Semilocal convergence of a continuation method in Banach spaces

This paper is concerned with the semilocal convergence of a continuation method between two third-order iterative methods, namely, the Halley’s and the convex acceleration of Newton’s method, also known as the Super-Halley’s method. This convergence analysis is discussed using the recurrence relations approach. This approach simplifies the analysis and leads to improved results. The convergence analysis is established under the assumption that the second Frëchet derivative satisfies Lipschitz continuity condition. An existence-uniqueness theorem is given. Also, a closed form of error bound is derived in terms of a real parameter α ∈ [0, 1]. Two numerical examples are worked out to demonstrate the efficacy of our approach. On comparing the existence and uniqueness region and error bounds for the solution obtained by our analysis with those obtained by using majorizing sequences [15], we observed that our analysis gives better results. Further, we have observed that for particular values of the α, our analysis reduces to those for the Halley’s method (α = 0) and the convex acceleration of Newton’s method (α = 1), respectively, with improved results.     

Convergence of a continuation method under the Hölder condition of the first derivative

The aim of this paper is to describe a continuation method combining the Chebyshev’s method and the convex acceleration of Newton’s method to solve nonlinear equations in Banach spaces. The semilocal convergence analysis of the method is established using recurrence relations under the assumption that the first Fréchet derivative satisfies the Hölder continuity condition. This condition is milder than the usual Lipschitz condition. The computation of second Fréchet derivative is also avoided. Two real valued functions and a real sequence are used to establish a convergence criterion of R-order (1+p), where p∈(0,1] is the order of the Hölder condition. An existence and uniqueness theorem along with the closed form of error bounds is derived in terms of a real parameter α∈[0,1]. Two numerical examples are worked out to demonstrate the efficacy of our convergence analysis. For both the examples, the convergence conditions hold for the Chebyshev’s method (α=0). However, for the convex acceleration of Newton’s method (α=1), these convergence conditions hold for the first example but fail for the second example. For particular values of α, our method reduces to the Chebyshev’s method (α=0) and the convex acceleration of Newton’s method (α=1).     

On A-invariant mean and A-almost convergence

The idea of A-invariant mean and A-almost convergence is due to J. P. Duran [8], which is a generalization of the usual notion of Banach limit and almost convergence. In this paper, we discuss some important properties of this method and prove that the space F(A) of A-almost convergent sequences is a BK space with ?? · ??_??, and also show that it is a nonseparable closed subspace of the space l _?? of bounded sequences.     

Construction of simple majorizing sequences for iterative methods

The interest of the majorizing (scalar) sequences lies in that, from their convergence, we can deduce the convergence of an iterative method in Banach spaces. We propose a new technique to construct majorizing sequences that generalizes that given by Kantorovich for Newton’s method.     

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.     

An improved semilocal convergence analysis for the Chebyshev method

We expand the applicability of the Chebyshev method for approximating a locally unique solution of nonlinear equations in a Banach space setting. Our majorizing sequences are finer than the known results in scientific literature and the convergence criteria can be weaker. Numerical examples are also presented which further validate the developed theoretical results.     

On the convergence of multiple Walsh-Fourier series of functions of bounded generalized variation

The convergence of multiple Walsh-Fourier series of functions of bounded generalized variation is investigated. The sufficient and necessary conditions on the sequence ?? = {?? _n } are found for the convergence of multiple Walsh-Fourier series of functions of bounded partial ??-variation.     

An improved convergence analysis of a one-step intermediate Newton iterative scheme for nonlinear equations

We revisit a one-step intermediate Newton method for the iterative computation of a zero of the sum of two nonlinear operators that was analyzed by Uko and Velásquez (Rev. Colomb. Mat. 35:21?C27, 2001). By utilizing weaker hypotheses of the Zabrejko-Nguen kind and a modified majorizing sequence we perform a semilocal convergence analysis which yields finer error bounds and more precise information on the location of the solution that the ones obtained in Rev. Colomb. Mat. 35:21?C27, 2001. This error analysis is obtained at the same computational cost as the analogous results of Uko and Velásquez (Rev. Colomb. Mat. 35:21?C27, 2001). We also give two generalizations of the well-known Kantorovich theorem on the solvability of nonlinear equations and the convergence of Newton??s method. Finally, we provide a numerical example to illustrate the predicted-by-theory performance of the Newton iterates involved in this paper.     

