonlinear programming and optimal control and we show that the $\varepsilon $-subgradient method contains as a special case a minimax algorithm due to Pshenichnyi [5].

Numerical Solutions by the Continuation Method

Abstract: The continuation method is developed with a special emphasis on its suitability for numerical solutions on fast computers. Four problems are treated in detail : finding roots of a polynomial, boundary value problems of nonlinear equations, identification of parameters and eigenvalue problems of linear ordinary differential operators. Numerical results are given for these problems. Finally, the continuation method is compared to iterative methods and several schemes which combine them are proposed.

Finite element method applied to analysis of flow over a spillway crest

Abstract: Numerical analysis of fluid flow over a spillway is treated in the present paper. A variational principle is introduced for a flow with a free surface boundary under gravity, where the stream function and the profile of the free boundary are independent quantities subjected to variation. A new iteration method is formulated by the combined use of the variational principle and the finite element method. A numerical example based on the iterative method is illustrated. Results thus obtained show good agreement with those obtained from an empirical formula.

Technique for implicit dynamic routing in rivers with tributaries

Abstract: The prediction of transient flow in a river having a major tributary poses a challenging problem for the streamflow forecaster. The interaction of storage and dynamic effects between the two rivers can be simulated efficiently by a mathematical model consisting of the two unsteady flow differential equations and of known stage time, discharge time, or stage discharge relationships at the extremities of the rivers. Numerical solutions of discharge and water surface elevation are obtained from the differential equations at specified time intervals by an implicit finite difference technique. This produces successive systems of nonlinear equations that are efficiently solved by the Newton-Raphson iterative method in combination with an extrapolation procedure and a specialized direct method for solving a system of linear equations. The length of the specified time interval is not limited by computational stability; however, accuracy constraints may limit its size. Some numerical results are presented to illustrate the interaction between a river and a tributary when they are subjected to a flood wave of long duration.

Iterative methods for solving linear equations

Abstract: This paper presents some of the original versions of the conjugate-gradient method for solving a system of linear equations of the formAx=k.

The Analysis of Designed Experiments with Censored Observations

Abstract: An observation is said to be "censored" if we know only that it is less than (or greater than) a certain known value. Sampford and Taylor [1959] proposed an iterative method of analysis for randomized block experiments with Type I censoring, and this method can also be used for other experimental designs. The method has been applied to simulated data. The results confirm that parameters can be estimated with only a small bias, and that it is possible to do an approximate t-test to compare treatment means. An approximation due to Tiku will reduce the labour of calculation.

An iterative method for large systems of linear structural equations

Abstract: A method is described for the solution of large sets of sparse equations arising in structural analysis. This method, called partial elimination, combines the concepts of elimination and iteration in such a way that good convergence rates can be obtained using a computer storage space not much greater than that required for other iterative methods.

Computer technique for solving 3-dimensional electron-optics and capacitance problems

Abstract: Electron-optics problems involving planar and axial symmetry are frequently analysed by using iterative procedures to solve Laplace's equation within a specified boundary. The computation time and storage requirement for these procedures may be prohibitive when it is necessary to extend the analysis to three dimensions for problems involving asymmetric fields. An alternative approach is described, known as the method of moments, which does not use an iterative method, but calculates the field directly from the charges induced on the electrodes by the excitation potentials. This technique provides a further facility, which permits the capacitances between arbitrary 3-dimensional electrodes to be obtained. The basic theory and operation of a computer program which employs this method are described. As a practical application, the program has been used to investigate the electron-optical properties of a mesh with rectangular apertures.

Formulation of a Nonlinear Predictor

Abstract: The classical linear prediction problem has received much attention both theoretically and empirically in the past few decades and a great deal is known about it (though there remain important open questions). See, e.g., discussions and lists of references in Parzen (1969) and Rozanov (1967). The nonlinear prediction problem is much more difficult and much less is known about it. See, e.g., Masani and Wiener (1959). We reduce a certain type of nonlinear prediction problem to the multiple linear prediction problem, see Wiener (1956) and Van Ness (1966). We can then take advantage of theoretical results and computational techniques which have been developed to solve the linear problem. The methods are applied to both real and generated data using an iterative method due to Parzen and fast Fourier transform techniques on a small computer (an IBM 1130).

Solution of Linear Equations—State-of-the-Art

Abstract: Most problems in structural analysis require at some stage the solution of linear equations which may require considerable execution times even on today's computers. This paper is a state-of-the-art review of research efforts directed towards computer algorithms to solve large equation systems accurately and economically. Many different solution techniques are analyzed, with emphasis on the physical interpretation of the various strategies. First, the standard Gauss algorithm is reviewed, followed by a consideration of modifications due to symmetry, the Cholesky algorithm, and a physical interpretation of the reduction process. Particular attention is given to specialized direct solution techniques of more recent data, such as various band solutions, partitioning methods (static condensation, substructuring), and frontal solutions. A short summary of iterative methods concludes this paper.

