Many algorithms have been proposed for finding a local minimum of a function f ( x ) f rom R" into R ~ using only function and gradient values. These include discrete Newton methods, quasi -Newton methods, and the conjugate gradient algorithm. For a discussion of these methods, see, for example [1]. There are classes of problems for which each of these methods is the most efficient. In this pap...

We consider the nonlinear systems of equations that result from discretizations of a prototype variational model for the equilibrium director field characterizing the orientational properties of a liquid crystal material. In the presence of pointwise unit-vector constraints and coupled electric fields, the numerical solution of such equations by Lagrange-Newton methods leads to linear systems w...

Numerical reservoir simulation models are used in the energy industry to forecast the behavior of hydrocarbon reservoirs and connected surface facilities over long production periods. The simulation of oil and gas reservoirs requires the construction and solution of large, sparse linear systems with millions or tens of millions of unknowns at each step of a Newton iteration. The problem size an...

Keywords: Sixth-order convergence Extraneous fixed point Asymptotic error constant Efficiency index Double-Newton method Basin of attraction a b s t r a c t A high-order family of two-point methods costing two derivatives and two functions are developed by introducing a two-variable weighting function in the second step of the classical double-Newton method. Their theoretical and computational ...

The smoothing Newton method for solving a system of nonsmooth equations F (x) = 0, which may arise from the nonlinear complementarity problem, the variational inequality problem or other problems, can be regarded as a variant of the smoothing method. At the kth step, the nonsmooth function F is approximated by a smooth function f(·, εk), and the derivative of f(·, εk) at xk is used as the Newto...

In this thesis we study algorithms for online convex optimization and their relation to approximate optimization. In the first part, we propose a new algorithm for a general online optimization framework called online convex optimization. Whereas previous efficient algorithms are mostly gradient-descent based, the new algorithm is inspired by the Newton-Raphson method for convex optimization, a...

In this paper we develop efficient algorithms to compute limit cycles and their isochrones (i.e. the sets of points with the same asymptotic phase) for planar vector fields. We formulate a functional equation for the parameterization of the invariant cycle and its isochrones and we show that it can be solved by means of a Newton method. Using the right transformations, we can solve the equation...

Various methods have been developed for computing the correlation matrix nearest in the Frobenius norm to a given matrix. We focus on a quadratically convergent Newton algorithm recently derived by Qi and Sun. Various improvements to the efficiency and reliability of the algorithm are introduced. Several of these relate to the linear algebra: the Newton equations are solved by minres instead of...

In the numerical solution of ODEs by implicit time-stepping methods, a system of (nonlinear) equations has to be solved each step. It is common practice to use xed-point iterations or, in the stii case, some modiied Newton iteration. The convergence rate of such methods depends on the stepsize. Similarly, a stepsize change may force a refactorization of the iteration matrix in the Newton solver...

5 Non-negative matrix factorization (NMF) has become a popular machine 6 learning approach to many problems in text mining, speech and image 7 processing, bio-informatics and seismic data analysis to name a few. In 8 NMF, a matrix of non-negative data is approximated by the low-rank 9 product of two matrices with non-negative entries. In this paper, the 10 approximation quality is measured by t...