Showing papers on "Iterative method published in 1977"
TL;DR: A particular class of regular splittings of not necessarily symmetric M-matrices is proposed, if the matrix is symmetric, this splitting is combined with the conjugate-gradient method to provide a fast iterative solution algorithm.
Abstract: A particular class of regular splittings of not necessarily symmetric M-matrices is proposed. If the matrix is symmetric, this splitting is combined with the conjugate-gradient method to provide a fast iterative solution algorithm. Comparisons have been made with other well-known methods. In all test problems the new combination was faster than the other methods.
1,614 citations
TL;DR: In this paper, an iterative method for solving nonsymmetric linear systems based on the Tchebychev polynomials in the complex plane is discussed, and the iteration is shown to converge whenever the eigenvalues of the linear system lie in the open right half complex plane.
Abstract: In this paper an iterative method for solving nonsymmetric linear systems based on the Tchebychev polynomials in the complex plane is discussed. The iteration is shown to converge whenever the eigenvalues of the linear system lie in the open right half complex plane. An algorithm is developed for finding optimal iteration parameters as a function of the convex hull of the spectrum.
292 citations
TL;DR: In this article, a unified treatment for iterative algorithms for the solution of the symmetric linear complementarity problem is given, which includes, as special cases, extensions of the Jacobi, Gauss-Seidel, and nonsymmetric and symmetric successive over-relaxation methods.
Abstract: A unified treatment is given for iterative algorithms for the solution of the symmetric linear complementarity problem:
$$Mx + q \geqslant 0, x \geqslant 0, x^T (Mx + q) = 0$$
, whereM is a givenn×n symmetric real matrix andq is a givenn×1 vector. A general algorithm is proposed in which relaxation may be performed both before and after projection on the nonnegative orthant. The algorithm includes, as special cases, extensions of the Jacobi, Gauss-Seidel, and nonsymmetric and symmetric successive over-relaxation methods for solving the symmetric linear complementarity problem. It is shown first that any accumulation point of the iterates generated by the general algorithm solves the linear complementarity problem. It is then shown that a class of matrices, for which the existence of an accumulation point that solves the linear complementarity problem is guaranteed, includes symmetric copositive plus matrices which satisfy a qualification of the type:
$$Mx + q > 0 for some x in R^n $$
. Also included are symmetric positive-semidefinite matrices satisfying this qualification, symmetric, strictly copositive matrices, and symmetric positive matrices. Furthermore, whenM is symmetric, copositive plus, and has nonzero principal subdeterminants, it is shown that the entire sequence of iterates converges to a solution of the linear complementarity problem.
285 citations
01 Apr 1977
TL;DR: This paper surveys the state of the art in sparse matrix research in January 1976, and discusses the solution of sparse simultaneous linear equations, including the storage of such matrices and the effect of paging on sparse matrix algorithms.
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.
242 citations
TL;DR: In this paper, the first bi-harmonic problem on general two-dimensional domains was solved using a mixed finite element method, where the continuous problem has been approximated by an appropriate mixed-finite element method.
Abstract: We describe in this report various methods, iterative and "almost direct," for solving the first biharmonic problem on general two-dimensional domains once the continuous problem has been approximated by an appropriate mixed finite element method. Using the approach described in this report we recover some well known methods for solving the first biharmonic equation as a system of coupled harmonic equations, but some of the methods discussed here are completely new, including a conjugate gradient type algorithm. In the last part of this report we discuss the extension of the above methods to the numerical solution of the two dimensional Stokes problem in p- connected domains (p $\geq$ 1) through the stream function-vorticity formulation.
234 citations
TL;DR: In this paper, an iterative prefiltering method is proposed as an approach to estimate the poles and zeros of the vocal-tract transfer function simultaneously rather than sequentially as in Shanks' method.
