Showing papers on "Convergence (routing) published in 1968"
TL;DR: Unified theory for convergence results based on nonlinear estimates for iteration function and on majorizing sequences concept is presented in this article, where the convergence results are based on the nonlinear estimate of the iteration function.
Abstract: Unified theory for convergence results based on nonlinear estimates for iteration function and on majorizing sequences concept
173 citations
TL;DR: The method is a variation of Newton's method incorporating Gaussian elimination in such a way that the most recent information is always used at each step of the algorithm, and it is proved that the iteration converges locally and that the convergence is quadratic in nature.
Abstract: In this paperwe present an iterative method for the numerical solution of(1.1). The method is a variation ofNewton'smethod incorporating Gaussian elimination in such a way that the most recent information is always used at each step of the algorithm. After specifying the method in terms of an iteration function, we prove that the iteration converges locally and that the convergence is quadratic in nature. Computer results are given and a comparison is made with Newton's method; these results illustrate the effectiveness of the method for nonlinear systems containing linear or mildly nonlinear equations. 2. Notation. We shall introduce most of the notation as needed;however
171 citations
01 Jan 1968
170 citations
TL;DR: In this paper, a new empirical size-dependent growth rate model is proposed which has properties superior to those of previous models, and experimental steady state data are presented to illustrate the application of the model to actual CMSMPR crystallization systems.
Abstract: Empirical size-dependent growth rate models are studied for their effect on the population density distributions from a continuous, mixed suspension, mixed product removal (CMSMPR) crystallizer. The growth rate models and/or their corresponding population density distributions are examined for continuity, convergence of moments, versatility, and their ability to fit experimental data.
A new empirical size-dependent growth rate model is proposed which has properties superior to those of previous models. Experimental steady state data are presented to illustrate the application of the model to actual CMSMPR crystallization systems.
131 citations
80 citations
TL;DR: The efficiency of learning for an m-state automaton in terms of expediency and convergence, under two distinct types of reinforcement schemes: one based on penalty probabilities and the other on penalty strengths, is discussed.
Abstract: A stochastic automaton responds to the penalties from a random environment through a reinforcement scheme by changing its state probability distribution in such a way as to reduce the average penalty received. In this manner the automaton is said to possess a variable structure and the ability to learn. This paper discusses the efficiency of learning for an m-state automaton in terms of expediency and convergence, under two distinct types of reinforcement schemes: one based on penalty probabilities and the other on penalty strengths. The functional relationship between the successive probabilities in the reinforcement scheme may be either linear or nonlinear. The stability of the asymptotic expected values of the state probability is discussed in detail. The conditions for optimal and expedient behavior of the automaton are derived. Reduction of the probability of suboptimal performance by adopting the Beta model of the mathematical learning theory is discussed. Convergence is discussed in the light of variance analysis. The initial learning rate is used as a measure of the overall convergence rate. Learning curves can be obtained by solving nonlinear difference equations relating the successive expected values. An analytic expression concerning the convergence behavior of the linear case is derived. It is shown that by a suitable choice of the reinforcement scheme it is possible to increase the separation of asymptotic state probabilities.
61 citations
Journal Article•
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TL;DR: In this paper, an equivalent relationship has been established between the problems of convergence or zero-input convergence of nonlinear systems and the problem of zero input convergence of linear systems, where the formulation of the problem and the analysis are done directly on a component-connection model instead of the usual input-output state model.
Abstract: An equivalent relationship has been established between the problems of convergence or zero-input convergence of nonlinear systems and the problem of zero-input convergence of linear systems. A special feature is that the formulation of the problem and the analysis are done directly on a component-connection model instead of the usual input-output state model. As a consequence, the results obtained can be easily interpreted in terms of given practical systems.
56 citations
TL;DR: In this article, the authors considered the convergence of the QR algorithm to Hessenberg matrices and showed that the Hessenberg form ion = 0, i > j + 1 is invariant under the QR transformation.
