Solving nonlinear equations by a new derivative free iterative method

We develop a new simple iteration formula, which does not require any derivatives of f(x), for solving a nonlinear equation f(x) = 0. It is proved that the convergence order of the new method is quadratic. Furthermore, the new method can approximate complex roots. By several numerical examples we show that the presented method will give desirable approximation to the root without a particularly good initial approximation and be efficient for all cases, regardless of the behavior of f(x).     

The improved linear multistep methods for differential equations with piecewise continuous arguments

This paper deals with the convergence of the linear multistep methods for the equation x′(t) = ax(t) + a₀x([t]). Numerical experiments demonstrate that the 2-step Adams-Bashforth method is only of order p = 0 when applied to the given equation. An improved linear multistep methods is constructed. It is proved that these methods preserve their original convergence order for ordinary differential equations (ODEs) and some numerical experiments are given.     

Transformation methods for finding multiple roots of nonlinear equations

In this paper, to estimate a multiple root p of an equation f(x) = 0, we transform the function f(x) to a hyper tangent function combined with a simple difference formula whose value changes from −1 to 1 as x passes through the root p. Then we apply the so-called numerical integration method to the transformed equation, which may result in a specious approximate root. Furthermore, in order to enhance the accuracy of the approximation we propose a Steffensen-type iterative method, which does not require any derivatives of f(x) nor is quite affected by an initial approximation. It is shown that the convergence order of the proposed method becomes cubic by simultaneous approximation to the root and its multiplicity. Results for some numerical examples show the efficiency of the new method.     

Newton type iteration for Tikhonov regularization of non-linear ill-posed Hammerstein type equations

An iterative method is investigated for a nonlinear ill-posed Hammerstein type operator equation KF(x)=f. We use a center-type Lipschitz condition in our convergence analysis instead of the usual Lipschitz condition. The adaptive method of Pereverzev and Schock (SIAM J. Numer. Anal. 43(5):2060–2076, 2005) is used for choosing the regularization parameter. The optimality of this method is proved under a general source condition involving the Fréchet derivative of F at some initial guess x ₀. A numerical example of nonlinear integral equation shows the efficiency of this procedure.     

Regularization of outflow problems in unsaturated porous media with dry regions

We study a porous medium with saturated, unsaturated, and dry regions, described by Richards' equation for the saturation s and the pressure p. Due to a degenerate permeability coefficient k(x,s) and a degenerate capillary pressure function pc(x,s), the equations may be of elliptic, parabolic, or of ODE-type. We construct a parabolic regularization of the equations and find conditions that guarantee the convergence of the parabolic solutions to a solution of the degenerate system. An example shows that the convergence fails in general. Our approach provides an existence result for the outflow problem in the case of x-dependent coefficients and a method for a numerical approximation.     

New higher order methods for solving nonlinear equations with multiple roots

For a nonlinear equation f(x)=0 having a multiple root we consider Steffensen’s transformation, T. Using the transformation, say, F_q(x)=T^qf(x) for integer q≥2, repeatedly, we develop higher order iterative methods which require neither derivatives of f(x) nor the multiplicity of the root. It is proved that the convergence order of the proposed iterative method is 1+2^q−2 for any equation having a multiple root of multiplicity m≥2. The efficiency of the new method is shown by the results for some numerical examples.     

A Newton-like method and its application

In this paper we prove an existence and uniqueness theorem for solving the operator equation F(x)+G(x)=0, where F is a Gateaux differentiable continuous operator while the operator G satisfies a Lipschitz-condition on an open convex subset of a Banach space. As corollaries, a theorem of Tapia on a weak Newton's method and the classical convergence theorem for modified Newton-iterates are deduced. An existence theorem for a generalized Euler-Lagrange equation in the setting of Sobolev space is obtained as a consequence of the main theorem. We also obtain a class of Gateaux differentiable operators which are nowhere Frechet differentiable. Illustrative examples are also provided.     

