Optimization plays a key role in machine learning. Recently, stochastic second-order methods have attracted much attention due to their low computational cost in each iteration. However, these algorithms might perform poorly especially if it is hard to approximate the Hessian well and efficiently. As far as we know, there is no effective way to handle this problem. In this paper, we resort to N...

Interval Newton methods are widely used to find reliable enclosures for the roots of the polynomial systems. But the computation process needs the evaluation of an enclosure for the derivative of the polynomial and evaluation of the polynomial at a particular point. Again, the subdivisions (if any are needed) will require repeated evaluation of the function values. We propose an alternative app...

Newton-Krylov methods, combination of Newton-like methods and Krylov subspace methods for solving the Newton equations, often need adequate preconditioning in order to be successful. Approximations of the Jacobian matrices are required to form preconditioners and this step is very often the dominant cost of Newton-Krylov methods. Therefore, working with preconditioners destroys in principle the...

In the Newton/log-barriermethod,Newton steps are taken for the log-barrier function for a xed value of the barrier parameter until a certain convergence criterion is satis ed. The barrier parameter is then decreased and the Newton process is repeated. A naive analysis indicates that Newton's method does not exhibit superlinear convergence to the minimizer of each instance of the log-barrier fun...

Newton iteration is known (under some precise conditions) to converge quadratically to zeros of non-degenerate systems of polynomials. This and other properties may be used to obtain theorems on the global complexity of solving systems of polynomial equations (See Shub and Smale in 6]), using a model of computability over the reals. However, it is not practical (and not desirable) to actually c...

The connection between closed Newton-Cotes, trigonometrically-fitted differential methods and symplectic integrators is investigated in this paper. It is known from the literature that several one step symplectic integrators have been obtained based on symplectic geometry. However, the investigation of multistep symplectic integrators is very poor. Zhu et al. (1996) presented the well known ope...

Newton-Krylov methods, primarily using the Jacobian-Free Newton-Krylov (JFNK) approximation, are examined as an alternative to the traditional power iteration method for the calculation of the fundamental eigenmode in reactor analysis applications based on diffusion theory. One JFNK approach can be considered an acceleration technique for the standard power iteration as it is “wrapped around” t...

The launch of the Chandra (NASA) and XMM-Newton (ESA) X-ray observatories in 1999 has revolutionized our view of the Universe, by providing astro-physical information about many classes of sources with unprecedent detail. The high throughput of XMM-Newton makes it the ideal instrument to provide low to moderate resolution spectroscopy of faint and extended sources. After 3 years of operations, ...

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...