Abstract: Some algorithms based upon a projection process onto the Krylov subspace K/sub m/ = Span(r/sub 0/, Ar/sub 0/,..., A/sup m-1/r/sub 0/) are developed, generalizing the method of conjugate gradients to unsymmetric systems. These methods are extensions of Arnoldi's algorithm for solving eigenvalue problems. The convergence is analyzed in terms of the distance of the solution to the subspace K/sub m/ and some error bounds are established showing in particular a similarity with the conjugate gradient method (for symmetric matrices) when the eigenvalues are real. Several numerical experiments are described and discussed.

Abstract: We describe and analyze two multilevel iterative procedures for solving linear systems arising from certain finite element discretizations of nonself-adjoint and indefinite elliptic partial differe...

Abstract: Subroutine PLTMG is a Fortran program for solving self-adjoint elliptic boundary value problems in general regions ofR 2. It is based on a piecewise linear triangle finite element method, an adaptive grid refinement procedure, and a multi-level iterative method to solve the resulting sets of linear equations. In this work we describe the method and present some numerical results and comparisons.

Abstract: Dispersion characteristics for open-typed dielectric waveguide structures operated at millimeter-and submillimeter-wave frequencies are calculated by a finite-element iterative procedure with a given criterion on the maximum field strength at the virtual boundary. Numerical results for a rectangular dielectric image guide are presented and compared with results from other methods. The strip dielectric guide and the insulated image guide with finite- or infinite-width substrates are also analyzed.

Abstract: We present an efficient numerical method for computing electromagnetic (EM) scattering of arbitrary three‐dimensional (3-D) local inhomogeneities buried in a uniform or two‐layered earth. In this scheme the inhomogeneity is enclosed by a volume whose conductivity is discretized by a finite‐element mesh and whose boundary is only a slight distance away from the inhomogeneity. The scheme uses two sets of independent equations. The first is a set of finite‐element equations derived from a variational integral, and the second is a mathematical expression for the fields at the boundany in terms of electric fields inside the boundary. The Green’s function is used to derive the second set of equations. An iterative algorithm has been developed to solve these two sets of equations. The solutions are the electric fields at nodes inside the finite‐element mesh. The scattered fields anywhere may then be obtained by performing volume integrations over the inhomogeneous region. The scheme is used for modeling 3-D inho...

Abstract: We present a study of electron—molecular-ion collisions. The scattering equations are solved using an iterative approach to the Schwinger variational principle. These equations are formulated using the Coulomb Green's function to properly treat the long-range Coulomb tail of the molecular-ion potential. We apply this approach to electron—hydrogen-molecular-ion collisions in the static-exchange approximation. We obtain elastic differential cross sections, and also use the continuum states from these calculations to compute the photoionization cross section of the hydrogen molecule. The iterative method used here converged rapidly in all calculations performed.

Abstract: Global and element residuals are introduced to determine a posteriori, computable, error bounds for finite element computations on a given mesh. The element residuals provide a criterion for determining where a finite element mesh requires refinement. This indicator is implemented in an algorithm in a finite element research program. There it is utilized to automatically refine the mesh for sample two-point problems exhibiting boundary layer and interior layer solutions. Results for both linear and nonlinear problems are presented. An important aspect of this investigation concerns the use of adaptive refinement in conjunction with iterative methods for system solution. As the mesh is being enriched through the refinement process, the solution on a given mesh provides an accurate starting iterate for the next mesh, and so on. A wide range of iterative methods are examined in a feasibility study and strategies for interweaving refinement and iteration are compared.

Abstract: The coordinate systems of each of the Galilean satellites are defined and coordinates of features seen in the Voyager pictures of these satellites are presented. The control nets of the satellites were computed by means of single block analytical triangulations. The normal equations were solved by the conjugate iterative method which is convenient and which converges rapidly as the initial estimates of the parameters are very good.

Abstract: The method of steepest descent is applied to the solution of electrostatic problems. The relation between this method and the Rayleigh-Ritz, Galerkin's, and the method of least squares is outlined. Also, explicit error formulas are given for the rate of convergence for this method. It is shown that this method is also suitable for solving singular operator equations. In that case this method monotonically converges to the solution with minimum norm. Finally, it is shown that the technique yields as a by-product the smallest eigenvalue of the operator in the finite dimensional space in which the problem is solved. Numerical results are presented only for the electrostatic case to illustrate the validity of this procedure which show excellent agreement with other available data.

