Weaker convergence for Newton's method under Hölder differentiability

We use more precise majorizing sequences than in earlier studies such as [J. Appell, E. De Pascale, J.V. Lysenko, and P.P. Zabrejko, New results on Newton–Kantorovich approximations with applications to nonlinear integral equations, Numer. Funct. Anal. Optim. 18 (1997), pp. 1–17; I.K. Argyros, Concerning the ‘terra incognita’ between convergence regions of two Newton methods, Nonlinear Anal. 62 (2005), pp. 179–194; F. Cianciaruso, A further journey in the ‘terra incognita’ of the Newton–Kantorovich method, Nonlinear Funct. Anal. Appl. 15 (2010), pp. 173–183; F. Cianciaruso and E. De Pascale, Newton–Kantorovich approximations when the derivative is Hölderian: Old and new results, Numer. Funct. Anal. Optim. 24 (2003), pp. 713–723; F. Cianciaruso, E. De Pascale, and P.P. Zabrejko, Some remarks on the Newton–Kantorovich approximations, Atti Sem. Mat. Fis. Univ. Modena 48 (2000), pp. 207–215; E. De Pascale and P.P. Zabrejko, Convergence of the Newton–Kantorovich method under Vertgeim conditions: A new improvement, Z. Anal. Anwendvugen 17 (1998), pp. 271–280; J.A. Ezquerro and M.A. Hernández, On the R-order of convergence of Newton's method under mild differentiability conditions, J. Comput. Appl. Math. 197 (2006), pp. 53–61; J.V. Lysenko, Conditions for the convergence of the Newton–Kantorovich method for nonlinear equations with Hölder linearizations (in Russian), Dokl. Akad. Nauk BSSR 38 (1994), pp. 20–24; P.D. Proinov, New general convergence theory for iterative processes and its applications to Newton–Kantorovich type theorems, J. Complexity 26 (2010), pp. 3–42; J. Rokne, Newton's method under mild differentiability conditions with error analysis, Numer. Math. 18 (1971/72), pp. 401–412; B.A. Vertgeim, On conditions for the applicability of Newton's method, (in Russian), Dokl. Akad. N., SSSR 110 (1956), pp. 719–722; B.A. Vertgeim, On some methods for the approximate solution of nonlinear functional equations in Banach spaces, Uspekhi Mat. Nauk 12 (1957), pp. 166–169 (in Russian); English transl.: Amer. Math. Soc. Transl. 16 (1960), pp. 378–382; P.P. Zabrejko and D.F. Nguen, The majorant method in the theory of Newton–Kantorovich approximations and the Pták error estimates, Numer. Funct. Anal. Optim. 9 (1987), pp. 671–684; A.I. Zinc?enko, Some approximate methods of solving equations with non-differentiable operators (Ukrainian), Dopovidi Akad. Nauk Ukraïn. RSR (1963), pp. 156–161] to provide a semilocal convergence analysis for Newton's method under Hölder differentiability conditions. Our sufficient convergence conditions are also weaker even in the Lipschitz differentiability case. Moreover, the results are obtained under the same or less computational cost. Numerical examples are provided where earlier conditions do not hold but for which the new conditions are satisfied.     

An improved convergence analysis for the Newton–Kantorovich method using recurrence relations

