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Showing papers on "Rate of convergence published in 1983"




Journal ArticleDOI
TL;DR: A new method is presented which, at each iteration, computes a direction of search by solving the Newton system of equations, projected, if necessary, into a linear manifold along which F is locally differentiable, and has quadratic convergence to a solutionx under given conditions.
Abstract: We consider the problem of minimizing a sum of Euclidean norms. $$F(x) = \sum olimits_{i = 1}^m {||r_i } (x)||$$ here the residuals {r i(x)} are affine functions fromR n toR 1 (n≥1≥2,m>-2). This arises in a number of applications, including single-and multi-facility location problems. The functionF is, in general, not differentiable atx if at least oner i (x) is zero. Computational methods described in the literature converge quite slowly if the solution is at such a point. We present a new method which, at each iteration, computes a direction of search by solving the Newton system of equations, projected, if necessary, into a linear manifold along whichF is locally differentiable. A special line search is used to obtain the next iterate. The algorithm is closely related to a method recently described by Calamai and Conn. The new method has quadratic convergence to a solutionx under given conditions. The reason for this property depends on the nature of the solution. If none of the residuals is zero at* x, thenF is differentiable at* x and the quadratic convergence follows from standard properties of Newton's method. If one of the residuals, sayr i * x), is zero, then, as the iteration proceeds, the Hessian ofF becomes extremely ill-conditioned. It is proved that this illconditioning, instead of creating difficulties, actually causes quadratic convergence to the manifold (xℛr i (x)=0}. If this is a single point, the solution is thus identified. Otherwise it is necessary to continue the iteration restricted to this manifold, where the usual quadratic convergence for Newton's method applies. If several residuals are zero at* x, several stages of quadratic convergence take place as the correct index set is constructed. Thus the ill-conditioning property accelerates the identification of the residuals which are zero at the solution. Numerical experiments are presented, illustrating these results.

113 citations


Journal ArticleDOI
TL;DR: In this article, a regionS of the complex plane is determined such that a two-step iterative method converges if the eigenvalues of an iteration operatorT are contained inS. For a givenS, optimal methods are described, and upper and lower bounds are derived for the associated asymptotic rate of convergence.
Abstract: Using the theory of Euler methods from summability theory, we investigate general iterative methods for solving linear systems of equations. In particular, for a given Euler method, a regionS of the complex plane is determined such that ak-step iterative method converges if the eigenvalues of an iteration operatorT are contained inS. For a givenS, optimal methods are described, and upper and lower bounds are derived for the associated asymptotic rate of convergence. Special attention is given to two-step methods with complex parameters.

89 citations


Journal ArticleDOI
TL;DR: In this article, Newton's method is analyzed in the neighborhood of irregular singularities, which include all minimizers at which the Hessian has a one-dimensional null space, and it is shown that depending on a parameter specific to the underlying optimization problem or system of simultaneous equations, the method is either to converge with a limiting ratio of about $\frac{2} {3}$, or to diverge from arbitrarily close starting points, or to behave in a certain sense chaotically.
Abstract: Recent work on Newton’s method at singularities of the Jacobian has established linear convergence under certain regularity assumptions. Here, Newton’s method is analyzed in the neighborhood of irregular singularities which include all minimizers at which the Hessian has a one-dimensional null space. Depending on a parameter specific to the underlying optimization problem or system of simultaneous equations, Newton’s method is found either to converge with a limiting ratio of about $\frac{2} {3}$, or to diverge from arbitrarily close starting points, or to behave in a certain sense chaotically.

86 citations


Journal ArticleDOI
TL;DR: In this paper, a variation bornee en termes de moyenne arithmetique de la suite des variations totales is presented. But this variation is not a polynome de Bernstein de f.

83 citations


01 Jan 1983
TL;DR: This paper considers a variant of the Byzantine Generals problem, in which processes start with arbitrary real values rather than Boolean values or values from some bounded range, and in which approximate, rather than exact, agreement is the desired goal.
Abstract: This paper considers a variant of the Byzantine Generals problem, in which processes start with arbitrary real values rather than Boolean values or values from some bounded range, and in which approximate, rather than exact, agreement is the desired goal. Algorithms are presented to reach approximate agreement in asynchronous, as well as synchronous systems. The asynchronous agreement algorithm is an interesting contrast to a result of Fischer et al, who show that exact agreement with guaranteed termination is not attainable in an asynchronous system with as few as one faulty process. The algorithms work by successive approximation, with a provable convergence rate that depends on the ratio between the number of faulty processes and the total number of processes. Lower bounds on the convergence rate for algorithms of this form are proved, and the algorithms presented are shown to be optimal.

