On the convergence of Wang-Zheng's method

The choice of initial conditions ensuring safe convergence of the implemented iterative method is one of the most important problems in solving polynomial equations. These conditions should depend only on the coefficients of a given polynomial P and initial approximations to the zeros of P. In this paper we state initial conditions with the described properties for the Wang-Zheng method for the simultaneous approximation of all zeros of P. The safe convergence and the fourth-order convergence of this method are proved.     

On the convergence of the midpoint method

The midpoint method is an iterative method for the solution of nonlinear equations in a Banach space. Convergence results for this method have been studied in [3, 4, 9, 12]. Here we show how to improve and extend these results. In particular, we use hypotheses on the second Fréchet derivative of the nonlinear operator instead of the third-derivative hypotheses employed in the previous results and we obtain Banach space versions of some results that were derived in [9, 12] only in the real or complex space. We also provide various examples that validate our results.      

On the convergence of the MAOR method

In order to solve a linear system Ax = b, Hadjidimos et al. (1992) defined a class of modified AOR (MAOR) method, whose special case implies the MSOR method. In this paper, some sufficient and/or necessary conditions for convergence of the MAOR and MSOR methods will be achieved, when A is a two-cyclic matrix and when A is a Hermitian positive-definite matrix, an H-, L- or M-matrix, and a strictly or irreducibly diagonally dominant matrix. The convergence results on the MSOR method are better than some known theorems. The optimum parameters and the optimum spectral radii of the MAOR and MSOR methods are obtained, which also answers the open problem given by Hadjidimos et al.     

On the cubic convergence of the Paardekooper method

We show a simple way how asymptotic convergence results can be conveyed from a simple Jacobi method to a block Jacobi method. Our pilot methods are the well known symmetric Jacobi method and the Paardekooper method for reducing a skew-symmetric matrix to the real Schur form. We show resemblance in the quadratic and cubic convergence estimates, but also discrepances in the asymptotic assumptions. By numerical tests we confirm that our asymptotic assumptions for the Paardekooper method are most general.     

On the convergence of the projected gradient method

We study the projected gradient algorithm for linearly constrained optimization. Wolfe (Ref. 1) has produced a counterexample to show that this algorithm can jam. However, his counterexample is only ¹( ⁿ ), and it is conjectured that the algorithm is convergent for ²-functions. We show that this conjecture is partly right. We also show that one needs more assumptions to prove convergence, since we present a family of counterexamples. We finally give a demonstration that no jamming can occur for quadratic objective functions.This work was supported by the Natural Sciences and Engineering Research Council of Canada     

On the linear convergence of the circumcentered-reflection method

In order to accelerate the Douglas–Rachford method we recently developed the circumcentered-reflection method, which provides the closest iterate to the solution among all points relying on successive reflections, for the best approximation problem related to two affine subspaces. We now prove that this is still the case when considering a family of finitely many affine subspaces. This property yields linear convergence and incites embedding of circumcenters within classical reflection and projection based methods for more general feasibility problems.     

On the convergence of the modified overrelaxation method

The modified overrelaxation (MSOR) method is applied to a linear system Ax=b, where A has property A. We get bounds for the spectral radius of the iteration matrix of this method, and we achieve convergence conditions for the MSOR method when A is strictly diagonally dominant. We extend our conclusions to another kind of matrices—H,L,M or Stieltjes. In the last section we use the vectorial norms, getting convergence conditions for the MSOR method, when A is a block-H matrix. We also generalize a theorem of Robert's for this kind of matrices.     

On the convergence of the cyclic Jacobi method

In this paper we develop a class of cyclic orderings and prove that the cyclic Jacobi method using any cyclic ordering from this class converges to the eigensystem of a symmetric matrix.     

On the convergence of the LJ search method

The convergence of the Luus-Jaakola search method for unconstrained optimization problems is established.Notation E _n Euclideann-space - f Gradient off(x) - ² f Hessian matrix - (·) ^T Transpose of (·) - I Index set {1, 2, ...,n} - [x _i1 ^*(j) ] Point around which search is made in the (j + 1)th iteration, i.e., [x _1l ^*(j) ,x _2l ^*(j) ,...,x _n1 ^*(j) ] - r _i ⁽ⁱ⁾ Range ofx _il ^*(i) in the (j + 1)th iteration - l ₁ min_i {r _i ⁽⁰⁾ } - l ₂ min_i {r _i ⁽⁰⁾ } - A _j Region of search in thejth iteration, i.e., {x E _n:x _il ^*(j-1) –0.5r _i ^(j-1) x _ix _il ^*(j-1) +0.5r _i ^(j-1) ,i I} - S _j Closed sphere with center origin and radius _j - Reduction factor in each iteration - 1– - (·) Gamma function Many discussions with Dr. S. N. Iyer, Professor of Electrical Engineering, College of Engineering, Trivandrum, India, are gratefully acknowledged. The author has great pleasure to thank Dr. K. Surendran, Professor, Department of Electrical Engineering, P.S.G. College of Technology, Coimbatore, India, for suggesting this work.     