We generate a sequence using the Newton–Kantorovich method in order to approximate a locally unique solution of an operator equation on a Banach space under Hölder continuity conditions. Using recurrence relations, Hölder as well as centre-Hölder continuity assumptions on the operator involved, we provide a semilocal convergence analysis with the following advantages over the elegant work by Hernánde? in (The Newton method for operators with Hölder continuous first derivative, J. Optim. Theory Appl. 109(3) (2001), pp. 631–648.) (under the same computational cost): finer error bounds on the distances involved, and a more precise information on the location of the solution. Our results also compare favourably with recent and relevant ones in (I.K. Argyros, Concerning the “terra incognita” between convergence regions of two Newton methods, Nonlinear Anal. 62 (2005), pp. 179–194; I.K. Argyros, Computational Theory of Iterative Methods, in Studies in Computational Mathematics, Vol. 15, C.K. Chui and L. Wuytack, eds., Elsevier Publ. Co., New York, USA, 2007; I.K. Argyros, On the gap between the semilocal convergence domain of two Newton methods, Appl. Math. 34(2) (2007), pp. 193–204; I.K. Argyros, On the convergence region of Newton's method under Hölder continuity conditions, submitted for publication; I.K. Argyros, Estimates on majorizing sequences in the Newton–Kantorovich method, submitted for publication; F. Cianciaruso and E. DePascale, Newton–Kantorovich approximations when the derivative is Hölderian: Old and new results, Numer. Funct. Anal. Optim. 24 (2003), pp. 713–723; F. Cianciaruso and E. DePascale, Estimates of majorizing sequences in the Newton–Kantorovich method, Numer. Funct. Anal. Optim. 27(5–6) (2006), pp. 529–538; F. Cianciaruso and E. DePascale, Estimates of majorizing sequences in the Newton–Kanorovich method: A further improvement, J. Math. Anal. Appl. 322 (2006), pp. 329–335; N.T. Demidovich, P.P. Zabreiko, and Ju.V. Lysenko, Some remarks on the Newton–Kantorovich mehtod for nonlinear equations with Hölder continuous linearizations, Izv. Akad. Nauk Belorus 3 (1993), pp. 22–26 (in Russian). (E. DePascale and P.P. Zabreiko, The convergence of the Newton–Kantorovich method under Vertgeim conditions, A new improvement, Z. Anal. Anwendvugen 17 (1998), pp. 271–280.) and (L.V. Kantorovich and G.P. Akilov, Functional Analysis in Normed Spaces, Pergamon Press, Oxford, 1982; J.V. Lysenko, Conditions for the convergence of the Newton–Kantorovich method for nonlinear equations with Hölder linearizations, Dokl. Akad. Nauk BSSR 38 (1994), pp. 20–24 (in Russian); B.A. Vertgeim, On some methods for the approximate solution of nonlinear functional equations in Banach spaces, Uspekhi Mat. Nauk 12 (1957), pp. 166–169 (in Russian); Amer. Math. Soc. Transl. 16 (1960), pp. 378–382. (English Trans.).)     

On local convergence of a Newton-type method in Banach space

In this study we are concerned with the local convergence of a Newton-type method introduced by us [I.K. Argyros and D. Chen, On the midpoint iterative method for solving nonlinear equations in Banach spaces, Appl. Math. Lett. 5 (1992), pp. 7–9.] for approximating a solution of a nonlinear equation in a Banach space setting. This method has also been studied by Homeier [H.H.H. Homeier, A modified Newton method for rootfinding with cubic convergence, J. Comput. Appl. Math. 157 (2003), pp. 227–230.] and Özban [A.Y. Özban, Some new variants of Newton's method, Appl. Math. Lett. 17 (2004), pp. 677–682.] in real or complex space. The benefits of using this method over other methods using the same information have been explained in [I.K. Argyros, Computational theory of iterative methods, in Studies in Computational Mathematics, Vol. 15, C.K. Chui and L. Wuytack, eds., Elsevier Science Inc., New York, USA, 2007.; I.K. Argyros and D. Chen, On the midpoint iterative method for solving nonlinear equations in Banach spaces, Appl. Math. Lett. 5 (1992), pp. 7–9.; H.H.H. Homeier, A modified Newton method for rootfinding with cubic convergence, J. Comput. Appl. Math. 157 (2003), pp. 227–230.; A.Y. Özban, Some new variants of Newton's method, Appl. Math. Lett. 17 (2004), pp. 677–682.]. Here, we give the convergence radii for this method under a type of weak Lipschitz conditions proven to be fruitful by Wang in the case of Newton's method [X. Wang, Convergence of Newton's method and inverse function in Banach space, Math. Comput. 68 (1999), pp. 169–186 and X. Wang, Convergence of Newton's method and uniqueness of the solution of equations in Banach space, IMA J. Numer. Anal. 20 (2000), pp. 123–134.]. Numerical examples are also provided.     

On the convergence region of Newton's method under Hölder continuity conditions

We approximate a locally unique solution of an equation in a Banach space setting using Newton's method under Hölder continuity assumptions. Using more precise estimates on the distances involved than before [F. Cianciaruso and E. DePascale, Estimates of majorizing sequences in the Newton–Kantorovich method, Numer. Funct. Anal. Optim. 27(5 and 6) (2006), pp. 529–538], we show that the convergence region is extended with finer error estimates and the location of the solution is precise under the same computational cost.     

