Computing multiple zeros by using a parameter in Newton-Secant method
University Mediterranea of Reggio Calabria（雷焦卡拉布里亚地中海大学）；Young Researchers and Elite Club, Hamedan Branch, Islamic Azad University（伊斯兰阿扎德大学哈马丹分校青年研究者与精英俱乐部）；Technische Universität Dresden（德累斯顿工业大学）；MEDAlics, Research Centre at the University Dante Alighieri（但丁阿利吉耶里大学MEDAlics研究中心）
Massimiliano Ferrara, Somayeh Sharifi, Mehdi Salimi
In this paper, we modify the Newton-Secant method with third order of convergence for finding multiple roots of nonlinear equations. Per iteration this method requires two evaluations of the function and one evaluation of its first derivative. This method has the efficiency index equal to $3^{\frac{1}{3}}\approx 1.44225$. We describe the analysis of the proposed method along with numerical experiments including comparison with existing methods. Moreover, the dynamics of the proposed method are shown with some comparisons to the other existing methods.
Optimal Newton-Secant like methods without memory for solving nonlinear equations with its dynamics
Technische Universität Dresden（德累斯顿工业大学）；Department of Mathematics, Hamedan Branch, Islamic Azad University（伊斯兰阿扎德大学哈马丹分校数学系）；Young Researchers and Elite Club, Hamedan Branch, Islamic Azad University（伊斯兰阿扎德大学哈马丹分校青年研究者与精英俱乐部）
Quasi-Newton techniques approximate the Newton step by estimating the Hessian using the so-called secant equations. Some of these methods compute the Hessian using several secant equations but produce non-symmetric updates. Other quasi-Newton schemes, such as BFGS, enforce symmetry but cannot satisfy more than one secant equation. We propose a new type of quasi-Newton symmetric update using several secant equations in a least-squares sense. Our approach generalizes and unifies the design of quasi-Newton updates and satisfies provable robustness guarantees.
In this paper, we introduce a new variant of the BFGS method designed to perform well when gradient measurements are corrupted by noise. We show that by treating the secant condition with a penalty method approach motivated by regularized least squares estimation, one can smoothly interpolate between updating the inverse Hessian approximation with the original BFGS update formula and not updating the inverse Hessian approximation. Furthermore, we find the curvature condition is smoothly relaxed as the interpolation moves towards not updating the inverse Hessian approximation, disappearing enti…
In large-scale optimization, when either forming or storing Hessian matrices are prohibitively expensive, quasi-Newton methods are often used in lieu of Newton's method because they only require first-order information to approximate the true Hessian. Multipoint symmetric secant (MSS) methods can be thought of as generalizations of quasi-Newton methods in that they attempt to impose additional requirements on their approximation of the Hessian. Given an initial Hessian approximation, MSS methods generate a sequence of possibly-indefinite matrices using rank-2 updates to solve nonconvex unconst…
We propose a new modification of Newton iteration for finding some nonnegative Z-eigenpairs of a nonnegative tensor. The method has local quadratic convergence to a nonnegative eigenpair of a nonnegative tensor, under the usual assumption guaranteeing the local quadratic convergence of the original Newton iteration.
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