Abstract: The convergence of the additive and linear ART algorithm with relaxation is proved in a new way and under weaker assumptions on the sequence of the relaation parameters than in earlier works. These algorithms are iterative methods for the reconstruction of digitized pictures from one-dimesional views. A second proof using elementary matrix algebra shows the geometric convergence of the linear ART algorithm with relaxation.

Abstract: In this paper we discuss an iterative technique for finding the algebraically smallest (or largest) eigenvalue of the generalized eigenvalue problem $A - \lambda M$, where A and M are real, symmetric, and M is positive definite. We assume that A and M are such that it is undesirable to factor the matrix $A - \sigma M$ for any value of $\sigma $. We prove that the algorithm is globally convergent, and that convergence is asymptotically quadratic. Finally, we discuss the modifications required in the algorithm to make it computationally feasible.

Abstract: Consider a polynomialP (z) of degreen whose zeros are known to lie inn closed disjoint discs, each disc containing one and only one zero Starting from the known simultaneous interval processes of the third and fourth order, based on Laguerre iterations, two generalised iterative methods in terms of circular regions are derived in this paper These interval methods make use of the definition of thek-th root of a disc The order of convergence of the proposed interval methods isk+2 (k≧1) Both procedures are suitable for simultaneous determination of interval approximations containing real or complex zeros of the considered polynomialP A criterion for the choice of the appropriatek-th root set is also given For one of the suggested methods a procedure for accelerating the convergence is proposed Starting from the expression for interval center, the generalised iterative method of the (k+2)-th order in standard arithmetic is derived

Abstract: For a number of years, the author has used the finite element method to investigate the collapse strength of thin plated steel structures [1–3]. The work has been directed primarily towards steel bridges which are usually fabricated from engineering steel for which the stress-strain curve exhibits a significant plateau. The collapse behaviour usually involves an interaction between material and geometric non-linearities and is influenced by initial geometric imperfections and residual welding stresses. The present communication describes a number of numerical techniques that the author has developed in order to analyse such structures. The topics covered include approximate yield criteria, accelerated iterative methods and incremental solutions using a ‘length constraint’.

Abstract: : A new iterative method is presented for solving non-symmetric linear systems of equations. The method requires that the symmetric of the matrix of the linear system be positive definite, and the method is efficient only if the symmetric part is easily invertible. The method is modeled on the conjugate gradient method for symmetric positive definite systems and has the finite termination property. The results from several numerical experiments are presented and compared with a similar method proposed by Concus, Golub, and Widlund.

Abstract: Factor analysis is a powerful method for the analysis of data that can be classified in two ways, but it can only be applied to complete data matrices. An iterative method is described that makes possible its application to incomplete data sets, and an example is given of its successful application to a case where about half of the data is missing.

Abstract: For a class of variational inequalities on a Hilbert space $H$ bifurcating solutions exist and may be characterized as critical points of a functional with respect to the intersection of the level surfaces of another functional and a closed convex subset $K$ of $H$. In a recent paper we have used a gradient-projection type algorithm to obtain the solutions for discretizations of the variational inequalities. A related but Newton-based method is given here. Global and asymptotically quadratic convergence is proved. Numerical results show that it may be used very efficiently in following the bifurcating branches and that it compares favorably with several other algorithms. The method is also attractive for a class of nonlinear eigenvalue problems ($K = H$) for which it reduces to a generalized Rayleigh-quotient iteration. So some results are included for the path following in turning point problems.

Abstract: Publisher Summary This chapter discusses the multilevel iterative method for nonlinear elliptic equations. It describes the solution of the quasilinear elliptic boundary value problem. The smoothing equation represents the use of some iterative method that can rapidly damp components of the error that oscillate on the order of hJ. It is found that if the problem has severe singularities, the procedure may result in the refinement of relatively few triangles. It is found that because one want the dimension of the spaces T, to increase geometrically, this adaptive procedure may be invoked several times to compute a single triangulation. The adaptive procedure is likely to produce triangulations that are highly nonuniform. For such triangulations, the convergence of the multilevel iterative method can still be proven, but the proof of the asymptotic optimal order work estimate may no longer hold. The function f(x,y) = 0 along the P − N junction is positive in the N-region and is negative in the P-region.