A wavelet Petrov–Galerkin method for solving integro-differential equations

In this paper, we solve integro-differential equation by using the Alpert multiwavelets as basis functions. We also use the orthogonality of the basis of the trial and test spaces in the Petrov–Galerkin method. The computations are reduced because of orthogonality. Thus the final system that we get from discretizing the integro-differential equation has a very small dimension and enough accuracy. We compare the results with [M. Lakestani, M. Razzaghi, and M. Dehghan, Semiorthogonal spline wavelets approximation for Fredholm integro-differential equations, Math. Probl. Eng. 2006 (2006), pp. 1–12, Article ID 96184] and [A. Ayad, Spline approximation for first-order Fredholm integro-differential equation, Stud. Univ. Babes-Bolyai. Math., 41(3), (1996), pp. 1–8] which used a much larger dimension system and got less accurate results. In [Z. Chen and Y. Xu, The Petrov–Galerkin and iterated Petrov–Galerkin methods for second kind integral equations, SIAM J. Numer. Anal. 35(1) (1998), pp. 406–434], convergence of Petrov–Galerkin method has been discussed with some restrictions on degrees of chosen polynomial basis, but in this paper convergence is obtained for every degree.     

Choice of θ and its effects on stability in the stochastic θ-method of stochastic delay differential equations

The second author's work [F. Wu, X. Mao, and L. Szpruch, Almost sure exponential stability of numerical solutions for stochastic delay differential equations, Numer. Math. 115 (2010), pp. 681–697] and Mao's papers [D.J. Higham, X. Mao, and C. Yuan, Almost sure and moment exponential stability in the numerical simulation of stochastic differential equations, SIAM J. Numer. Anal. 45 (2007), pp. 592–607; X. Mao, Y. Shen, and G. Alison, Almost sure exponential stability of backward Euler–Maruyama discretizations for hybrid stochastic differential equations, J. Comput. Appl. Math. 235 (2011), pp. 1213–1226] showed that the backward Euler–Maruyama (BEM) method may reproduce the almost sure stability of stochastic differential equations (SDEs) without the linear growth condition of the drift coefficient and the counterexample shows that the Euler–Maruyama (EM) method cannot. Since the stochastic θ-method is more general than the BEM and EM methods, it is very interesting to examine the interval in which the stochastic θ-method can capture the stability of exact solutions of SDEs. Without the linear growth condition of the drift term, this paper concludes that the stochastic θ-method can capture the stability for θ∈(1/2, 1]. For θ∈[0, 1/2), a counterexample shows that the stochastic θ-method cannot reproduce the stability of the exact solution. Finally, two examples are given to illustrate our conclusions.     

Solving inverse eigenvalue problems by a projected Newton method

The inverse eigenvalue problem for symmetric matrices (IEP) can be formulated as a system of two matrix equations. For solving the system a variation of Newton's method is used which has been proposed by Fusco and Zecca [Calcolo XXIII (1986), pp. 285–303] for the simultaneous computation of eigenvalues and eigenvectors of a given symmetric matrix. An iteration step of this method consists of a Newton step followed by an orthonormalization with the consequence that each iterate satisfies one of the given equations. The method is proved to convergence locally quadratically to regular solutions. The algorithm and some numerical examples are presented. In addition, it is shown that the so-called Method III proposed by Friedland, Nocedal, and Overton [SIAM J. Numer. Anal., 24 (1987), pp. 634–667] for solving IEP may be constructed similarly to the method presented here.     

Fourth-order convergence theorem by using majorizing functions for super-Halley method in Banach spaces

In this paper, we study the semilocal convergence of a multipoint fourth-order super-Halley method for solving nonlinear equations in Banach spaces. We establish the Newton–Kantorovich-type convergence theorem for the method by using majorizing functions. We also get the error estimate. In comparison with the results obtained in Wang et al. [X.H. Wang, C.Q. Gu, and J.S. Kou, Semilocal convergence of a multipoint fourth-order super-Halley method in Banach spaces, Numer. Algorithms 56 (2011), pp. 497–516], we can provide a larger convergence radius. Finally, we report some numerical applications to demonstrate our approach.     