Abstract: Kopec, Oppenheim, and Tribolet have described a homomorphic technique for producing, from a speech signal, a minimum-phase estimate of the vocal-tract impulse response. Once such an estimate has been obtained, the problem of modeling the vocal tract with a pole-zero model is a classical one in nonlinear estimation theory. It is shown in this paper that Shanks' method, Kalman's method, and the iterative prefiltering method are all different linearizations of the same nonlinear problem, and the iterative prefiltering method is proposed as an approach to estimating the poles and zeros of the vocal-tract transfer function simultaneously. A simulation is described which shows the advantage of estimating poles and zeros simultaneously rather than sequentially as in Shanks' method. A preliminary example of application to real speech is also given.
171 citations
01 Jan 1977
147 citations
TL;DR: In this paper, an interative approach is proposed for the numerical analysis of elastic-plastic continua, which 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.
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.
129 citations
TL;DR: In this paper, 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, which can search an arbitrary n-dimensional rectangle for a safe starting region for a quadratically convergent iterative method.
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.
128 citations
TL;DR: This work establishes the lattice theoric foundations upon which the synthesis of invariant assertions is based and shows that an approximation of the optimal solution to a fixpoint system of equations can be obtained by strengthening the term of a chaotic iteration sequence.
Abstract: The problem of discovering invariant assertions of programs is explored in light of the fixpoint approach in the static analysis of programs, Cousot [1977a], Cousot[1977b].In section 2 we establish the lattice theoric foundations upon which the synthesis of invariant assertions is based. We study the resolution of a fixpoint system of equations by Jacobi's successive approximations method. Under continuity hypothesis we show that any chaotic iterative method converges to the optimal solution. In section 3 we study the deductive semantics of programs. We show that a system of logical forward equations can be associated with a program using the predicate transformer rules which define the semantics of elementary instructions. The resolution of this system of semantic equations by chaotic iterations leads to the optimal invariants which exactly define the semantics of this program. Therefore these optimal invariants can be used for total correctness proofs (section 4). Next we show that usually a system of inequations is used as a substitute for the system of equations. Hence the solutions to this system of inequations are approximate invariants which can only be used for proofs of partial correctness (section 5). In section 6 we show that symbolic execution of programs consists in fact in solving the semantic equations associated with this program. The construction of the symbolic execution tree corresponds to the chaotic successive approximations method. Therefore symbolic execution permits optimal invariant assertions to be discovered provided that one can pass to the limit, that is consider infinite paths in the symbolic execution tree. Induction nrinciDles can be used for that purpose. In section 7 we show how difference equations can be utilized to discover the general term of the sequence of successive approximations so that optimal invariants are obtained by a mere passage to the limit. In section 8 we show that an approximation of the optimal solution to a fixpoint system of equations can be obtained by strengthening the term of a chaotic iteration sequence. This formalizes the synthesis of approximate invariants by heuristic methods. Various examples provide a helpful intuitive support to the technical sections.
121 citations
TL;DR: In this article, a method for accelerating the convergent iterative procedures of solving the system of linear equations X = AX + f is presented, where the number of eigenvalues of A that are greater in absolute value than unity is not very large.
TL;DR: It is proved that the iterative method can produce a solution to the equations in O(N) arithmetical operations where N is the number of unknowns.
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.
01 Jan 1977
TL;DR: 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, this paper is 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: 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.
TL;DR: Using SIRT, direct 3-D reconstruction is shown to be superior to serial 2- D reconstruction from coaxial projections when the range of viewing angles is limited and the concept of tomographlc projections is introduced and shown to greatly simplify the calculations.
TL;DR: In this article, the authors considered the growth of branching processes in random environment and showed that this process either "explodes" at an exponential rate or becomes extinct w.p.1.
Abstract: In this paper, growth of branching processes in random environment is considered. In particular it is shown that this process either "explodes" at an exponential rate or else becomes extinct w.p.1. A classification theorem outlining the cases of "explosion or extinction" is given. To prove these theorems, the associated branching process (the process conditioned on each particle having infinite descent) and the reduced branching process (the particles of the process having infinite descent) are introduced. The method of proof used, in general, is direct probabilistic computation, in contrast with the classical functional iteration method.