Rectangular and Annular Modal Analyses of Multimode Waveguide Bends

Abstract: Spurious modes in multimode H-plane waveguide bends of nonuniform curvature are computed on the basis of rectangular and annular waveguide modal analyses. The sets of coupled differential equations for the wave amplitudes are solved numerically using both an iterative approach and the Runge-Kutta method. The advantages and limitations of the different approaches to this problem are considered in detail.

Iterative solution of block tridiagonal systems on parallel or vector computers

Abstract: Two iterative algorithms for the solution of block tridiagonal systems on a parallel or vector computer are analyzed. One is a new Parallel Gauss iteration and the second is Jacobi iteration. The results obtained by comparing the algorithms on a model problem are reported.

Solution of Linear Systems of Equations

Abstract: In this paper we consider various iterative methods for solving systems of linear algebraic equations. We shall be primarily concerned with large systems with sparse matrices such as arise in the solution of elliptic boundary value problems in two dimensions by finite difference methods. Our discussion is divided into two parts: First, we review the well-known facts about such methods as the Jacobi, Gauss-Seidel, and successive overrelaxation methods. A treatment of various acceleration techniques is included. In the second part of the paper we consider the symmetric overrelaxation method and describe some practical procedures which can be used to obtain very rapid convergence. By showing that the method can be very effective in many cases and by outlining a definite procedure for its application, we hope to encourage its wider usage and to stimulate further research.

Iterative method for calculating hot carrier distributions in semiconductors

Abstract: A method is described for iteratively solving the Boltzmann equation with the angular dependence of the distribution function expanded in Legendre polynomials. Compared with earlier integral equation methods this approach is shown to have a number of advantages which can lead to an increase in computing efficiency by as much as two orders of magnitude. Numerical results for p-Ge and n-GaAs have been calculated to show the convergence properties of the present approach.

Component-wise error analysis of iterative methods practiced on a floating-point system

Abstract: : In the present paper the author gives a component-wise error estimate for an approximate solution obtained by practicing an iterative method on a floating-point system. Further, in case of Newton's method, the author gives a more precise error estimate and also a simple convenient rule for stopping the iterative process. (Author)

On the construction of weak orders from fragmentary information

Abstract: An iterative method is proposed for constructing a weak order from a partial order on a set of stimuli that is based on individual pairwise comparison data. The method generalizes Duncan Luce's construction of the weak order induced by a semiorder. Various aspects of the iterative procedure are discussed, including its rationale, the number of iterations required to obtain a weak order, and the extent to which the data support additions to the initial partial order as a function of the number of iterations performed before the additions occur.

Application of Sparse Matrix Techniques to Reservoir Simulation.

Abstract: Publisher Summary This chapter discusses the application of sparse matrix techniques to reservoir simulation. It presents computing time requirements of sparse Gaussian elimination for some typical problems of reservoir simulation. In many reservoir simulation problems, the solution of a system of nonlinear parabolic partial differential equations describes multiphase flow in two or three space dimensions. The chapter focuses on two-dimensional problems. The most common technique is to approximate the domain by a rectilinear mesh or grid and to approximate the partial differential equations by five point difference equations together with suitable linearizations. In reservoir simulation, systems of the form have usually been solved with iterative rather than elimination methods. This saves both time and storage. However, selecting an efficient iterative method, optimal acceleration parameters, and a good stopping criterion is difficult and expensive. It has been found that in some situations, iterative methods do not converge to an acceptable solution within a reasonable number of iterations because of the increasing complexity of simulation problems.

Iteration Methods for Finding all Zeros of

Abstract: Durand and Kerner independently have proposed a quadratically convergent iteration method for finding all zeros of a polynomial simultaneously. Here, a new deriva- tion of their iteration equation is given, and a second, cubically convergent iteration method is proposed. A relatively simple procedure for choosing the initial approximations is described, which is applicable to either method.

Modified Nodal Iterative Load Flow Algorithm to Handle Series Capacitive Branches

Abstract: This paper presents a modification of the well known Nodal Iterative load flow algorithm. The modification is shown to be capable of giving a significant improvement in convergence in routine applications and to give good convergence in a wide range of cases involving series capacitive branches. The modification causes no significant increase in program complexity.

Identification of parameters by the continuation method.

Abstract: Aerodynamic coefficients are determined in practice by adjusting them so that the solution of the nonlinear equations of motion will best fit the experimental free-flight data. The numerical iteration method previously proposed by Chapman and Kirk is efficient, but convergence crucially depends on the starting solution. Here it is proposed to find this starting solution by the noniterative continuation method. The equations of motion are imbedded in a one parameter family of problems and the aerodynamuc coefficients found by integration of a rather complicated initial-value problem. As an example, the problem of extracting the aerodynamic coefficients from a simulated ballistic range flight of a model of the Gemini capsule is solved. Comparison also is made between the continuation and iteration methods.