A non-monotone pattern search approach for systems of nonlinear equations

In this paper, a new pattern search is proposed to solve the systems of nonlinear equations. We introduce a new non-monotone strategy which includes a convex combination of the maximum function of some preceding successful iterates and the current function. First, we produce a stronger non-monotone strategy in relation to the generated strategy by Gasparo et al. [Nonmonotone algorithms for pattern search methods, Numer. Algorithms 28 (2001), pp. 171–186] whenever iterates are far away from the optimizer. Second, when iterates are near the optimizer, we produce a weaker non-monotone strategy with respect to the generated strategy by Ahookhosh and Amini [An efficient nonmonotone trust-region method for unconstrained optimization, Numer. Algorithms 59 (2012), pp. 523–540]. Third, whenever iterates are neither near the optimizer nor far away from it, we produce a medium non-monotone strategy which will be laid between the generated strategy by Gasparo et al. [Nonmonotone algorithms for pattern search methods, Numer. Algorithms 28 (2001), pp. 171–186] and Ahookhosh and Amini [An efficient nonmonotone trust-region method for unconstrained optimization, Numer. Algorithms 59 (2012), pp. 523–540]. Reported are numerical results of the proposed algorithm for which the global convergence is established.     

Fast multilevel augmentation method for nonlinear integral equations

In this study, we extend the multilevel augmentation method for Hammerstein equations established in Chen et al. [Fast multilevel augmentation methods for solving Hammerstein equations, SIAM J. Numer. Anal. 47 (2009), pp. 2321–2346] to solve nonlinear Urysohn integral equations. Under certain differentiability assumptions on the kernel function, we show that the method enjoys the optimal convergence order and linear computational complexity. Finally, numerical experiments are presented to confirm the theoretical results and illustrate the efficiency of the method.     

A non-monotone inexact regularized smoothing Newton method for solving nonlinear complementarity problems

In the paper [S.P. Rui and C.X. Xu, A smoothing inexact Newton method for nonlinear complementarity problems, J. Comput. Appl. Math. 233 (2010), pp. 2332–2338], the authors proposed an inexact smoothing Newton method for nonlinear complementarity problems (NCP) with the assumption that F is a uniform P function. In this paper, we present a non-monotone inexact regularized smoothing Newton method for solving the NCP which is based on Fischer–Burmeister smoothing function. We show that the proposed algorithm is globally convergent and has a locally superlinear convergence rate under the weaker condition that F is a P ₀ function and the solution of NCP is non-empty and bounded. Numerical results are also reported for the test problems, which show the effectiveness of the proposed algorithm.     

Convergence of the modified SOR–Newton method for non-smooth equations

Motivated by Chen [On the convergence of SOR methods for nonsmooth equations. Numer. Linear Algebra Appl. 9 (2002), pp. 81–92], in this paper, we further investigate a modified SOR–Newton (MSOR–Newton) method for solving a system of nonlinear equations F(x)=0, where F is strongly monotone and locally Lipschitz continuous but not necessarily differentiable. The convergence interval of the parameter in the MSOR–Newton method is given. Compared with that of the SOR–Newton method, this interval can be enlarged. Furthermore, when the B-differential of F(x) is difficult to compute, a simple replacement can be used, which can reduce the computational load. Numerical examples show that at the same cost of computational complexity, this MSOR–Newton method can converge faster than the corresponding SOR–Newton method by choosing a suitable parameter.     

Efficient Steffensen-type algorithms for solving nonlinear equations

We introduce a Steffensen-type method (STTM) for solving nonlinear equations in a Banach space setting. Then, we present a local convergence analysis for (STTM) using recurrence relations. Numerical examples validating our theoretical results are also provided in this study to show that (STTM) is faster than other methods [I.K. Argyros, J. Ezquerro, J.M. Gutiérrez, M. Hernández, and S. Hilout, On the semilocal convergence of efficient Chebyshev-Secant-type methods, J. Comput. Appl. Math. 235 (2011), pp. 3195–3206; J.A. Ezquerro and M.A. Hernández, An optimization of Chebyshev's method, J. Complexity 25 (2009), pp. 343–361] using similar convergence conditions.     

