On the semilocal convergence of inexact Newton methods in Banach spaces

We provide two types of semilocal convergence theorems for approximating a solution of an equation in a Banach space setting using an inexact Newton method [I.K. Argyros, Relation between forcing sequences and inexact Newton iterates in Banach spaces, Computing 63 (2) (1999) 134–144; I.K. Argyros, A new convergence theorem for the inexact Newton method based on assumptions involving the second Fréchet-derivative, Comput. Appl. Math. 37 (7) (1999) 109–115; I.K. Argyros, Forcing sequences and inexact Newton iterates in Banach space, Appl. Math. Lett. 13 (1) (2000) 77–80; I.K. Argyros, Local convergence of inexact Newton-like iterative methods and applications, Comput. Math. Appl. 39 (2000) 69–75; I.K. Argyros, Computational Theory of Iterative Methods, in: C.K. Chui, L. Wuytack (Eds.), in: Studies in Computational Mathematics, vol. 15, Elsevier Publ. Co., New York, USA, 2007; X. Guo, On semilocal convergence of inexact Newton methods, J. Comput. Math. 25 (2) (2007) 231–242]. By using more precise majorizing sequences than before [X. Guo, On semilocal convergence of inexact Newton methods, J. Comput. Math. 25 (2) (2007) 231–242; Z.D. Huang, On the convergence of inexact Newton method, J. Zheijiang University, Nat. Sci. Ed. 30 (4) (2003) 393–396; L.V. Kantorovich, G.P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1982; X.H. Wang, Convergence on the iteration of Halley family in weak condition, Chinese Sci. Bull. 42 (7) (1997) 552–555; T.J. Ypma, Local convergence of inexact Newton methods, SIAM J. Numer. Anal. 21 (3) (1984) 583–590], we provide (under the same computational cost) under the same or weaker hypotheses: finer error bounds on the distances involved; an at least as precise information on the location of the solution. Moreover if the splitting method is used, we show that a smaller number of inner/outer iterations can be obtained.     

Majorizing functions and two-point Newton-type methods

The semi-local convergence of a Newton-type method used to solve nonlinear equations in a Banach space is studied. We also give, as two important applications, convergence analyses of two classes of two-point Newton-type methods including a method mentioned in [5] and the midpoint method studied in [1], [2] and [12]. Recently, interest has been shown in such methods [3] and [4].     

On the convergence of modified Newton methods for solving equations containing a non-differentiable term

We provide a semilocal convergence analysis for certain modified Newton methods for solving equations containing a non-differentiable term. The sufficient convergence conditions of the corresponding Newton methods are often taken as the sufficient conditions for the modified Newton methods. That is why the latter methods are not usually treated separately from the former. However, here we show that weaker conditions, as well as a finer error analysis than before can be obtained for the convergence of modified Newton methods. Numerical examples are also provided.     

On the convergence of Newton's method for a class of nonsmooth operators

We provide an analog of the Newton–Kantorovich theorem for a certain class of nonsmooth operators. This class includes smooth operators as well as nonsmooth reformulations of variational inequalities. It turns out that under weaker hypotheses we can provide under the same computational cost over earlier works [S.M. Robinson, Newton's method for a class of nonsmooth functions, Set-Valued Anal. 2 (1994) 291–305] a semilocal convergence analysis with the following advantages: finer error bounds on the distances involved and a more precise information on the location of the solution. In the local case not examined in [S.M. Robinson, Newton's method for a class of nonsmooth functions, Set-Valued Anal. 2 (1994) 291–305] we can show how to enlarge the radius of convergence and also obtain finer error estimates. Numerical examples are also provided to show that in the semilocal case our results can apply where others [S.M. Robinson, Newton's method for a class of nonsmooth functions, Set-Valued Anal. 2 (1994) 291–305] fail, whereas in the local case we can obtain a larger radius of convergence than before [S.M. Robinson, Newton's method for a class of nonsmooth functions, Set-Valued Anal. 2 (1994) 291–305].     

On the semilocal convergence of efficient Chebyshev-Secant-type methods

We introduce a three-step Chebyshev-Secant-type method (CSTM) with high efficiency index for solving nonlinear equations in a Banach space setting. We provide a semilocal convergence analysis for (CSTM) using recurrence relations. Numerical examples validating our theoretical results are also provided in this study.     

Improved local convergence of Newton’s method under weak majorant condition

We provide a local convergence analysis for Newton’s method under a weak majorant condition in a Banach space setting. Our results provide under the same information a larger radius of convergence and tighter error estimates on the distances involved than before [14]. Special cases and numerical examples are also provided in this study.     