Abstract: 0. Introduction. The QR algorithm was developed by Francis (1960) to find the eigenvalues (or roots) of real or complex matrices. We shall consider it here in the context of exact arithmetic. Sufficient conditions for convergence, listed in order of increasing generality have been given by Francis [1], Kublanovskaja [3], Parlett [4], and Wilkinson [8]. It seems that necessary and sufficient conditions would be very complicated for a general matrix. One of the many merits of Francis' paper was the observation that the Hessenberg form ion = 0, i > j + 1) is invariant under the QR transformation and the algorithm is usually applied to Hessenberg matrices which are unreduced, that is Oij 9a 0, i = j + 1. The properties of this form combine with those of the algorithm in such a way that a complete convergence theory can be stated quite simply. The aim is to produce a sequence of unitarily similar matrices whose limit is upper triangular. Elementwise convergence to a particular triangular matrix is not necessary for determining eigenvalues; block triangular form with 1X1 and 2X2 blocks on the diagonal is sufficient. Definition. A sequence {77(s) = (A(*yO> s = 1, 2, • • • } of n X n Hessenberg matrices is said to \"converge\" whenever hf+xjh^'/j-x —* 0, for each,/ = 2, ■ • -, n — 1. Theorem 1. The basic QR algorithm applied to an unreduced Hessenberg matrix 77 produces a sequence of Hessenberg matrices which \"converges\" if, and only if, among each set of H's eigenvalues with equal magnitude, there are at most two of even and two of odd multiplicity. This is a special case, tailored to computer programs, of the main theorem. In general let coi > o>2 > • • • > o>r > 0 be the distinct nonzero magnitudes occurring among the roots of H. Of the roots of magnitude o>¿ let pQ) have even multiplicities
52 citations
TL;DR: In this paper, the spectrum of density fluctuations in a simple gas is calculated by solving the linearized Boltzmann equation as an initial value problem, based on the method of polynomial expansion and on the use of generalized kinetic models.
Abstract: The spectrum of density fluctuations in a simple gas is calculated by solving the linearized Boltzmann equation as an initial‐value problem. The analysis is based on the method of polynomial expansion and on the use of generalized kinetic models. The numerical convergence of both types of solutions is studied, and it is shown that the method of kinetic models is capable of giving very accurate solutions to the Boltzmann equation at any wavelength to mean‐free‐path ratio. Explicit results are obtained for two repulsive interactions, the rigid‐sphere potential and the Maxwell molecule potential. It is found that density fluctuations are not very sensitive to the details of the repulsive part of the intermolecular interaction.
37 citations
TL;DR: In this paper, the authors generalize this result to general dissipative approximations, using a completely different technique, and give a fairly complete classification of all stable difference approximation according to the influence which the influence has on the convergence.
Abstract: Assume that the approximation (1.3) is stable. What can we say about the convergence of v,(t) towards u(x, t), as h -> O? For two special cases this question has been answered in an interesting paper by S. Parter [1]. He has shown that the estimates of Theorem 1 hold for the LaxWendroff scheme and the Friedrichs scheme. We want to generalize this result to general dissipative approximations, using a completely different technique. Furthermore, we shall give a fairly complete classification of all stable difference approximations according to the influence which the
36 citations
TL;DR: In this article, the degenerate case an = 0 for certain values of n is considered and the value Sn(0) is called the nth approximant and the numbers an and frw are referred to as the elements of the continued fraction.
Abstract: (1.3) 5 „ ( » ) =S n _i(0) , w è 2. A continued fraction is a sequence of constants {5n(0)} obtained from a c.f.g, sequence {Sn(z)}. We shall subsequently extend this definition to allow for the degenerate case an = 0 for certain values of n. The value Sn(0) is called the nth approximant and the numbers an and frw are referred to as the elements of the continued fraction. To exhibit the elements explicitly we write
01 Jan 1968
TL;DR: In this paper, a new examination of the theory of pseudo-potentials, necessitated by the discovery that smoothness of the pseudo-wave function is not a relevant criterion for their effectiveness, provides an understanding of the analytic properties of pseudopotentials in terms of their pseudo-core energies Ecprime.
