In this paper, we present a new family of two-step Newton-type iterative methods with memory for solving nonlinear equations. In order to obtain a Newton-type method with memory, we first present an optimal two-parameter fourth-order Newton-type method without memory. Then, based on the two-parameter method without memory, we present a new two-parameter Newton-type method with memory. Using two...

It has long been known that variational inequality problems can be reformulated as nonsmooth equations. Recently, locally high-order convergent Newton methods for nonsmooth equations have been well established via the concept of semismoothness. When the constraint set of the variational inequality problem is a rectangle, several locally convergent Newton methods for the reformulated nonsmooth e...

The variable fractional-delay (FD) filter structure by Tassart and Depalle performs Lagrange interpolation in an efficient way. We point out that this structure directly corresponds to Newton’s interpolation (backward difference) formula, hence we prefer to refer to it as the Newton FD filter. This structure does not function correctly when the fractional delay is made time-variant, e.g., in sa...

A novel type of fractals (i.e., Zhang fractals) is yielded via solving time-varying or static nonlinear equations in complex domain by discrete-time complex-valued Zhang dynamics (DTCVZD). The DTCVZD model that uses different types of activation functions can generate various Zhang fractals. These fractals are different from the conventional Newton fractals discovered 30 years ago (since 1983) ...

Computational experience with several limited-memory quasi-Newton and truncated Newton methods for unconstrained nonlinear optimization is described. Comparative tests were conducted on a well-known test library [J. on several synthetic problems allowing control of the clustering of eigenvalues in the Hessian spectrum, and on some large-scale problems in oceanography and meteorology. The result...

The local superlinear convergence of the generalized Newton method for solving systems of nonsmooth equations has been proved by Qi and Sun under the semismooth condition and nonsingularity of the generalized Jacobian at the solution. Unlike the Newton method for systems of smooth equations, globalization of the generalized Newton method seems dif-cult to achieve in general. However, we show th...

Recently, Li et al. (Comput. Optim. Appl. 26:131–147, 2004) proposed a regularized Newton method for convex minimization problems. The method retains local quadratic convergence property without requirement of the singularity of the Hessian. In this paper, we develop a truncated regularized Newton method and show its global convergence. We also establish a local quadratic convergence theorem fo...

This paper studies Newton-type methods for minimization of partly smooth convex functions. Sequential Newton methods are provided using local parameterizations obtained from U -Lagrangian theory and from Riemannian geometry. The Hessian based on the U -Lagrangian depends on the selection of a dual parameter g; by revealing the connection to Riemannian geometry, a natural choice of g emerges for...

This paper studies Newton-type methods for minimization of partly smooth convex functions. Sequential Newton methods are provided using local parameterizations obtained from U-Lagrangian theory and from Riemannian geometry. The Hessian based on the ULagrangian depends on the selection of a dual parameter g; by revealing the connection to Riemannian geometry, a natural choice of g emerges for wh...