Kantorovich-type semilocal convergence analysis for inexact Newton methods

We provide a new semilocal convergence analysis for generating an inexact Newton method converging to a solution of a nonlinear equation in a Banach space setting. Our analysis is based on our idea of recurrent functions. Our results are compared favorably to earlier ones by others and us (Argyros (2007, 2009) [5] and [6], Argyros and Hilout (2009) [7], Guo (2007) [15], Shen and Li (2008) [18], Li and Shen (2008) [19], Shen and Li (2009) [20]). Numerical examples are provided to show that our results apply, but not earlier ones [15], [18], [19] and [20].     

On convergence of the inexact Rayleigh quotient iteration with MINRES

For the Hermitian inexact Rayleigh quotient iteration (RQI), we present a new general theory, independent of iterative solvers for shifted inner linear systems. The theory shows that the method converges at least quadratically under a new condition, called the uniform positiveness condition, that may allow the residual norm _ξk≥1${\xi }_k\ge 1$ of the inner linear system at outer iteration k+1$k+1$ and can be considerably weaker than the condition _ξk≤ξ<1${\xi }_k\le \xi <1$ with ξ$\xi$ a constant not near one commonly used in the literature. We consider the convergence of the inexact RQI with the unpreconditioned and tuned preconditioned MINRES methods for the linear systems. Some attractive properties are derived for the residuals obtained by MINRES. Based on them and the new general theory, we make a refined analysis and establish a number of new convergence results. Let ‖_rk‖$\Vert r_k\Vert$ be the residual norm of approximating eigenpair at outer iteration k$k$. Then all the available cubic and quadratic convergence results require _ξk=O(‖_rk‖)${\xi }_k=O\left(\Vert r_k\Vert \right)$ and _ξk≤ξ${\xi }_k\le \xi$ with a fixed ξ$\xi$ not near one, respectively. Fundamentally different from these, we prove that the inexact RQI with MINRES generally converges cubically, quadratically and linearly provided that _ξk≤ξ${\xi }_k\le \xi$ with a constant ξ<1$\xi <1$ not near one, _ξk=1−O(‖_rk‖)${\xi }_k=1-O\left(\Vert r_k\Vert \right)$ and _ξk=1−O(‖_rk‖²)${\xi }_k=1-O\left({\Vert r_k\Vert }^2\right)$, respectively. The new convergence conditions are much more relaxed than ever before. The theory can be used to design practical stopping criteria to implement the method more effectively. Numerical experiments confirm our results.     

Extending the Newton-Kantorovich hypothesis for solving equations

The famous Newton-Kantorovich hypothesis (Kantorovich and Akilov, 1982 [3], Argyros, 2007 [2], Argyros and Hilout, 2009 [7]) has been used for a long time as a sufficient condition for the convergence of Newton’s method to a solution of an equation in connection with the Lipschitz continuity of the Fréchet-derivative of the operator involved. Here, using Lipschitz and center-Lipschitz conditions, and our new idea of recurrent functions, we show that the Newton-Kantorovich hypothesis can be weakened, under the same information. Moreover, the error bounds are tighter than the corresponding ones given by the dominating Newton-Kantorovich theorem (Argyros, 1998 [1]; [2] and [7]; Ezquerro and Hernández, 2002 [11]; [3]; Proinov 2009, 2010 [16] and [17]).Numerical examples including a nonlinear integral equation of Chandrasekhar-type (Chandrasekhar, 1960 [9]), as well as a two boundary value problem with a Green’s kernel (Argyros, 2007 [2]) are also provided in this study.     

Majorizing sequences for iterative methods

We provide convergence results for very general majorizing sequences of iterative methods. Using our new concept of recurrent functions, we unify the semilocal convergence analysis of Newton-type methods (NTM) under more general Lipschitz-type conditions. We present two very general majorizing sequences and we extend the applicability of (NTM) using the same information before Chen and Yamamoto (1989) [13], Deuflhard (2004) [16], Kantorovich and Akilov (1982) [19], Miel (1979) [20], Miel (1980) [21] and Rheinboldt (1968) [30]. Applications, special cases and examples are also provided in this study to justify the theoretical results of our new approach.     

On a class of Newton-like methods for solving nonlinear equations

We provide a semilocal convergence analysis for a certain class of Newton-like methods considered also in [I.K. Argyros, A unifying local-semilocal convergence analysis and applications for two-point Newton-like methods in Banach space, J. Math. Anal. Appl. 298 (2004) 374–397; I.K. Argyros, Computational theory of iterative methods, in: C.K. Chui, L. Wuytack (Eds.), Series: Studies in Computational Mathematics, vol. 15, Elsevier Publ. Co, New York, USA, 2007; J.E. Dennis, Toward a unified convergence theory for Newton-like methods, in: L.B. Rall (Ed.), Nonlinear Functional Analysis and Applications, Academic Press, New York, 1971], in order to approximate a locally unique solution of an equation in a Banach space.     

