相关论文: On optimization and solution of roots of a functio…

Some fractional Newton methods have been proposed in order to find roots of nonlinear equations using fractional derivatives. In this paper we introduce a fractional Newton method with order $\alpha+1$ and compare with another fractional…

A new method of root finding is formulated that uses a numerical iterative process involving three points. A given function y = f(x) whose roots are desired is fitted and approximated by a polynomial function of the form P(x)= a(x-b)^N that…

We use Newton's method to find all roots of several polynomials in one complex variable of degree up to and exceeding one million and show that the method, applied to appropriately chosen starting points, can be turned into an algorithm…

The following document presents some novel numerical methods valid for one and several variables, which using the fractional derivative, allow to find solutions for some non-linear systems in the complex space using real initial conditions.…

Many problems in applied mathematics require root finding algorithms. Unfortunately, root finding methods have limitations. Firstly, regarding the convergence, there is a trade-off between the size of it's domain and it's rate. Secondly the…

Finding roots of equations is at the heart of most computational science. A well-known and widely used iterative algorithm is the Newton's method. However, its convergence depends heavily on the initial guess, with poor choices often…

Functional iterations such as Newton's are a popular tool for polynomial root-finding. We consider realistic situation where some (e.g., better-conditioned) roots have already been approximated and where further computations is directed to…

Root-finding method is an iterative process that constructs a sequence converging to a solution of an equation. Householder's method is a higher-order method that requires higher order derivatives of the reciprocal of a function and has…

We investigate Newton's method as a root finder for complex polynomials of arbitrary degree. While polynomial root finding continues to be one of the fundamental tasks of computing, with essential use in all areas of theoretical…

The accuracy of the numerical solution of a fractional differential equation depends on the differentiability class of the solution. The derivatives of the solutions of fractional differential equations often have a singularity at the…

We give a new improvement over Newton's method for root-finding, when the function in question is doubly differentiable. It generally exhibits faster and more reliable convergence. It can be also be thought of as a correction to Halley's…

When Newton's method, or Halley's method is used to approximate the $p${th} root of $1-z$, a sequence of rational functions is obtained. In this paper, a beautiful formula for these rational functions is proved in the square root case,…

The Newton, Gauss--Newton and Levenberg--Marquardt methods all use the first derivative of a vector function (the Jacobian) to minimise its sum of squares. When the Jacobian matrix is ill-conditioned, the function varies much faster in some…

This paper demonstrates new methods and implementations of nonlinear solvers with higher-order of convergence, which is achieved by efficiently computing higher-order derivatives. Instead of computing full derivatives, which could be…

This paper introduces a new functional expansion framework that extends classical ideas beyond the Taylor series. Unlike traditional Taylor expansions based on local polynomial approximations, the proposed approach arises from exact…

We consider the problem of numerically identifying roots of a target function - under the constraint that we can only measure the derivatives of the function at a given point, not the function itself. We describe and characterize two…

The fractional calculus of variations and fractional optimal control are generalizations of the corresponding classical theories, that allow problem modeling and formulations with arbitrary order derivatives and integrals. Because of the…

A zero-finding technique for solving nonlinear equations more efficiently than they usually are with traditional iterative methods in which the order of convergence is improved is presented. The key idea in deriving this procedure is to…

We describe a method for calculating the roots of special functions satisfying second order linear ordinary differential equations. It exploits the recent observation that the solutions of a large class of such equations can be represented…

We extend Robust Optimization to fractional programming, where both the objective and the constraints contain uncertain parameters. Earlier work did not consider uncertainty in both the objective and the constraints, or did not use Robust…