On iterative methods for solving equations with covering mappings

In this paper we propose an iterative method for solving the equation Υ(x, x) = y, where the mapping Υ acts in metric spaces and is covering in the first argument and Lipschitzian in the second one. Each subsequent element x _i+1 of the sequence of iterations is defined by the previous one as a solution to the equation Υ(x, x _i) = y _i, where y _i can be an arbitrary point sufficiently close to y. Conditions for convergence and error estimates are obtained. The method proposed is an iterative development of the Arutyunov method for finding coincidence points of mappings. In order to determine x _i+1 in practical implementation of the method in linear normed spaces, it is proposed to perform one step by using the Newton–Kantorovich method. The thus-obtained method of solving the equation of the form Υ(x, u) = ψ(x) ? φ(u) coincides with the iterative method proposed by A.I. Zinchenko,M.A. Krasnosel’skii, and I.A. Kusakin.     

A unifying local-semilocal convergence analysis and applications for two-point Newton-like methods in Banach space

We provide a local as well as a semilocal convergence analysis for two-point Newton-like methods in a Banach space setting under very general Lipschitz type conditions. Our equation contains a Fréchet differentiable operator F and another operator G whose differentiability is not assumed. Using more precise majorizing sequences than before we provide sufficient convergence conditions for Newton-like methods to a locally unique solution of equation F(x)+G(x)=0. In the semilocal case we show under weaker conditions that our error estimates on the distances involved are finer and the information on the location of the solution at least as precise as in earlier results. In the local case a larger radius of convergence is obtained. Several numerical examples are provided to show that our results compare favorably with earlier ones. As a special case we show that the famous Newton-Kantorovich hypothesis is weakened under the same hypotheses as the ones contained in the Newton-Kantorovich theorem.     

A new semilocal convergence theorem for Newton's method

A new semilocal convergence theorem for Newton's method is established for solving a nonlinear equation F(x) = 0, defined in Banach spaces. It is assumed that the operator F is twice Fréchet differentiable, and F″ satisfies a Lipschitz type condition. Results on uniqueness of solution and error estimates are also given. Finally, these results are compared with those that use Kantorovich conditions.     

An iterative row-action method for interval convex programming

The iterative primal-dual method of Bregman for solving linearly constrained convex programming problems, which utilizes nonorthogonal projections onto hyperplanes, is represented in a compact form, and a complete proof of convergence is given for an almost cyclic control of the method. Based on this, a new algorithm for solving interval convex programming problems, i.e., problems of the form minf(x), subject to γ≤Ax≤δ, is proposed. For a certain family of functionsf(x), which includes the norm ∥x∥ and thex logx entropy function, convergence is proved. The present row-action method is particularly suitable for handling problems in which the matrixA is large (or huge) and sparse.     

High accuracy cubic spline approximation for two dimensional quasi-linear elliptic boundary value problems

We report a new 9 point compact discretization of order two in y- and order four in x-directions, based on cubic spline approximation, for the solution of two dimensional quasi-linear elliptic partial differential equations. We describe the complete derivation procedure of the method in details and also discuss how our discretization is able to handle Poisson’s equation in polar coordinates. The convergence analysis of the proposed cubic spline approximation for the nonlinear elliptic equation is discussed in details and we have shown under appropriate conditions the proposed method converges. Some physical examples and their numerical results are provided to justify the advantages of the proposed method.     

Mixed problem for the wave equation with a summable potential and nonzero initial velocity

The resolvent approach in the Fourier method, combined with Krylov’s ideas concerning convergence acceleration for Fourier series, is used to obtain a classical solution of a mixed problem for the wave equation with a summable potential, fixed ends, a zero initial position, and an initial velocity ψ(x), where ψ(x) is absolutely continuous, ψ'(x) ∈ L ₂[0,1], and ψ(0) = ψ(1) = 0. In the case ψ(x) ∈ L[0,1], it is shown that the series of the formal solution converges uniformly and is a weak solution of the mixed problem.     

Mixed problem for the wave equation with integrable potential in the case of two-point boundary conditions of distinct orders

We study a mixed problem for the wave equation with integrable potential and with two-point boundary conditions of distinct orders for the case in which the corresponding spectral problem may have multiple spectrum. Based on the resolvent approach in the Fourier method and the Krylov convergence acceleration trick for Fourier series, we obtain a classical solution u(x, t) of this problem under minimal constraints on the initial condition u(x, 0) = ?(x). We use the Carleson–Hunt theorem to prove the convergence almost everywhere of the formal solution series in the limit case of ?(x) ∈ L _p[0, 1], p > 1, and show that the formal solution is a generalized solution of the problem.     

