Related papers: Root Finding by High Order Iterative Methods Based…
We consider the theoretical and numerical aspects of the quadrature rules associated with a sequence of polynomials generated by a special $R_{II}$ recurrence relation. We also look into some methods for generating the nodes (which lie on…
We generalize Newton-type methods for minimizing smooth functions to handle a sum of two convex functions: a smooth function and a nonsmooth function with a simple proximal mapping. We show that the resulting proximal Newton-type methods…
Newton's method is a fundamental technique in optimization with quadratic convergence within a neighborhood around the optimum. However reaching this neighborhood is often slow and dominates the computational costs. We exploit two…
We find a family of convergent schemes of nodes for non-complete interpolatory quadrature rules.
Evaluating or finding the roots of a polynomial $f(z) = f_0 + \cdots + f_d z^d$ with floating-point number coefficients is a ubiquitous problem. By using a piecewise approximation of $f$ obtained with a careful use of the Newton polygon of…
An inexact Newton type method for numerical minimization of convex piecewise quadratic functions is considered and its convergence is analyzed. Earlier, a similar method was successfully applied to optimizaton problems arising in numerical…
In many iterative optimization methods, fixed-point theory enables the analysis of the convergence rate via the contraction factor associated with the linear approximation of the fixed-point operator. While this factor characterizes the…
Given a function f: [a,b] -> R, if f(a) < 0 and f(b)> 0 and f is continuous, the Intermediate Value Theorem implies that f has a root in [a,b]. Moreover, given a value-oracle for f, an approximate root of f can be computed using the…
We conjecture that the roots of a degree-n univariate complex polynomial are located in a union of n-1 annuli, each of which is centered at a root of the derivative and whose radii depend on higher derivatives. We prove the conjecture for…
We develop and analyze iterative methods for computing the principal square root of third-order tensors under the T-product framework. Tensor extensions of the Newton iteration (quadratic convergence) and the Denman--Beavers iteration…
We are concerned with the reconstruction of a sound-soft obstacle using far field measurements of the scattered waves associated with incident plane waves sent from one direction but at multiple frequencies. We define, for each frequency,…
In this paper, we develop a regularized higher-order Taylor based method for solving composite (e.g., nonlinear least-squares) problems. At each iteration, we replace each smooth component of the objective function by a higher-order Taylor…
Relaxed Newton's method is a one-parameter family of root-finding methods that generalizes the classical Newton's method. When viewed as a rational map on the Riemann sphere, this family exhibits rich and subtle global dynamics that depend…
This paper proposes and develops a new Newton-type algorithm to solve subdifferential inclusions defined by subgradients of extended-real-valued prox-regular functions. The proposed algorithm is formulated in terms of the second-order…
We develop a computationally efficient algorithm for the automatic regularization of nonlinear inverse problems based on the discrepancy principle. We formulate the problem as an equality constrained optimization problem, where the…
We propose an approach to constructing iterative methods for finding polynomial roots simultaneously. One feature of this approach is using the fundamental theorem of symmetric polynomials. Within this framework, we reconstruct many of the…
Combinatorial classes T that are recursively defined using combinations of the standard multiset, sequence, directed cycle and cycle constructions, and their restrictions, have generating series T(z) with a positive radius of convergence;…
In this paper we construct high order numerical methods for solving third and fourth orders nonlinear functional differential equations (FDE). They are based on the discretization of iterative methods on continuous level with the use of the…
Newton's method is the most widespread high-order method, demanding the gradient and the Hessian of the objective function. However, one of the main disadvantages of Newtons method is its lack of global convergence and high iteration cost.…
The iterative problem of solving nonlinear equations is studied. A new Newton like iterative method with adjustable parameters is designed based on the dynamic system theory. In order to avoid the derivative function in the iterative…