Abstract: A new examination of the theory of pseudo-potentials, necessitated by the discovery that smoothness of the pseudo-wave function is not a relevant criterion for their effectiveness, provides an understanding of the analytic properties of pseudo-potentials in terms of their pseudo-core energies Ecprime. In particular, it is found for Hermitian pseudo-potentials that, if Ecprime greater, similar , the Born series cannot have good convergence properties, but, for Ecprime = 0, the series has good asymptotic convergence. The new pseudo-potential, with Ecprime = 0, is the correct form to use at all energies, but differs substantially from other forms only at higher energies. Its effectiveness is demonstrated by a model calculation.
TL;DR: The problem of identifying a linear discrete system is considered where the input-output data is noise-corrupted and an iterative algorithm is suggested which converges in a statistical metric through the principle of random contraction mapping.
Abstract: The problem of identifying a linear discrete system is considered where the input-output data is noise-corrupted. An iterative algorithm is suggested which converges in a statistical metric. This convergence is obtained through the principle of random contraction mapping.
TL;DR: ChERTOCK as discussed by the authors proposed and illustrated a more general iterative scheme than the classical Neumann series and derived conditions for its convergence, and showed that the convergence of such a scheme is possible.
Abstract: A preferred method of solving the integral equation for sound radiation problems is by some iterative method. This paper proposes and illustrates a more general iterative scheme than the classical Neumann series and derives conditions for its convergence. I Il I II I 111 1 eg Reprinted from QUARTERLY OF APPLIED MATHEIMATICS Vol. XXVI, No. 2, July 1968 CONVERGENCE OF ITERATIVE SOLUTIONS TO INTEGRAL EQUATIONS FOR SOUND RADIATION* BY GEORGE CHERTOCK, (Naval Ship Research and Development Center, Washington, D. C.)
TL;DR: In this paper, a procedure for computing the Marcum Q function by means of Laguerre polynomials is proposed. And a recursive formula is given, and the results of some tests of convergence are also indicated.
Abstract: A procedure is suggested to compute the Marcum Qfunction by means of Laguerre polynomials; a recursive formula is given, and the results of some tests of convergence are also indicated.
TL;DR: In this article, it was shown that an earlier result of Ezeilo [1] can be extended to the more general equation (1.1), and this is achieved without any extra conditions on h′(x) and h″(x).
Abstract: In this paper it is shown that an earlier result of Ezeilo [1] can be extended to the more general equation (1.1), and this is achieved without any extra conditions on h′(x) and h″(x). A result on the existence of a unique periodic solution of (1.1) is also obtained as an application of the convergence result.
TL;DR: In this article, the authors investigated the rate of convergence of a version of the method of intermediate problems, which enables the eigenvalues sought to be confined within arbitrarily narrow intervals so that guaranteed results are obtained.
Abstract: IT is well-known (see, for example, [1]), that Ritz's method gives upper bounds for the eigenvalues of a selfadjoint positive definite operator with a discrete spectrum. In some cases lower limits for the eigenvalues of such an operator can be obtained by the intermediate problem method [2–6], The joint application of both methods enables the eigenvalues sought to be confined within arbitrarily narrow intervals so that guaranteed results are obtained. This paper investigates the rate of convergence of a version of the method of intermediate problems. This version was first proposed by Aronszajn [3], but we will confine ourselves to a later treatment of it given in [5]. A brief description of this version, necessary for an understanding of the subsequent reasoning, is given in section 1. More detailed information can be obtained in [5]. An estimate of the rate of convergence is obtained in Section 2. In the derivation of this estimate we have investigated some methods which are applied in [7, 8] in the investigation of the convergence of projection methods.
TL;DR: In this article, the convergence of R. Grammel's method for the solution of boundary-eigenvalue-problems is discussed. And it is further shown that GMM can be extended to the case where it is difficult to find the exactGreen function.
Abstract: The paper deals with the convergence ofR. Grammel's method for the solution of boundary-eigenvalue-problems. It is further shown thatGrammel's method can be extended to the case where it is difficult to find the exactGreen function. In this case one may use a different, ‘related’Green function if one adds certain supplemental terms to the properGrammel equations.