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In this article, we consider two parametric classes of derivative free iterative methods of high order of convergence in Banach spaces. We study local and semilocal convergence results in Banach spaces for both families of iterative processes. We carry out a numerical application to solve a nonlinear and nondifferentiable Fredholm integral equation in the Banach space of continuous functions with the maximum norm. [ABSTRACT FROM AUTHOR]

This paper tests for convergence in labour productivity at the local level in 10 sub‐Saharan countries, disaggregated into 1136 administrative entities. This work combines nighttime lights data and a unique set of population censuses to produce local measures of growth, employment and sectoral shares. We find evidence of unconditional convergence across sectors in the range of 2%. However, convergence is heterogeneous and conditional on both manufacturing and services employment shares. Convergence is also associated with proximity to the main city, moderate population density, low land suitability and relatively moderate temperature. Finally, the within effect dominates the between effect. [ABSTRACT FROM AUTHOR]

The secant method is the most used iterative method to solve an operator equation where the operator involved is nondifferentiable. A known problem that arises when applying this method is its accessibility. Then, we try to improve it by using the technique of decomposition of the operator involved, so that the operator is in turn the sum of two operators, one differentiable and the other continuous but not differentiable. For this, we use a family of Newton‐secant‐type iterative methods that arises from Newton's method and from a well‐known family of secant‐type iterative methods. We study the accessibility of the new methods in two different ways: From the convergence balls of the methods, obtained from a local study of the convergence, and from a dynamic study of the methods. Some examples related to chemistry are also presented to prove the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we study the simplest quadratic matrix equation: Q(X)=X2+BX+C=0$$ \mathcal{Q}(X)={X}^2+ BX+C=0 $$. We transform this equation into an equivalent fixed‐point equation, and based on it, we construct the Krasnoselskij method. From this transformation, we can obtain iterative schemes more accurate than the successive approximation method. Moreover, under suitable conditions, we establish different results for the existence and localization of a solution for this equation with the Krasnoselskij method. Finally, we see numerically that the predictor–corrector iterative scheme, with the Krasnoselskij method as a predictor and the Newton method as corrector method, can improve the numerical application of the Newton method when approximating a solution of the quadratic matrix equation. [ABSTRACT FROM AUTHOR]

There is a plethora of k‐step solvers for equations involving operators on Banach spaces. Their convergence is estimated by adopting hypotheses on high‐order derivatives which are not even in these iterative solvers. In addition, no computable error bounds or information on the uniqueness of the solution based on Lipschitz‐type functions are given. Moreover, the choice of the initial guess is like shooting in the dark. Finally, the criteria of convergence differ from solver to solver, so no comparison can be made between their convergence domains. The novelty of our work is that we address these problems by introducing a generalized k‐step solver containing all previous k‐step solvers. Moreover, we address all previously stated problems in the earlier papers and utilizing weaker conditions that require only the continuity of the involved operator. Applications are also presented where we test the convergence criteria. [ABSTRACT FROM AUTHOR]

In the paper, we consider the local‐in‐time and the global‐in‐time convergence of infinity‐ion‐mass limit for bipolar Euler–Maxwell systems by setting the mass of an electron me=1 and letting the mass of an ion mi → +∞. We use the method of asymptotic expansions to handle the local‐in‐time convergence problem and find that the limiting process from bipolar models to unipolar models is actually decoupling but not the vanishing of equations for the corresponding the other particle. Moreover, when the initial data are sufficiently close to the constant equilibrium state, we also establish the corresponding global‐in‐time convergence. [ABSTRACT FROM AUTHOR]

In this paper, we study the quadratic matrix equations. To improve the application of iterative schemes, we use a transform of the quadratic matrix equation into an equivalent fixed‐point equation. Then, we consider an iterative process of Chebyshev‐type to solve this equation. We prove that this iterative scheme is more efficient than Newton's method. Moreover, we obtain a local convergence result for this iterative scheme. We finish showing, by an application to noisy Wiener‐Hopf problems, that the iterative process considered is computationally more efficient than Newton's method. [ABSTRACT FROM AUTHOR]

SUMMARY In this paper we consider the Pad'e family of iterations for computing the matrix sign function and the Padé family of iterations for computing the matrix p-sector function. We prove that all the iterations of the Padé family for the matrix sign function have a common convergence region. It completes a similar result of Kenney and Laub for half of the Padé family. We show that the iterations of the Padé family for the matrix p-sector function are well defined in an analogous common region, depending on p. For this purpose we proved that the Padé approximants to the function (1− z)−σ, 0<σ<1, are a quotient of hypergeometric functions whose poles we have localized. Furthermore we proved that the coefficients of the power expansion of a certain analytic function form a positive sequence and in a special case this sequence has the log-concavity property. Copyright © 2011 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]