TL;DR: The weighted mean scheme as discussed by the authors 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.
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
TL;DR: The Natural Iteration (NI) method for ternary alloys has been proposed in this article for the pair approximation of the cluster-variation method, which converges regardless of the choice of the initial state and the free energy always decreases monotonically as the iteration progresses.
TL;DR: This paper presents a study of algorithms for searching high dimensional sets and presents a new systematic algorithm for this purpose and four criteria for measuring the dispersion of a point set are discussed and applied.
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...
01 Jan 1977
TL;DR: In this paper, the authors consider the problem of finding the solution of an elliptic partial differential equation with auxiliary conditions, which select among all possible solutions, a uniquely determined function, provided that the data is properly posed.
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.
TL;DR: In this paper, a general scheme for the construction of iterative processes for finding approximately the generalized solution x0 of the equation Fx = 0 with a strongly monotonic (possibly discontinuous) operator in Hilbert space is given, and successive approximations xk are shown to convergence to x0 at a rate |χ k − χ 0 |=O (k − 1 2 ).
Abstract: A GENERAL scheme for the construction of iterative processes for finding approximately the generalized solution x0 of the equation Fx = 0 with a strongly monotonic (possibly discontinuous) operator in Hilbert space is given, and the successive approximations xk are shown to convergence to x0 at a rate |χ k − χ 0 |=O (k − 1 2 ) . The iterative processes given in [1, 2], and also in [3], fit into our general scheme; their convergence is proved without assuming that the operator is bounded.
TL;DR: In this article, a Block-Stodola eigensolution method is presented for large algebraic eigenystems of the form AU = λBU where A is real but non-symmetric.
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.
TL;DR: Optimized iteration methods for the solution of large-scale fast reactor finite difference diffusion theory calculations are presented, and the performance of a computer code employing these methods is compared with that of several existing production diffusion theory codes for a range of typical problems.
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...
TL;DR: In this paper, a Galerkin finite element formulation of diffusion processes based on a diagonal capacity matrix is analyzed from the standpoint of local stability and convergence, and the accelerated point iterative method is adopted and is shown to converge when the conductance matrix is locally diagonally dominant.
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.
TL;DR: In this paper, an explanation of the nature of the convergence of a sequence of vectors of the general method of accelerating the iterative process for the solution of a system of linear equations is given.
Abstract: AN EXPLANATION is given of the nature of the convergence of a sequence of vectors of the general method of accelerating the convergence of the iterative process for the solution of a system of linear equations.
TL;DR: In this paper, a general iterative process that generalizes Stearns' K-transfer schemes is proposed and shown to converge to the nucleolus, and stability and finite convergence properties are shown to hold.
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.
TL;DR: In this article, the authors compared 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.
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.
TL;DR: In this paper, an iterative algorithm for solving nonlinear inverse problems in remote sensing of density profiles of a simple ocean model by using acoustic pulses is developed, where the adiabatic sound velocity is assumed to be proportional to the inverse square root of the density.
TL;DR: 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.
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.
TL;DR: It is shown that in most cases the algorithms developed in the paper may be efficiently executed on a parallel processor system.
Abstract: A parallel processor system and its mode of operation are described. A notation for writing programs on it is introduced. Methods for iterative solution of a set of linear equations are then discussed. The well-known algorithms of Jacobi and Gauss–Seidel are parallelized despite the apparent inherent sequentiality of the latter. New, parallel methods for the iterative solution of linear equations are introduced and their convergence is discussed. A measure of speedup is computed for all methods. It shows that in most cases the algorithms developed in the paper may be efficiently executed on a parallel processor system.
TL;DR: In this paper, an iterative method for solving the matrix equation XA+AY=F is discussed, and algorithms and techniques for accelerating convergence are outlined; the method compares favourably with existing techniques.
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.