On the order of convergence of Adomian method

In this paper, we contemplate the order of convergence of the Decomposition method, and we apply the results to some problems.     

On the convergence of He's variational iteration method

In this work we will consider He's variational iteration method for solving second-order initial value problems. We will discuss the use of this approach for solving several important partial differential equations. This method is based on the use of Lagrange multipliers for identification of optimal value of a parameter in a functional. This procedure is a powerful tool for solving the large amount of problems. Using the variational iteration method, it is possible to find the exact solution or an approximate solution of the problem. This technique provides a sequence of functions which converges to the exact solution of the problem. Our emphasis will be on the convergence of the variational iteration method. In the current paper this scheme will be investigated in details and efficiency of the approach will be shown by applying the procedure on several interesting and important models.     

On the global convergence of Chebyshev's iterative method

In [A. Melman, Geometry and convergence of Euler's and Halley's methods, SIAM Rev. 39(4) (1997) 728–735] the geometry and global convergence of Euler's and Halley's methods was studied. Now we complete Melman's paper by considering other classical third-order method: Chebyshev's method. By using the geometric interpretation of this method a global convergence theorem is performed. A comparison of the different hypothesis of convergence is also presented.     

On the convergence of a dual-primal substructuring method

Summary.   In the Dual-Primal FETI method, introduced by Farhat et al. [5], the domain is decomposed into non-overlapping subdomains, but the degrees of freedom on crosspoints remain common to all subdomains adjacent to the crosspoint. The continuity of the remaining degrees of freedom on subdomain interfaces is enforced by Lagrange multipliers and all degrees of freedom are eliminated. The resulting dual problem is solved by preconditioned conjugate gradients. We give an algebraic bound on the condition number, assuming only a single inequality in discrete norms, and use the algebraic bound to show that the condition number is bounded by for both second and fourth order elliptic selfadjoint problems discretized by conforming finite elements, as well as for a wide class of finite elements for the Reissner-Mindlin plate model. Received January 20, 2000 / Revised version received April 25, 2000 / Published online December 19, 2000     

On the convergence of an inexact Newton-type method

In this paper we give local convergence results of an inexact Newton-type method for monotone equations under a local error bound condition. This condition may hold even for problems with non-isolated solutions, and it therefore is weaker than the standard non-singularity condition.     

On the convergence and divergence of Bairstow's method

Summary Necessary and sufficient condition of algebraic character is given for the invertibility of the Jacobian matrix in Bairstow's method. This leads to a sufficient condition for local quadratic convergence. Results also yield the rank of the Jacobian, when it is singular. In the second part of the paper we give several examples for two step cyclization and more complicated kind of divergence. Some of the polynomials in divergence examples are proved to be irreducible over the field of rational numbers.     

On the convergence of the Newton/log-barrier method

In the Newton/log-barrier method, Newton steps are taken for the log-barrier function for a fixed value of the barrier parameter until a certain convergence criterion is satisfied. The barrier parameter is then decreased and the Newton process is repeated. A naive analysis indicates that Newton’s method does not exhibit superlinear convergence to the minimizer of each instance of the log-barrier function until it reaches a very small neighborhood, namely within O(μ²) of the minimizer, where μ is the barrier parameter. By analyzing the structure of the barrier Hessian and gradient in terms of the subspace of active constraint gradients and the associated null space, we show that this neighborhood is in fact much larger –O(μ^σ) for any σ∈(1,2] – thus explaining why reasonably fast local convergence can be attained in practice. Moreover, we show that the overall convergence rate of the Newton/log-barrier algorithm is superlinear in the number of function/derivative evaluations, provided that the nonlinear program is formulated with a linear objective and that the schedule for decreasing the barrier parameter is related in a certain way to the step length and convergence criteria for each Newton process. Received: October 10, 1997 / Accepted: September 10, 2000?Published online February 22, 2001     

On the convergence of the symmetric successive overrelaxation method

The symmetric successive overrelaxation (SSOR) iterative method is applied to the solution of the system of linear equations $\text{A}\text{x}\text{=}\text{b}$, where A is an n×n nonsingular matrix. We give bounds for the spectral radius of the SSOR iterative matrix when A is an Hermitian positive definite matrix, and when A is a nonsingular M-matrix. Then, we discuss the convergence of the SSOR iterative method associated with a new splitting of the matrix A which extends the results of Varga and Buoni [1].     

On the convergence of the heat balance integral method

Convergence properties are established for the piecewise linear heat balance integral solution of a benchmark moving boundary problem, thus generalising earlier results [Numer. Heat Transfer 8 (1985) 373]. A convergence rate of O(n⁻¹) is identified with minor effects at large values of the Stefan number β (slow interface movement). The correct O(n^−1/2) behaviour for incident heat flux is recovered for β → 0 (pure heat conduction) as previously found [Numer. Heat Transfer 8 (1985) 373–382]. Numerical illustrations support the theoretical findings.