No Arabic abstract
Newtons method for polynomial root finding is one of mathematics most well-known algorithms. The method also has its shortcomings: it is undefined at critical points, it could exhibit chaotic behavior and is only guaranteed to converge locally. Based on the {it Geometric Modulus Principle} for a complex polynomial $p(z)$, together with a {it Modulus Reduction Theorem} proved here, we develop the {it Robust Newtons method} (RNM), defined everywhere with a step-size that guarantees an {it a priori} reduction in polynomial modulus in each iteration. Furthermore, we prove RNM iterates converge globally, either to a root or a critical point. Specifically, given $varepsilon $ and any seed $z_0$, in $t=O(1/varepsilon^{2})$ iterations of RNM, independent of degree of $p(z)$, either $|p(z_t)| leq varepsilon$ or $|p(z_t) p(z_t)| leq varepsilon$. By adjusting the iterates at {it near-critical points}, we describe a {it modified} RNM that necessarily convergence to a root. In combination with Smales point estimation, RNM results in a globally convergent Newtons method having a locally quadratic rate. We present sample polynomiographs that demonstrate how in contrast with Newtons method RNM smooths out the fractal boundaries of basins of attraction of roots. RNM also finds potentials in computing all roots of arbitrary degree polynomials. A particular consequence of RNM is a simple algorithm for solving cubic equations.
We propose a inexact Newton method for solving inverse eigenvalue problems (IEP). This method is globalized by employing the classical backtracking techniques. A global convergence analysis of this method is provided and the R-order convergence property is proved under some mild assumptions. Numerical examples demonstrate that the proposed method is very effective for solving the IEP with distinct eigenvalues.
The paper proposes and justifies a new algorithm of the proximal Newton type to solve a broad class of nonsmooth composite convex optimization problems without strong convexity assumptions. Based on advanced notions and techniques of variational analysis, we establish implementable results on the global convergence of the proposed algorithm as well as its local convergence with superlinear and quadratic rates. For certain structural problems, the obtained local convergence conditions do not require the local Lipschitz continuity of the corresponding Hessian mappings that is a crucial assumption used in the literature to ensure a superlinear convergence of other algorithms of the proximal Newton type. The conducted numerical experiments of solving the $l_1$ regularized logistic regression model illustrate the possibility of applying the proposed algorithm to deal with practically important problems.
This paper proposes a novel stochastic version of damped and regularized BFGS method for addressing the above problems.
In current work, non-familiar shifted Lucas polynomials are introduced. We have constructed a computational wavelet technique for solution of initial/boundary value second order differential equations. For this numerical scheme, we have developed weight function and Rodrigues formula for Lucas polynomials. Further, Lucas polynomials and their properties are used to propose shifted Lucas polynomials and then utilization of shifted Lucas polynomials provides us shifted Lucas wavelet. We furnished the operational matrix of differentiation and the product operational matrix of the shifted Lucas wavelets. Moreover, convergence and error analysis ensure accuracy of the proposed method. Illustrative examples show that the present method is numerically fruitful, effective and convenient for solving differential equations
In this paper, we develop a first order (in time) numerical scheme for the binary fluid surfactant phase field model. The free energy contains a double-well potential, a nonlinear coupling entropy and a Flory-Huggins potential. The resulting coupled system consists of two Cahn-Hilliard type equations. This system is solved numerically by finite difference spatial approximation, in combination with convex splitting temporal discretization. We prove the proposed scheme is unique solvable, positivity-preserving and unconditionally energy stable. In addition, an optimal rate convergence analysis is provided for the proposed numerical scheme, which will be the first such result for the binary fluid-surfactant system. Newton iteration is used to solve the discrete system. Some numerical experiments are performed to validate the accuracy and energy stability of the proposed scheme.