Abstract: This paper surveys the state of the art in sparse matrix research in January 1976. Much of the survey deals with the solution of sparse simultaneous linear equations, including the storage of such matrices and the effect of paging on sparse matrix algorithms. In the symmetric case, relevant terms from graph theory are defined. Band systems and matrices arising from the discretization of partial differential equations are treated as separate cases. Preordering techniques are surveyed with particular emphasis on partitioning (to block triangular form) and tearing (to bordered block triangular form). Methods for solving the least squares problem and for sparse linear programming are also reviewed. The sparse eigenproblem is discussed with particular reference to some fairly recent iterative methods. There is a short discussion of general iterative techniques, and reference is made to good standard texts in this field. Design considerations when implementing sparse matrix algorithms are examined and finally comments are made concerning the availability of codes in this area.

Abstract: A search procedure based on interval computation is given for finding safe starting regions in n dimensions for iterative methods for solving systems of nonlinear equations. The procedure can search an arbitrary n-dimensional rectangle for a safe starting region for a quadratically convergent iterative method. The procedure is more powerful than continuation methods.

Abstract: An interative approach is proposed for the numerical analysis of elastic–plastic continua. This approach gives after convergence an implicit scheme of integration of the evolution problem, and is concerned with elastic-perfectly plastic materials and with hardening standard materials. Under a generalized assumption of positive hardening, the proof of convergence of the iterative solutions is given. Some numerical examples by the finite element method are also discussed.

Abstract: This paper is concerned with difficult global flow problems which require the symbolic evaluation of programs. We use, as is common in global flow analysis, a model in which the expressions computed are specified, but the flow of control is indicated only by a directed graph whose nodes are blocks of assignment statements. We show that if such a program model is interpreted in the domain of integer arithmetic then many natural global flow problems are unsolvable. We then develop a direct (non-iterative) method for finding general symbolic values for program expressions. Our method gives results similar to an iterative method due to Kildall and a direct method due to Fong, Kam, and Ullman. By means of a structure called the global value graph which compactly represents both symbolic values and the flow of these values through the program, we are able to obtain results that are as strong as either of these algorithms at a lower time cost, while retaining applicability to all flow graphs.

Abstract: An iterative method of multiple grid type is proposed for solving general finite element systems. It is proved that the method can produce a solution to the equations in O(N) arithmetical operations where N is the number of unknowns.

Abstract: Publisher Summary Equilibrium problems in two-dimensional, and higher, continua give rise to elliptic partial differential equations. An alternative argument employs the maximum (minimum) modulus theorem. When a partial differential equation has accompanying auxiliary conditions, which select among all possible solutions, a uniquely determined function, the data is called properly posed, provided that the solution depends continuously on this data. Methods of solution for general computational problems fall into two categories—the direct and iterative procedures. Direct methods, of which the solution of a tridiagonal system is typical, give the exact answer in a finite number of steps, if there were no round-off error. The algorithm for such a procedure is complicated and non-repetitive. Many direct methods for linear systems are available. Iterative methods consist of repeated application of a simple algorithm. They yield the answer as a limit of a sequence, even without consideration of round-off errors.

Abstract: The weighted-mean scheme is a method for constructing finite-difference approximations of second-order partial differential equations of the advection-diffusion type using only the center and adjacent points in each space direction. The scheme tends to a centered-difference formulation for strongly diffusive cases and to an upstream formulation for strongly advective cases. The error of approximation is O (h 2 ) or better, when h tends to zero, and the scheme assures stability and convergence to all iterative methods no matter how large the grid size. The scheme thus makes it possible to choose the biggest grid size suitable for each specific problem thereby reducing the computing time considerably. DOI: 10.1111/j.2153-3490.1977.tb00763.x

Abstract: This paper presents a study of algorithms for searching high dimensional sets and presents a new systematic algorithm for this purpose. The context used for this study (and its original motivation) is the generation of starting points for algorithms to optimize functions of several variables. Such algorithms involve “local” iterative methods, and convergence analysis, etc. assumes that a starting point is sufficiently close to the solution. Such points are not available for many real problems, but rather one knows reasonable bounds on the solution. Thus, a general purpose program for real problems would be based on a polyalgorithm which combines several local methods and a global search method.The current practice is to use random searching. The new algorithm is compared with random searching in three distinct ways. First, four criteria for measuring the dispersion of a point set are discussed and applied. Second, a probabilistic model is developed and used to measure abilities to generate points in certa...

Abstract: A Galerkin finite element formulation of diffusion processes based on a diagonal capacity matrix is analysed from the standpoint of local stability and convergence. The theoretical analysis assumes that the conductance matrix is locally diagonally dominant, and it is shown that one can always construct a finite element network of linear triangles satisfying this condition. Time derivatives are replaced by finite differences, leading to a mixed explicit-implicit system of algebraic equations which can be efficiently solved by a point iterative technique. In this work the accelerated point iterative method is adopted and is shown to converge when the conductance matrix is locally diagonally dominant. Several examples are included in Part II of this paper to demonstrate the efficiency of the new approach.

Abstract: Optimized iteration methods for the solution of large-scale fast reactor finite difference diffusion theory calculations are presented, along with their theoretical basis. The computational and dat...

Abstract: A Block–Stodola eigensolution method is presented for large algebraic eigensystems of the form AU = λBU where A is real but non-symmetric. The steps in this method parallel those of a previous technique for the case when both A and B were real and symmetric. The essence of the technique is simultaneous iteration using a group of trial vectors instead of only one vector as is the case in the classical Stodola–Vianello iteration method. The problem is then transformed into a subspace where a direct solution of the reduced algebraic eigenvalue problem is sought. The main advantage is the significant reduction of computational effort in extracting a subset of eigenvalues and corresponding eigenvectors. Theorems from linear algebra serve to underlie the basis of the present technique. Complex eigendata that emerge during iteration can be handled without doubling the size of the problem. Higher order eigenvalue problems are reducible to first order form for which this technique is applicable. The treatment of the quadratic eigenvalue problem illustrates the details of this extension.