AN ITERATIVE ALGORITHM FOR MAXIMAL MONOTONE MULTIVALUED OPERATOR EQUATIONS

1 IntroductionThe multivalued operator equations occur in various applications, e.g., mecha11ical systeimwith dry and viscous damping, electrical networks with switches, oscil1ations in viscoelastic-ity, optimization probIems with uonsmooth data, dynanilcal systems with nondifferentiablepotential, and optimal colltroI problellls. There have been a number of results, for instance,[1l-[6l, oll the solutions of multivallled operator equations. Amoug theln, R.T.Rockafellar[1]gave a prorimal poin…     

Circulant preconditioners for pricing options

We use the normalized preconditioned conjugate gradient method with Strang’s circulant preconditioner to solve a nonsymmetric Toeplitz system A_nx=b, which arises from the discretization of a partial integro-differential equation in option pricing. By using the definition of family of generating functions introduced in [16], we prove that Strang’s circulant preconditioner leads to a superlinear convergence rate under certain conditions. Numerical results exemplify our theoretical analysis.     

On the convergence of the von Neumann difference approximation to hyperbolic initial boundary value problems

The general implicit finite-difference approximation of second order devised by von Neumann is applied to the initial boundary value problem for a somewhat generalized wave equation. By means of eigenvalue expansion it is shown that the method is uniform convergent of order $$\log \left( {\Delta x^{ - 1} } \right)O\left( {\Delta t^2 + \Delta x^2 } \right)$$ Δt, Δx mesh widths). Moreover, the convergence on the linet=T reveals to be proportional toT.     

Properties and numerical testing of a parallel global optimization algorithm

In the present paper, we have considered three methods with which to control the error in the homotopy analysis of elliptic differential equations and related boundary value problems, namely, control of residual errors, minimization of error functionals, and optimal homotopy selection through appropriate choice of auxiliary function H(x). After outlining the methods in general, we consider three applications. First, we apply the method of minimized residual error in order to determine optimal values of the convergence control parameter to obtain solutions exhibiting central symmetry for the Yamabe equation in three or more spatial dimensions. Secondly, we apply the method of minimizing error functionals in order to obtain optimal values of the convergnce control parameter for the homotopy analysis solutions to the Brinkman?CForchheimer equation. Finally, we carefully selected the auxiliary function H(x) in order to obtain an optimal homotopy solution for Liouville??s equation.     

An accelerated Newton method for equations with semismooth Jacobians and nonlinear complementarity problems

We discuss local convergence of Newton’s method to a singular solution x ^* of the nonlinear equations F(x) =  0, for $F:{\mathbb{R}}^n \rightarrow {\mathbb{R}}^n$ . It is shown that an existing proof of Griewank, concerning linear convergence to a singular solution x ^* from a starlike domain around x ^* for F twice Lipschitz continuously differentiable and x ^* satisfying a particular regularity condition, can be adapted to the case in which F′ is only strongly semismooth at the solution. Further, Newton’s method can be accelerated to produce fast linear convergence to a singular solution by overrelaxing every second Newton step. These results are applied to a nonlinear-equations reformulation of the nonlinear complementarity problem (NCP) whose derivative is strongly semismooth when the function f arising in the NCP is sufficiently smooth. Conditions on f are derived that ensure that the appropriate regularity conditions are satisfied for the nonlinear-equations reformulation of the NCP at x ^*.     

Analytic continuations of a general algebraic function by means of Puiseux series

A complete list of power series (centered at the point x = 0) is obtained for the solution y(x) of the general reduced algebraic equation $y^n x_s y^{n_s } + ... + x_1 y^{n_1 } - 1 = 0$ . The domains of convergence of these series are described in terms of the amoeba of the discriminant of the equation. Sectorial domains through which one selected series is analytically continued to the other series are explicitly described.