The Ulm method under mild differentiability conditions

The Ulm method is considered to approximate a solution of a nonlinear operator equation F(x) = 0. We study the convergence of this method when F′ is ω-conditioned and prove that the R-order of convergence is at least 1 + p if ω is quasi-homogeneous of type ω(tz)≤ t ^p ω(z), for z > 0, tϵ[0,1] and pϵ[0,1]. Preparation of this paper was partly supported by the Ministry of Education and Science (MTM 2005-03091).     

Solving nonlinear integral equations of Fredholm type with high order iterative methods

The application of high order iterative methods for solving nonlinear integral equations is not usual in mathematics. But, in this paper, we show that high order iterative methods can be used to solve a special case of nonlinear integral equations of Fredholm type and second kind. In particular, those that have the property of the second derivative of the corresponding operator have associated with them a vector of diagonal matrices once a process of discretization has been done.     

On the stability of a class of convergence acceleration methods for power series

In this paper we study the problem of evaluating the sum of a power series whose terms are given numerically with a moderate accuracy. For a large class of divergent series a sum may be defined using analytic continuation. This sum may be estimated using the values of a finite number of terms. However, it is established here that the accuracy of this estimate will generally deteriorate if we use an ever-growing number of terms. A result on the stability of product quadrature is also obtained as a corollary of our main stability theorem.Dedicated to professor Germund Dahlquist, on the occasion of his 60th birthday     

The effect of non-optimal bases on the convergence of Krylov subspace methods

Summary There are many examples where non-orthogonality of a basis for Krylov subspace methods arises naturally. These methods usually require less storage or computational effort per iteration than methods using an orthonormal basis (optimal methods), but the convergence may be delayed. Truncated Krylov subspace methods and other examples of non-optimal methods have been shown to converge in many situations, often with small delay, but not in others. We explore the question of what is the effect of having a non-optimal basis. We prove certain identities for the relative residual gap, i.e., the relative difference between the residuals of the optimal and non-optimal methods. These identities and related bounds provide insight into when the delay is small and convergence is achieved. Further understanding is gained by using a general theory of superlinear convergence recently developed. Our analysis confirms the observed fact that in exact arithmetic the orthogonality of the basis is not important, only the need to maintain linear independence is. Numerical examples illustrate our theoretical results.This revised version was published online in June 2005 due to a typesetting mistake in the footnote on page 7.     

A unified approach for constructing fast two-step Newton-like methods

We introduce some new very general ways of constructing fast two-step Newton-like methods to approximate a locally unique solution of a nonlinear operator equation in a Banach space setting. We provide existence-uniqueness theorems as well as an error analysis for the iterations involved using Newton-Kantorovich-type hypotheses and the majorant method. Our results depend on the existence of a Lipschitz function defined on a closed ball centered at a certain point and of a fixed radius and with values into the positive real axis. Special choices of this function lead to favorable comparisons with results already in the literature. Some applications to the solution of nonlinear integral equations appearing in radiative transfer as well as to the solution of integral equations of Uryson-type are also provided.     

On the convergence of splittings for semidefinite linear systems

Recently, Lee et al. [Young-ju Lee, Jinbiao Wu, Jinchao Xu, Ludmil Zikatanov, On the convergence of iterative methods for semidefinite linear systems, SIAM J. Matrix Anal. Appl. 28 (2006) 634-641] introduce new criteria for the semi-convergence of general iterative methods for semidefinite linear systems based on matrix splitting. The new conditions generalize the classical notion of P-regularity introduced by Keller [H.B. Keller, On the solution of singular and semidefinite linear systems by iterations, SIAM J. Numer. Anal. 2 (1965) 281-290]. In view of their results, we consider here stipulations on a splitting A=M-N, which lead to fixed point systems such that, the iterative scheme converges to a weighted Moore-Penrose solution to the system Ax=b. Our results extend the result of Lee et al. to a more general case and we also show that it requires less restrictions on the splittings than Keller’s P-regularity condition to ensure the convergence of iterative scheme.     

On Newton-type methods for multiple roots with cubic convergence

We introduce two families of Newton-type methods for multiple roots with cubic convergence. A further Newton-type method for multiple roots with cubic convergence is presented that is related to quadrature. We also provide numerical tests that show that these new methods are competitive to other known methods for multiple roots.     

On some one-point hybrid iterative methods

When we choose an iterative process for solving a nonlinear equation, the region of accessibility of the iterative process is certainly useful. We know that the higher the order of convergence of the iterative process, the smaller the region of accessibility. In this paper, we present a simple modification of the classic third-order iterative processes, so as to consider, for each of them, the same region of accessibility as that of the Newton method, that is to say a method of order of convergence two.