71 citations


Journal ArticleDOI
TL;DR: In this paper, the optimal rate of convergence estimates for both semidiscrete and second order in time fully discrete schemes were obtained for the Korteweg-de Vries equation.
Abstract: Standard Galerkin approximations, using smooth splines on a uniform mesh, to 1-periodic solutions of the Korteweg-de Vries equation are analyzed. Optimal rate of convergence estimates are obtained for both semidiscrete and second order in time fully discrete schemes. At each time level, the resulting system of nonlinear equations can be solved by Newton's method. It is shown that if a proper extrapolation is used as a starting value, then only one step of the Newton iteration is required.

65 citations


Journal ArticleDOI
TL;DR: The rate of convergence of the transfer matrix finite-size scaling (or phenomenological renormalization) method is studied in this article, where it is shown both heuristically and numerically that the convergence of estimates for exponents, etc, is governed asymptotically by the leading irrelevant-variable scaling exponent.
Abstract: The rate of convergence of the transfer matrix finite-size scaling (or phenomenological renormalisation) method is studied. It is shown both heuristically and numerically that the convergence of estimates for exponents, etc, is governed asymptotically by the leading irrelevant-variable scaling exponent. The more rapid apparent convergence rates observed in many practical calculations for two-dimensional lattices of widths up to ten lattice spacings are attributed to cancellation between various correction terms.

53 citations


Journal ArticleDOI
TL;DR: The concept of the sampling window is introduced for the central interpolation of finite energy band-limited functions and does significantly reduce the truncation-error bound.
Abstract: The concept of the sampling window is introduced for the central interpolation of finite energy band-limited functions. The sampling window does not increase the rate of convergence of the truncation error series, as do various convergence factors, but does significantly reduce the truncation-error bound.

47 citations


Book ChapterDOI
01 Jan 1983
TL;DR: In this paper, the authors discuss the optimal uniform rate of convergence for nonparametric estimators of a density function or its derivatives, which is defined as the lower and upper bound of the convergence rate.
Abstract: Publisher Summary This chapter discusses optimal uniform rate of convergence for the nonparametric estimators of a density function or its derivatives. It describes the optimal uniform rate of convergence of an arbitrary estimator of T(f) for various choices of F. It is called the optimal rate of convergence, if it is both a lower and an achievable rate of convergence.

Proceedings ArticleDOI
01 Jan 1983
TL;DR: In this paper, a finite-volume formulation of the mass conservation equation for two-dimensional and axisymmetric flows is presented. But the method is based on a finite volume formulation of a Cartesian coordinate system, and is an extension of the method of Purvis and Burkhalter (1979).
Abstract: A method is developed for solving the full-potential equation for two-dimensional and axisymmetric flow which retains the grid and boundary condition simplicity of the transonic small-disturbance codes. The method is based on a finite-volume formulation of the mass conservation equation in a Cartesian coordinate system, and is an extension of the method of Purvis and Burkhalter (1979). This finite-volume approach, combined with the simple boundary treatment, is shown to result in a highly robust method applicable to a wide range of geometries and flow conditions. The accuracy of the method is demonstrated for general geometries in two-dimensional and axisymmetric flows. The use of this method results in significant gains in convergence rate over the vertical-line over-relaxation scheme by incorporating an AF2-type algorithm (Ballhaus et al., 1978). It is suggested that the simplicity of this method shold allow a relatively easy extension to complex geometries in three-dimensional flows, and complex two-dimensional configurations such as multielement airfoils should be amenable to this method.

Proceedings ArticleDOI
01 Jan 1983
TL;DR: In this article, the convergence rate of modal analysis solutions of step-type waveguide discontinuity problems is studied, and convergence rate depends on the ratio between the modal terms retained in different regions.
Abstract: Convergence of modal analysis solutions of step-type waveguide discontinuity problems is studied. The convergence rate depends on the ratio between the number of modal terms retained in different regions. Guidelines for accurate and efficient computations are indicated.

