On an Aitken–Newton type method

We study the solving of nonlinear equations by an iterative method of Aitken type, which has the interpolation nodes controlled by the Newton method. We obtain a local convergence result which shows that the q-convergence order of this method is 6 and its efficiency index is $\sqrt[5]{6},$ which is higher than the efficiency index of the Aitken or Newton methods. Monotone sequences are obtained for initial approximations farther from the solution, if they satisfy the Fourier condition and the nonlinear mapping satisfies monotony and convexity assumptions on the domain.     

On the Gauss–Newton method

We provide a new semilocal convergence analysis of the Gauss–Newton method (GNM) for solving nonlinear equation in the Euclidean space. Using a combination of center-Lipschitz, Lipschitz conditions, and our new idea of recurrent functions, we provide under the same or weaker hypotheses than before (Ben-Israel, J. Math. Anal. Appl. 15:243–252, 1966; Chen and Nashed, Numer. Math. 66:235–257, 1993; Deuflhard and Heindl, SIAM J. Numer. Anal. 16:1–10, 1979; Guo, J. Comput. Math. 25:231–242, 2007; Häußler, Numer. Math. 48:119–125, 1986; Hu et al., J. Comput. Appl. Math. 219:110–122, 2008; Kantorovich and Akilov, Functional Analysis in Normed Spaces, Pergamon, Oxford, 1982), a finer convergence analysis. The results can be extended in case outer or generalized inverses are used. Numerical examples are also provided to show that our results apply, where others fail (Ben-Israel, J. Math. Anal. Appl. 15:243–252, 1966; Chen and Nashed, Numer. Math. 66:235–257, 1993; Deuflhard and Heindl, SIAM J. Numer. Anal. 16:1–10, 1979; Guo, J. Comput. Math. 25:231–242, 2007; Häußler, Numer. Math. 48:119–125, 1986; Hu et al., J. Comput. Appl. Math. 219:110–122, 2008; Kantorovich and Akilov, Functional Analysis in Normed Spaces, Pergamon, Oxford, 1982).     

On finite difference approximation of a matrix-vector product in the Jacobian-free Newton–Krylov method

The Jacobian-free Newton–Krylov (JFNK) method is a special kind of Newton–Krylov algorithm, in which the matrix-vector product is approximated by a finite difference scheme. Consequently, it is not necessary to form and store the Jacobian matrix. This can greatly improve the efficiency and enlarge the application area of the Newton–Krylov method. The finite difference scheme has a strong influence on the accuracy and robustness of the JFNK method. In this paper, several methods for approximating the Jacobian-vector product, including the finite difference scheme and the finite difference step size, are analyzed and compared. Numerical results are given to verify the effectiveness of different finite difference methods.     

On a Moser–Steffensen Type Method for Nonlinear Systems of Equations

Mediterranean Journal of Mathematics - This paper is devoted to the construction and analysis of a Moser–Steffensen iterative scheme. The method has quadratic convergence without evaluating...     

On the Hodge–Newton filtration for p-divisible groups of Hodge type

Mathematische Zeitschrift - A p-divisible group, or more generally an F-crystal, is said to be Hodge–Newton reducible if its Newton polygon and Hodge polygon have a nontrivial contact point....     

On a nonsmooth version of Newton’s method using locally lipschitzian operators

In this study we are concerned with the problem of approximating a locally unique solution of an operator equation in Banach space using Newton’s method. The differentiability of the operator involved is not assumed. We provide a semilocal convergence analysis utilized to solve problems that were not covered before. Numerical examples are also provided to justify our approach.     

On a nonsmooth version of Newton’s method using locally lipschitzian operators

In this study we are concerned with the problem of approximating a locally unique solution of an operator equation in Banach space using Newton’s method. The differentiability of the operator involved is not assumed. We provide a semilocal convergence analysis utilized to solve problems that were not covered before. Numerical examples are also provided to justify our approach.     

Extending the applicability of the Gauss–Newton method under average Lipschitz–type conditions

We extend the applicability of the Gauss–Newton method for solving singular systems of equations under the notions of average Lipschitz–type conditions introduced recently in Li et al. (J Complex 26(3):268–295, 2010). Using our idea of recurrent functions, we provide a tighter local as well as semilocal convergence analysis for the Gauss–Newton method than in Li et al. (J Complex 26(3):268–295, 2010) who recently extended and improved earlier results (Hu et al. J Comput Appl Math 219:110–122, 2008; Li et al. Comput Math Appl 47:1057–1067, 2004; Wang Math Comput 68(255):169–186, 1999). We also note that our results are obtained under weaker or the same hypotheses as in Li et al. (J Complex 26(3):268–295, 2010). Applications to some special cases of Kantorovich–type conditions are also provided in this study.     

On solutions to a FitzHugh–Rinzel type model

Ricerche di Matematica - A ternary autonomous dynamical system of FitzHugh–Rinzel type is analyzed. The system, at start, is reduced to a nonlinear integro differential equation. The...     