Semilocal convergence of a continuation method with Hölder continuous second derivative in Banach spaces

In this paper, the semilocal convergence of a continuation method combining the Chebyshev method and the convex acceleration of Newton’s method used for solving nonlinear equations in Banach spaces is established by using recurrence relations under the assumption that the second Frëchet derivative satisfies the Hölder continuity condition. This condition is mild and works for problems in which the second Frëchet derivative fails to satisfy Lipschitz continuity condition. A new family of recurrence relations are defined based on two constants which depend on the operator. The existence and uniqueness regions along with a closed form of the error bounds in terms of a real parameter α∈[0,1]$\alpha \in [0,1]$ for the solution x^∗$x^{\ast }$ is given. Two numerical examples are worked out to demonstrate the efficacy of our approach. On comparing the existence and uniqueness regions for the solution obtained by our analysis with those obtained by using majorizing sequences under Hölder continuity condition on F^″$F^{″}$, it is found that our analysis gives improved results. Further, we have observed that for particular values of the α$\alpha$, our analysis reduces to those for the Chebyshev method (α=0$\alpha =0$) and the convex acceleration of Newton’s method (α=1)$(\alpha =1)$ respectively with improved results.     

Estimating upper bounds on the limit points of majorizing sequences for Newton’s method

We present new sufficient conditions for the semilocal convergence of Newton’s method to a locally unique solution of an equation in a Banach space setting. Upper bounds on the limit points of majorizing sequences are also given. Numerical examples are provided, where our new results compare favorably to earlier ones such as Argyros (J Math Anal Appl 298:374–397, 2004), Argyros and Hilout (J Comput Appl Math 234:2993-3006, 2010, 2011), Ortega and Rheinboldt (1970) and Potra and Pták (1984).     

Monotone convergence of Newton-like methods for M-matrix algebraic Riccati equations

For the algebraic Riccati equation whose four coefficient matrices form a nonsingular M-matrix or an irreducible singular M-matrix K, the minimal nonnegative solution can be found by Newton’s method and the doubling algorithm. When the two diagonal blocks of the matrix K have both large and small diagonal entries, the doubling algorithm often requires many more iterations than Newton’s method. In those cases, Newton’s method may be more efficient than the doubling algorithm. This has motivated us to study Newton-like methods that have higher-order convergence and are not much more expensive each iteration. We find that the Chebyshev method of order three and a two-step modified Chebyshev method of order four can be more efficient than Newton’s method. For the Riccati equation, these two Newton-like methods are actually special cases of the Newton–Shamanskii method. We show that, starting with zero initial guess or some other suitable initial guess, the sequence generated by the Newton–Shamanskii method converges monotonically to the minimal nonnegative solution.We also explain that the Newton-like methods can be used to great advantage when solving some Riccati equations involving a parameter.     

Diagonalization of matrices over the domain of principal ideals with minimal polynomial m(��) = (�� ? ��)(�� ? ��), �� �� ��

The necessary and sufficient conditions of diagonalization of matrices over a domain of principal ideals with minimal polynomial m(??) = (?? ? ??)(?? ? ??), ?? ?? ?? are obtained. On the basis of the obtained results, the conditions under which the matrices have common eigenvectors are indicated.     

On some very strong compactness conditions

The aim of this paper is to consider compactness notions by utilizing ??-sets, V-sets, locally closed sets, locally open sets, ??-closed sets and ??-open sets. We completely characterize these variations of compactness, and also provide various interesting examples that support our results.     

On the limit behavior of the magnetic Zakharov system

In this paper,we prove that the solutions of magnetic Zakharov system converge to those of generalized Zakharov system in Sobolev space H s,s > 3/2,when parameter β→∞.Further,when parameter (α,β) →∞ together,we prove that the solutions of magnetic Zakharov system converge to those of Schro¨dinger equation with magnetic effect in Sobolev space H s,s > 3/2.Moreover,the convergence rate is also obtained.     

