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 the convergence of an inexact Gauss–Newton trust-region method for nonlinear least-squares problems with simple bounds

We introduce an inexact Gauss–Newton trust-region method for solving bound-constrained nonlinear least-squares problems where, at each iteration, a trust-region subproblem is approximately solved by the Conjugate Gradient method. Provided a suitable control on the accuracy to which we attempt to solve the subproblems, we prove that the method has global and asymptotic fast convergence properties. Some numerical illustration is also presented.     

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....     

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 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.     

Global convergence of Newton’s method on an interval

The solution of an equation f(x)= given by an increasing function f on an interval I and right-hand side , can be approximated by a sequence calculated according to Newtons method. In this article, global convergence of the method is considered in the strong sense of convergence for any initial value in I and any feasible right-hand side. The class of functions for which the method converges globally is characterized. This class contains all increasing convex and increasing concave functions as well as sums of such functions on the given interval. The characterization is applied to Keplers equation and to calculation of the internal rate of return of an investment project.An earlier version was presented at the Joint National Meeting of TIMS and ORSA, Las Vegas, May 7–9, 1990. Financial support from Økonomisk Forskningsfond, Bodø, Norway, is gratefully acknowledged. The author thanks an anonymous referee for helpful comments and suggestions.     

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.     

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 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: \(0, 1, \infty \).     

On the iteratively regularized Gauss–Newton method in Banach spaces with applications to parameter identification problems

In this paper we propose an extension of the iteratively regularized Gauss–Newton method to the Banach space setting by defining the iterates via convex optimization problems. We consider some a posteriori stopping rules to terminate the iteration and present the detailed convergence analysis. The remarkable point is that in each convex optimization problem we allow non-smooth penalty terms including $L^1$ and total variation like penalty functionals. This enables us to reconstruct special features of solutions such as sparsity and discontinuities in practical applications. Some numerical experiments on parameter identification in partial differential equations are reported to test the performance of our method.     

On the attraction of Newton’s method to critical lagrange multipliers

The attraction of dual trajectories of Newton’s method for the Lagrange system to critical Lagrange multipliers is analyzed. This stable effect, which has been confirmed by numerical practice, leads to the Newton-Lagrange method losing its superlinear convergence when applied to problems with irregular constraints. At the same time, available theoretical results are of “negative” character; i.e., they show that convergence to a noncritical multiplier is not possible or unlikely. In the case of a purely quadratic problem with a single constraint, a “positive” result is proved for the first time demonstrating that the critical multipliers are attractors for the dual trajectories. Additionally, the influence exerted by the attraction to critical multipliers on the convergence rate of direct and dual trajectories is characterized.     

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.     

Multigrid Newton–Krylov method for radiation in diffusive semitransparent media

We present a fast multigrid solver for simplified _PN$P_{\mathrm{N}}$ (_SPN${\mathrm{SP}}_{\mathrm{N}}$) approximations to the diffusive radiation in non-grey semitransparent media. The method consists on reformulating the equations as a nonlinear fixed point problem in the temperature only. Given a mesh hierarchy, time and space discretizations are performed using second-order implicit and finite differencing methods, respectively. At each mesh level, a Newton–Krylov algorithm is applied to the discrete equations. As a smoother on the coarse meshes we propose the Atkinson–Brakhage operator. Numerical results are shown for glass cooling process using different geometry enclosures. The _SPN${\mathrm{SP}}_{\mathrm{N}}$ approximations capture the correct asymptotic behavior of the numerical solution with a computational cost lower than using the full radiative transfer equations.     

An improved local convergence analysis for Newton–Steffensen-type method

We provide a local convergence analysis for Newton–Steffensen-type algorithm for solving nonsmooth perturbed variational inclusions in Banach spaces. Under new center–conditions and the Aubin continuity property, we obtain the linear local convergence of Newton–Steffensen method. Our results compare favorably with related obtained in (Argyros and Hilout, 2007 submitted; Hilout in J. Math. Anal. Appl. 339:753–761, 2008).     

Gauss–Newton method for convex composite optimizations on Riemannian manifolds

A notion of quasi-regularity is extended for the inclusion problem ${F(p)\in C}$ , where F is a differentiable mapping from a Riemannian manifold M to ${\mathbb R^n}$ . When C is the set of minimum points of a convex real-valued function h on ${\mathbb R^n}$ and DF satisfies the L-average Lipschitz condition, we use the majorizing function technique to establish the semi-local convergence of sequences generated by the Gauss-Newton method (with quasi-regular initial points) for the convex composite function h ? F on Riemannian manifold. Two applications are provided: one is for the case of regularities on Riemannian manifolds and the other is for the case when C is a cone and DF(p ₀)(·) ? C is surjective. In particular, the results obtained in this paper extend the corresponding one in Wang et?al. (Taiwanese J Math 13:633?C656, 2009).     

On the local convergence of Newton’s method under generalized conditions of Kantorovich

Following an idea similar to that given by Dennis and Schnabel (1996) in [2], we prove a local convergence result for Newton’s method under generalized conditions of Kantorovich type.