On a normed version of a Rogers–Shephard type problem

A translation body of a convex body is the convex hull of two of its translates intersecting each other. In the 1950s, Rogers and Shephard found the extremal values, over the family of n-dimensional convex bodies, of the maximal volume of the translation bodies of a given convex body. In our paper, we introduce a normed version of this problem, and for the planar case, determine the corresponding quantities for the four types of volumes regularly used in the literature: Busemann, Holmes–Thompson, and Gromov’s mass and mass*. We examine the problem also for higher dimensions, and for centrally symmetric convex bodies.     

On a result by Dennis and Schnabel for Newton’s method: Further improvements

We improve local convergence results for Newton’s method by defining a more precise domain where the Newton iterates lie than in earlier studies using Dennis and Schnabel-type techniques. A numerical example is presented to show that the new convergence radii are larger and new error bounds are more precise than the earlier ones.     

On a model of thermoviscoelasticity of Jeffreys–Oldroyd type

In the plane case, the initial–boundary value problem for a thermoelastic medium model with a rheological relation determined by the Jeffreys–Oldroyd model is shown to be nonlocally weakly solvable. The study is based on separating the system, reducing it to an operator equation, and performing an iterative process.     

On a time-depending Monge–Ampère type equation

In this paper, we prove a comparison result between a solution u(x,t)$u(x,t)$, x∈Ω⊂R²$x\in \Omega \subset {\mathbb{R}}^2$, t∈(0,T)$t\in (0,T)$, of a time depending equation involving the Monge–Ampère operator in the plane and the solution of a conveniently symmetrized parabolic equation. To this aim, we prove a derivation formula for the integral of a smooth function g(x,t)$g(x,t)$ over sublevel sets of u$u$, {x∈Ω:u(x,t)<?}$\{x\in \Omega :u(x,t)<?\}$, ?∈R$?\in \mathbb{R}$, having the same perimeter in R²${\mathbb{R}}^2$.     

Harmonic number identities via the Newton–Andrews method

By means of the Bell polynomials, we establish explicit expressions of the higher-order derivatives of the binomial coefficient \(\binom{x+n}{m}\) and its reciprocal \(\binom{x+n}{m}^{-1}\) , and extend the application field of the Newton–Andrews method. As examples, we apply the results to the Chu–Vandermonde–Gauss formula and the Dougall–Dixon theorem and obtain a series of harmonic number identities. This paper generalizes some works presented before and provides a way to establish infinite harmonic number identities.     

Improved local convergence analysis of the Gauss–Newton method under a majorant condition

We present a local convergence analysis of Gauss-Newton method for solving nonlinear least square problems. Using more precise majorant conditions than in earlier studies such as Chen (Comput Optim Appl 40:97–118, 2008), Chen and Li (Appl Math Comput 170:686–705, 2005), Chen and Li (Appl Math Comput 324:1381–1394, 2006), Ferreira (J Comput Appl Math 235:1515–1522, 2011), Ferreira and Gonçalves (Comput Optim Appl 48:1–21, 2011), Ferreira and Gonçalves (J Complex 27(1):111–125, 2011), Li et al. (J Complex 26:268–295, 2010), Li et al. (Comput Optim Appl 47:1057–1067, 2004), Proinov (J Complex 25:38–62, 2009), Ewing, Gross, Martin (eds.) (The merging of disciplines: new directions in pure, applied and computational mathematics 185–196, 1986), Traup (Iterative methods for the solution of equations, 1964), Wang (J Numer Anal 20:123–134, 2000), we provide a larger radius of convergence; tighter error estimates on the distances involved and a clearer relationship between the majorant function and the associated least squares problem. Moreover, these advantages are obtained under the same computational cost.     

On the complex Falk–Langemeyer method

Numerical Algorithms - A new algorithm for the simultaneous diagonalization of two complex Hermitian matrices is derived. It is a proper generalization of the known Falk–Langemeyer algorithm...     

On a regularized Levenberg–Marquardt method for solving nonlinear inverse problems

We consider a regularized Levenberg–Marquardt method for solving nonlinear ill-posed inverse problems. We use the discrepancy principle to terminate the iteration. Under certain conditions, we prove the convergence of the method and obtain the order optimal convergence rates when the exact solution satisfies suitable source-wise representations.     

On Chudnovsky–Ramanujan type formulae

In a well known 1914 paper, Ramanujan gave a number of rapidly converging series for \(1/\pi \) which are derived using modular functions of higher level. Chudnovsky and Chudnovsky in their 1988 paper derived an analogous series representing \(1/\pi \) using the modular function J of level 1, which results in highly convergent series for \(1/\pi \), often used in practice. In this paper, we explain the Chudnovsky method in the context of elliptic curves, modular curves, and the Picard–Fuchs differential equation. In doing so, we also generalize their method to produce formulae which are valid around any singular point of the Picard–Fuchs differential equation. Applying the method to the family of elliptic curves parameterized by the absolute Klein invariant J of level 1, we determine all Chudnovsky–Ramanujan type formulae which are valid around one of the three singular points: \(0, 1, \infty \).