On the moving boundary formulation for parabolic problems on unbounded domains

The aim of this paper is to propose an original numerical approach for parabolic problems whose governing equations are defined on unbounded domains. We are interested in studying the class of problems admitting invariance property to Lie group of scalings. Thanks to similarity analysis the parabolic problem can be transformed into an equivalent boundary value problem governed by an ordinary differential equation and defined on an infinite interval. A free boundary formulation and a convergence theorem for this kind of transformed problems are available in [R. Fazio, A novel approach to the numerical solution of boundary value problems on infinite intervals, SIAM J. Numer. Anal. 33 (1996), pp. 1473–1483]. Depending on its scaling invariance properties, the free boundary problem is then solved numerically using either a noniterative, or an iterative method. Finally, the solution of the parabolic problem is retrieved by applying the inverse map of similarity.     

The Bruck's ergodic iteration method for the Ky Fan inequality over the fixed point set

We introduce a new iteration algorithm for solving the Ky Fan inequality over the fixed point set of a nonexpansive mapping, where the cost bifunction is monotone without Lipschitz-type continuity. The algorithm is based on the idea of the ergodic iteration method for solving multi-valued variational inequality which is proposed by Bruck [On the weak convergence of an ergodic iteration for the solution of variational inequalities for monotone operators in Hilbert space, J. Math. Anal. Appl. 61 (1977), pp. 159–164] and the auxiliary problem principle for equilibrium problems P.N. Anh, T.N. Hai, and P.M. Tuan. [On ergodic algorithms for equilibrium problems, J. Glob. Optim. 64 (2016), pp. 179–195]. By choosing suitable regularization parameters, we also present the convergence analysis in detail for the algorithm and give some illustrative examples.     

A new superconvergent method for systems of nonlinear singular boundary value problems

A new superconvergent method based on a sextic spline is described and analysed for the solution of systems of nonlinear singular two-point boundary value problems (BVPs). It is well known that the optimal orders of convergence could not be achieved using standard formulation of a sextic spline for the solution of BVPs. Based on the method used in our earlier research papers [J. Rashidinia and M. Ghasemi, B-spline collocation for solution of two-point boundary value problems, J. Comput. Appl. Math. 235 (2011), pp. 2325–2342; J. Rashidinia, M. Ghasemi, and R. Jalilian, An o(h ⁶) numerical solution of general nonlinear fifth-order two point boundary value problems, Numer. Algorithms 55(4) (2010), pp. 403–428], we construct a new O(h ⁸) locally superconvergent method for the solution of general nonlinear two-point BVPs up to order 6. The error bounds and the convergence properties of the method have been proved theoretically. Then, the method is extended to solve the system of nonlinear two-point BVPs. Some test problems are given to demonstrate the applicability and the superconvergent properties of the proposed method numerically. It is shown that the method is very efficient and applicable for stiff BVPs too.     

Unified convergence analysis for Picard iteration in <Emphasis Type="Italic">n</Emphasis>-dimensional vector spaces

In this paper, we provide three types of general convergence theorems for Picard iteration in n-dimensional vector spaces over a valued field. These theorems can be used as tools to study the convergence of some particular Picard-type iterative methods. As an application, we present a new semilocal convergence theorem for the one-dimensional Newton method for approximating all the zeros of a polynomial simultaneously. This result improves in several directions the previous one given by Batra (BIT Numer Math 42:467–476, 2002).     

Spline collocation method with four parameters for solving a system of fourth-order boundary-value problems