Abstract: Some aspects of the convergence of iterative processes are examined in a general context and a specific iterative process that generalizesStearns' K-transfer schemes is evolved. This yields a simplified proof ofStearns' convergence theorem and an iterative scheme that converges to the nucleolus. Stability and finite convergence properties are shown to hold and various known results on the nucleolus derive as by-products.

Abstract: This paper compares the efficiency of line successive overrelaxation (LSOR) with a two-dimensional correction procedure (2DC), the iterative, alternating direction implicit procedure (ADI), and the strongly implicit procedure (SIP) to solve finite-difference equations used to simulate several groundwater reservoirs. Three of the reservoirs are linear, two are isotropic areal problems, and the third is an anisotropic cross-section simulation. The fourth is a nonlinear water table aquifer with areas of thin saturation. SIP is generally the best method for the linear simulations and with the addition of another iteration parameter is the only method that gives an adequate rate of convergence for the water table problem. LSOR with 2DC is competitive with SIP on isotropic and anisotropic linear problems that are dominated by no-flow boundaries. ADI is generally more efficient than LSOR if a good set of iteration parameters are used, but this advantage is offset by the relative ease of finding the best acceleration parameter for LSOR.

Abstract: It is shown how any combinational function that can be described by a flow table—or equivalently—is realizable in iterative form—can be realized in tree form. The propagation delay is then proportional to the logarithm of n, the number of inputs, while the logic complexity is a linear function of n. These results are related to various implementations of high-speed binary adders and a proposed new high-speed adder circuit.

Abstract: An iterative method for solving the matrix equationXA+AY=F is discussed Algorithms and techniques for accelerating convergence are outlined. The method compares favourably with existing techniques.

Abstract: A combinatorial problem related to storage allocation is analyzed. The problem falls into a class of NP-complete, one-dimensional bin-packing problems. We propose an iterative approximation algorithm and show that it is superior to an earlier heuristic presented for this problem. The bulk of the paper is devoted to the proof of a worst-case performance bound.

Abstract: : This paper presents the Modified Conjugate Residual (MCR) Method, a stabilized version of Luenberger's Method of Conjugate Residuals, for solving large sparse systems of linear equations. This iterative method has special significance when the system is not positive definite so that methods like Conjugate Gradients are inapplicable. In the special case when the system is positive definite, MCR reduces to one of the family of general conjugate gradient methods discussed by Hestenes.

Abstract: Uhrin's iterative method for the computer analysis of electron paramagnetic resonance (E.P.R.) spectra of ions in crystals with crystal-field symmetry, orthorhombic or higher, has been investigated in detail for the case of coincident principal axes of g, crystal field and hyperfine interaction tensors; some new features of the method are presented. The method is found to be self-convergent for the special case of using experimental data for one orientation of the Zeeman field. Using this feature a modification of the method to determine all the crystal-field parameters from the field positions of the allowed transitions for minimum orientations of the Zeeman field is discussed. The computation time needed for the method is much less than that required for the parameter fitting methods. Further, the choice of the magnitudes and signs of the initial parameters is less stringent in comparison to least squares and parameter fitting methods.

Abstract: Optimal output feedback control of linear multivariable systems for quadratic performance indices is considered. An iterative algorithm for computing the optimal feedback gains is proposed that does not require a stabilizing output feedback law for initialization, rather any stabilizing state feedback law suffices. An illustrative numerical example is included.

Abstract: The problem of how to speed up the convergence of currently available iterative methods for transonic flow computations with minimal alterations in computer programing and storage is considered. A cyclic iterative procedure applying nonlinear sequence transformations akin to those of Aitken and Shanks is developed. Based on the "power method," the errors in these sequence transformations are studied. Examples testing the procedure for model Dirichlet problems and for transonic thin airfoil problems show that savings in computer time of a factor of two to five, or more, is generally possible, depending on accuracy requirements and the particular iterative procedure used. I. Introduction M ANY current methods of fluid dynamic computations make use of relaxation procedures. There are several aspects of the computation that considerably limit the usefulness and potentiality of these programs. One is the slow convergence with respect to iterations of the flowfield calculation, and, hence, costly computer time. This paper presents studies on how to speed up the convergence of currently available iterative procedures with minimal alterations in computer programing and storage requirements. To see the need of the convergence acceleration, we may take as an example the finite-difference solution to Dirichlet's problem on a unit square. The convergence rate of practical iterative procedures (Jacob!, Gauss-Seidel, Successive Over

Abstract: M. Henon has demonstrated [1] that the simple deterministic mapping, xk+1=axk+bx2 k-xk-1, possessing a conservation theorem similar to Liouville’s Theorem, can exhibit the same “stochastic” behavior which, in Hamiltonian dynamics, is considered necessary for establishing an “approach to equiLibrium” or “transport laws”. While this mapping is so simple that it could be studied on a pocket-calculator its analytic solution is not available. This is probably due to the fact that it is so successful in imitating the behavior of a “non- integrable” Hamiltonian system of differential equations.

Abstract: In this investigation, a technique to determine effectively an optimum shape of an axisymmetric body is suggested. The technique is a sequential search method consisting of the following two steps. In the first step, the superiority or inferiority of a given shape, which satisfies the design constraints, is judged by the deviation from the stress provided with a design object. In the second step, this given shape is modified by the proportional transformation method of the elements used in the finite element method. By the iteration of these steps the optimum shape will be obtained. As the application of this iterative method the optimum shapes of the thick-walled vessels under internal pressure will be obtained.