Journal ArticleDOI
TL;DR: It is shown that nice proofs of convergence and asymptotic expansions are known for one-step methods for ordinary differential equations can be generalized in a natural way to “extended” one- step methods for Volterra integral equations of the second kind.
Abstract: Nice proofs of convergence and asymptotic expansions are known for one-step methods for ordinary differential equations. It is shown that these proofs can be generalized in a natural way to “extended” one-step methods for Volterra integral equations of the second kind. Furthermore, the convergence of “mixed” one-step methods is investigated. For both types general Volterra–Runge–Kutta methods are considered as examples.

Journal ArticleDOI
TL;DR: In this article, it was shown that for m-dependent point processes, the bounds on the rate of convergence of the point processes of exceedances of one or several levels are roughly of the order where ρ is the maximal correlation, ρ =sup {0, r 1, r 2, …}.
Abstract: Let {ξ; t = 1, 2, …} be a stationary normal sequence with zero means, unit variances, and covariances let be independent and standard normal, and write . In this paper we find bounds on which are roughly of the order where ρ is the maximal correlation, ρ =sup {0, r 1 , r 2, …}. It is further shown that, at least for m-dependent sequences, the bounds are of the right order and, in a simple example, the errors are evaluated numerically. Bounds of the same order on the rate of convergence of the point processes of exceedances of one or several levels are obtained using a ‘representation' approach (which seems to be of rather wide applicability). As corollaries we obtain rates of convergence of several functionals of the point processes, including the joint distribution function of the k largest values amongst ξ1, …, ξn.

Journal ArticleDOI
TL;DR: A block-by-block method based on interpolatory quadrature rules is applied to delay Volterra integrodifferential equations with variable delay and an error bound is obtained and a rate of convergence is found.
Abstract: A block-by-block method based on interpolatory quadrature rules is applied to delay Volterra integrodifferential equations with variable delay. An error bound is obtained and a rate of convergence is found. Our numerical results are compared with those obtained by applying Neves's [19] algorithm and a cyclic linear multistep method of McKee [17]; they are also compared with those presented in Kemper [11].

Journal ArticleDOI
TL;DR: In this article, the authors proposed a three-step method with convergence rate 10.81525 which is much better than the six-order method of the Neta family of methods.
Abstract: Neta's three step sixth order family of methods for solving nonlinear equations require 3 function and 1 derivative evaluation per iteration. Using exactly the same information another three step method can be obtained with convergence rate 10.81525 which is much better than the sixth order.

Journal ArticleDOI
TL;DR: In this article, a hybrid numerical technique is developed for the treatment of axisymmetric unsteady spray equations, where an Eulerian mesh is employed for the parabolic gas-phase subsystem of equations while a Lagrangian scheme (or method of characteristics) is utilized for the droplet equations.

Journal ArticleDOI
TL;DR: A new procedure for direct minimization of the RHF energy is presented, which presents advantages over SCF methods with respect to convergence rate and computational cost.
Abstract: We present a new procedure for direct minimization of the RHF energy, which presents advantages over SCF methods with respect to convergence rate and computational cost. In this procedure we combine several techniques with the aim of obtaining best directions and step lengths for the iterative search for a minimum of the energy. In this article we develop the theory. Therefore, we analyze the variational function; we present a short description of the minimization techniques and we discuss in detail the way in which they are to be used. The computational aspects of the procedure will be treated in the following article.

Journal ArticleDOI
TL;DR: In this paper, the energy relaxation of energetic electrons is determined with a moment method solution of the hard sphere Lorentz-Fokker-Planck equation, and the convergence of the expansion of the electron speed distribution function in speed polynomials is rapid.
Abstract: The energy relaxation of energetic electrons is determined with a moment method solution of the hard sphere Lorentz–Fokker–Planck equation. The convergence of the expansion of the electron speed distribution function in speed polynomials is rapid and the energy relaxation can be written explicitly as a sum of a small number of exponential terms characterized by the lowest eigenvalues of the Lorentz–Fokker–Planck equation. The decay of the directed velocity is examined with a discrete ordinate method. The rate of convergence vs the number of quadrature points is extremely rapid and for particular values of the initial electron energy, an exact result can be obtained.