In this paper, a spline collocation method is applied to solve a system of fourth-order boundary-value problems associated with obstacle, unilateral and contact problems. The presented method is dependent on four collocation points to be satisfied by four parameters θ _j ∈(0, 1], j=1(1) 4 in each subinterval. It turns out that the proposed method when applied to the concerned system is a fourth-order convergent method and gives numerical results which are better than those produced by other spline methods [E.A. Al-Said and M.A. Noor, Finite difference method for solving fourth-order obstacle problems, Int. J. Comput. Math. 81(6) (2004), pp. 741–748; F. Geng and Y. Lin, Numerical solution of a system of fourth order boundary value problems using variational iteration method, Appl. Math. Comput. 200 (2008), pp. 231–241; J. Rashidinia, R. Mohammadi, R. Jalilian, and M. Ghasemi, Convergence of cubic-spline approach to the solution of a system of boundary-value problems, Appl. Math. Comput. 192 (2007), pp. 319–331; S.S. Siddiqi and G. Akram, Solution of the system of fourth order boundary value problems using non polynomial spline technique, Appl. Math. Comput. 185 (2007), pp. 128–135; S.S. Siddiqi and G. Akram, Numerical solution of a system of fourth order boundary value problems using cubic non-polynomial spline method, Appl. Math. Comput. 190(1) (2007), pp. 652–661; S.S. Siddiqi and G. Akram, Solution of the system of fourth order boundary value problems using cubic spline, Appl. Math. Comput. 187(2) (2007), pp. 1219–1227; Siraj-ul-Islam, I.A. Tirmizi, F. Haq, and S.K. Taseer, Family of numerical methods based on non-polynomial splines for solution of contact problems, Commun. Nonlinear Sci. Numer. Simul. 13 (2008), pp. 1448–1460]. Moreover, the absolute stability properties appear that the method is A-stable. Two numerical examples (one for each case of boundary conditions) are given to illustrate practical usefulness of the method developed.     

Analysis of the nonlinear Uzawa algorithm for symmetric saddle point problems

In this paper we discuss the convergence behaviour of the nonlinear Uzawa algorithm for solving saddle point problems presented in a recent paper of Cao [Z.H. Cao, Fast Uzawa algorithm for generalized saddle point problems, Appl. Numer. Math. 46 (2003), pp. 157–171]. For a general case, the results on the convergence of the algorithm are given.     

Taylor polynomial solution of hyperbolic type partial differential equations with constant coefficients

The purpose of this study is to give a Taylor polynomial approximation for the solution of hyperbolic type partial differential equations with constant coefficients. The technique used is an improved Taylor matrix method, which has been given for solving ordinary differential, integral and integro-differential equations [M. Gülsu and M. Sezer, A method for the approximate solution of the high-order linear difference equations in terms of Taylor polynomials, Int. J. Comput. Math. 82(5) (2005), pp. 629–642; M. Gülsu and M. Sezer, On the solution of the Riccati equation by the Taylor matrix method, Appl. Math. Comput. 188 (2007), pp. 446–449; A. Karamete and M. Sezer, A Taylor collocation method for the solution of linear integro-differential equations, Int. J. Comput. Math. 79(9) (2002), pp. 987–1000; N. Kurt and M. Çevik, Polynomial solution of the single degree of freedom system by Taylor matrix method, Mech. Res. Commun. 35 (2008), pp. 530–536; N. Kurt and M. Sezer, Polynomial solution of high-order linear Fredholm integro-differential equations with constant coefficients, J. Franklin Inst. 345 (2008), pp. 839–850; ?. Nas, S. Yalçinba?, and M. Sezer, A method for approximate solution of the high-order linear Fredholm integro-differential equations, Int. J. Math. Edu. Sci. Technol. 27(6) (1996), pp. 821–834; M. Sezer, Taylor polynomial solution of Volterra integral equations, Int. J. Math. Edu. Sci. Technol. 25(5) (1994), pp. 625–633; M. Sezer, A method for approximate solution of the second order linear differential equations in terms of Taylor polynomials, Int. J. Math. Edu. Sci. Technol. 27(6) (1996), pp. 821–834; M. Sezer, M. Gülsu, and B. Tanay, A matrix method for solving high-order linear difference equations with mixed argument using hybrid Legendre and Taylor polynomials, J. Franklin Inst. 343 (2006), pp. 647–659; S. Yalçinba?, Taylor polynomial solutions of nonlinear Volterra–Fredholm integral equation, Appl. Math. Comput. 127 (2002), pp. 196–206; S. Yalçinba? and M. Sezer, The approximate solution of high-order linear Volterra–Fredholm integro-differential equations in terms of Taylor polynomials, Appl. Math. Comput. 112 (2000), pp. 291–308]. Some numerical examples, which consist of initial and boundary conditions, are given to illustrate the reliability and efficiency of the method. Also, the results obtained are compared by the known results; the error analysis is performed and the accuracy of the solution is shown.