Majorizing sequences for Newton’s method from initial value problems

The most restrictive condition used by Kantorovich for proving the semilocal convergence of Newton’s method in Banach spaces is relaxed in this paper, providing we can guarantee the semilocal convergence in situations that Kantorovich cannot. To achieve this, we use Kantorovich’s technique based on majorizing sequences, but our majorizing sequences are obtained differently, by solving initial value problems.     

The Z \to c\bar c \to \gamma \gamma *, Z \to b\bar b \to \gamma \gamma * triangle diagrams and the Z → γψ, Z → γγ decays

We expounded an approach for studying the Z ?? ??? and Z ?? ???? decay based on the sum rules for the $Z \to c\bar c \to \gamma \gamma *$ and $Z \to b\bar b \to \gamma \gamma *$ amplitudes and their derivatives. We calculate the branching ratios of the Z ?? ??? and Z ?? ???? decays under different suppositions about the saturation of the sum rules. We find the lower bounds of ?? _?? BR(Z ?? ???) = 1.95 · 10 ^?7 and ?? _?? BR(Z ?? ????) = 7.23 · 10^?7 and discuss deviations from the lower bounds including the possibility of BR[Z ?? ??J/??(1S)] ?? BR[Z ?? ????(1S)] ?? 10 ^?6 , which is probably measurable at the LHC. Moreover, we calculate the angle distributions in the Z ?? ??? and Z ?? ???? decays.     

Existence and regularity of positive solutions of elliptic equations of Schr?dinger type

We prove the existence of positive solutions with optimal local regularity of the homogeneous equation of Schr?dinger type $$ - {\rm{div}}(A\nabla u) - \sigma u = 0{\rm{ in }}\Omega $$ for an arbitrary open ?? ? ? ⁿ under only a form-boundedness assumption on ?? ?? D??(??) and ellipticity assumption on A ?? L ^??(??) ^n×n . We demonstrate that there is a two-way correspondence between form boundedness and existence of positive solutions of this equation as well as weak solutions of the equation with quadratic nonlinearity in the gradient $$ - {\rm{div}}(A\nabla u) = (A\nabla v) \cdot \nabla v + \sigma {\rm{ in }}\Omega $$ As a consequence, we obtain necessary and sufficient conditions for both formboundedness (with a sharp upper form bound) and positivity of the quadratic form of the Schr?dinger type operator H = ?div(A?·)-?? with arbitrary distributional potential ?? ?? D??(??), and give examples clarifying the relationship between these two properties.     

An improved unifying convergence analysis of Newton’s method in Riemannian manifolds

Using more precise majorizing sequences we provide a finer convergence analysis than before [1], [7] of Newton’s method in Riemannian manifolds with the following advantages: weaker hypotheses, finer error bounds on the distances involved and a more precise information on the location of the singularity of the vector field.     

An interior-point piecewise linear penalty method for nonlinear programming

We present an interior-point penalty method for nonlinear programming (NLP), where the merit function consists of a piecewise linear penalty function and an ? ₂-penalty function. The piecewise linear penalty function is defined by a set of break points that correspond to pairs of values of the barrier function and the infeasibility measure at a subset of previous iterates and this set is updated at every iteration. The ? ₂-penalty function is a traditional penalty function defined by a single penalty parameter. At every iteration the step direction is computed from a regularized Newton system of the first-order equations of the barrier problem proposed in Chen and Goldfarb (Math Program 108:1?C36, 2006). Iterates are updated using a line search. In particular, a trial point is accepted if it provides a sufficient reduction in either of the penalty functions. We show that the proposed method has the same strong global convergence properties as those established in Chen and Goldfarb (Math Program 108:1?C36, 2006). Moreover, our method enjoys fast local convergence. Specifically, for each fixed small barrier parameter???, iterates in a small neighborhood (roughly within o(??)) of the minimizer of the barrier problem converge Q-quadratically to the minimizer. The overall convergence rate of the iterates to the solution of the nonlinear program is Q-superlinear.     

On the distribution of multiples of real numbers

We introduce new series (of the variable ??) that enable to measure the irregularity of distribution of the sequence of fractional parts {n??}. A detailed analysis of the convergence and divergence of these series is done, depending mainly on the convergents of ??. As a by product, we obtain new Fourier series of square integrable functions that converge almost everywhere but at no rational number.