Journal ArticleDOI
TL;DR: Crisfield's method is a geometrically appealing variant of Newton's method used to track equilibrium curves in nonlinear structural analysis as discussed by the authors, and a rigorous proof of quadratic convergence in one dimension is given, and numerical results for a complicated 21-dimensional lamella dome problem.

Journal ArticleDOI
TL;DR: In this paper, it was shown that Kublanovskaya's algorithm does not constitute a Newton iteration, though it can be justified through its equivalence with the generalized Rayleigh quotient.
Abstract: In SIAM J. Numer. Anal., 7 (1970), pp. 532–537, Kublanovskaya proposed a quadratically convergent algorithm for computing the eigenvalues of the generalized latent value problem for $\lambda $-matrices. In this contribution we show that Kublanovskaya’s algorithm does not constitute a Newton iteration, though it can be justified through its equivalence with the generalized Rayleigh quotient. An alternative iteration scheme that can provide a higher order of convergence is proposed.

Journal ArticleDOI
TL;DR: In this article, a coarse-grained information propagation method is used to accelerate the convergence of a simple, explicit fine-grid solution procedure on a sequence of successively coarser grids.

Journal ArticleDOI
TL;DR: The existence of local a posteriori error indicators for the p-version of the finite element method is demonstrated and it is shown that it is possible to construct reliable error indicators from the residuums and tractions which have the same rate of convergence as the strain energy.
Abstract: The existence of local a posteriori error indicators for the p-version of the finite element method is demonstrated through numerical examples. It is shown that it is possible to construct reliable error indicators from the residuums and tractions which have the same rate of convergence as the strain energy. The error indicators contribution of an element can be estimated by considering only the element and its immediate neighbors. The optimal sequence of p-distributions can be closely followed by the indicators even when there are only a very few elements, the elements vary greatly in size, the polynomial orders vary, the mesh grading is poor, and the smoothness indices and relative errors vary between wide limits.


Journal ArticleDOI
TL;DR: In this article, the authors give sufficient conditions for convergence of the chord method for a class of singular problems and show that the rate of convergence is sublinear in the presence of noise.
Abstract: We give sufficient conditions for convergence of the chord method for a class of singular problems. The rate of convergence is sublinear.

Journal ArticleDOI
TL;DR: In this paper, the authors applied fourth-order compact differencing to the steady solution of two-dimensional viscous incompressible flows at moderate Reynolds number, where the physical region where the fluid flow occurs is mapped onto a rectangle by means of the boundary-fitted coordinates transformation method.

Book ChapterDOI
TL;DR: In this paper, the authors discuss the convergence of the augmented Lagrangian method in quadratic programming and show that the convergence can be achieved in a finite number of iterations.
Abstract: Publisher Summary This chapter discusses the augmented Lagrangian methods in quadratic programming The advantage of the augmented Lagrangian is that, because of the presence of the term (q,Bv), the exact Solution of problem can be determined without making r tend to infinity, unlike ordinary penalization methods where this has the effect of causing deterioration in the conditioning of the systems to be solved The conjugate gradient method is especially attractive for solving quadratic problems because theoretically it converges in a finite number of iterations (≤M) and because moreover in the general case it leads to quadratic convergence It would be too lengthy and inappropriate to study here the convergence of this method For M "large" quadratic convergence becomes a greater attraction than convergence in a finite number of iterations

Journal ArticleDOI
TL;DR: This study leads to a monoparametric family of second order schemes of de Pillis' type and a method of selecting the optimal one is presented, which effectively shows the superiority of the optimal second order scheme over the fastest first order ones like the SOR and AOR schemes.
Abstract: This paper analyzes and studies the second order scheme of de Pillis' type, which is used for the solution of a linear system. This study leads to a monoparametric family of second order schemes of the aforementioned type and a method of selecting the optimal one is presented. In addition a number of concluding remarks is made and various applications and examples are given, which effectively show in some cases the superiority of our optimal second order scheme over the fastest first order ones like the SOR and AOR schemes. Many points are also made for the possibility of improving on the convergence rates of the optimal scheme of this paper, which suggest further research in this area.

Journal ArticleDOI
TL;DR: In this paper, a double iterative scheme was proposed to estimate the minimal eigenvalue of large sparse positive definite matrices, arising from the finite element integration of flow and elasticity equations.