scholarly journals The numerical stability of a Laguerre-like method for the simultaneous inclusion of polynomial zeros

2006 ◽  
pp. 93-109
Author(s):  
Miodrag Petkovic ◽  
Dusan Milosevic
Keyword(s):  
Convergence Rate ◽  
Iterative Method ◽  
Fourth Order ◽  
Numerical Stability ◽  
Convergence Order ◽  
Polynomial Zeros ◽  
Rounding Errors

The numerical stability of the fourth order iterative method of Laguerre's type for the simultaneous inclusion of polynomial zeros is analyzed in the presence of rounding errors. We state conditions under which the convergence order of the considered method is preserved. If these conditions are relaxed the convergence rate reduces to three.

scholarly journals Iterative method for solving the second boundary value problem (BVP) for Biharmonic type equation

2016 ◽  
Vol 14 (4) ◽  
pp. 66-72
Author(s):  
Đặng Quang Á
Keyword(s):  
Differential Equations ◽  
Convergence Rate ◽  
Iterative Method ◽  
Boundary Value Problem ◽  
Fourth Order ◽  
Boundary Value ◽  
Type Equation ◽  
Second Order ◽  
Second Order Equations

Solving BVPs for the fourth order differential equations by the reduction of them to BVPs for the  second order equations with the aim to use the achievements for the latter ones attracts attention from many researchers. In this paper, using the technique developed by  ourselves in recent works, we construct iterative method for the second BVP for  biharmonic type equation. The convergence rate of  the method is established.

scholarly journals Iterative Method for Solving the Second Boundary Value Problem for Biharmonic-Type Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Dang Quang A. ◽  
Nguyen Van Thien
Keyword(s):  
Differential Equations ◽  
Convergence Rate ◽  
Iterative Method ◽  
Boundary Value Problem ◽  
Fourth Order ◽  
Boundary Value ◽  
Type Equation ◽  
Numerical Examples ◽  
Optimal Value ◽  
Second Order Equations

Solving boundary value problems (BVPs) for the fourth-order differential equations by the reduction of them to BVPs for the second-order equations with the aim to use the achievements for the latter ones attracts attention from many researchers. In this paper, using the technique developed by ourselves in recent works, we construct iterative method for the second BVP for biharmonic-type equation, which describes the deflection of a plate resting on a biparametric elastic foundation. The convergence rate of the method is established. The optimal value of the iterative parameter is found. Several numerical examples confirm the efficiency of the proposed method.

scholarly journals Multi-Step Preconditioned Newton Methods for Solving Systems of Nonlinear Equations

2017 ◽  
Author(s):  
Fayyaz Ahmad ◽  
Malik Zaka Ullah ◽  
Ali Saleh Alshomrani ◽  
Shamshad Ahmad ◽  
Aisha M. Alqahtani ◽  
...  
Keyword(s):  
Iterative Method ◽  
Nonlinear Equations ◽  
Numerical Stability ◽  
Order Of Convergence ◽  
Convergence Order ◽  
Systems Of Nonlinear Equations ◽  
System Of Nonlinear Equations ◽  
Newton Methods ◽  
Rapid Convergence ◽  
Base Method

The study of different forms of preconditioners for solving a system of nonlinear equations, by using Newton’s method, is presented. The preconditioners provide numerical stability and rapid convergence with reasonable computation cost, whenever chosen accurately. Different families of iterative methods can be constructed by using a different kind of preconditioners. The multi-step iterative method consists of a base method and multi-step part. The convergence order of base method is quadratic and each multi-step add an additive factor of one in the previously achieved convergence order. Hence the convergence of order of an m-step iterative method is m + 1. Numerical simulations confirm the claimed convergence order by calculating the computational order of convergence. Finally, the numerical results clearly show the benefit of preconditioning for solving system of nonlinear equations.

scholarly journals Local and Semilocal Convergence of Wang-Zheng’s Method for Simultaneous Finding Polynomial Zeros

Symmetry ◽  
2019 ◽  
Vol 11 (6) ◽  
pp. 736
Author(s):  
Slav I. Cholakov
Keyword(s):  
Iterative Method ◽  
Error Estimates ◽  
Fourth Order ◽  
Semilocal Convergence ◽  
Polynomial Zeros ◽  
Convergence Theorems ◽  
Appl Math

In 1984, Wang and Zheng (J. Comput. Math. 1984, 1, 70–76) introduced a new fourth order iterative method for the simultaneous computation of all zeros of a polynomial. In this paper, we present new local and semilocal convergence theorems with error estimates for Wang–Zheng’s method. Our results improve the earlier ones due to Wang and Wu (Computing 1987, 38, 75–87) and Petković, Petković, and Rančić (J. Comput. Appl. Math. 2007, 205, 32–52).

scholarly journals Two solutions to Kirchhoff-type fourth-order implusive elastic beam equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Jian Liu ◽  
Wenguang Yu
Keyword(s):  
Iterative Method ◽  
Variational Methods ◽  
Critical Point ◽  
Fourth Order ◽  
Elastic Beam ◽  
Kirchhoff Type ◽  
Newton Iterative Method ◽  
Beam Equations ◽  
Critical Point Theorems ◽  
Elastic Beam Equations

AbstractIn this paper, the existence of two solutions for superlinear fourth-order impulsive elastic beam equations is obtained. We get two theorems via variational methods and corresponding two-critical-point theorems. Combining with the Newton-iterative method, an example is presented to illustrate the value of the obtained theorems.

An Accelerated Iterative Method with Third-Order Convergence

2012 ◽  
Vol 220-223 ◽  
pp. 2658-2661
Author(s):  
Zhong Yong Hu ◽  
Liang Fang ◽  
Lian Zhong Li
Keyword(s):  
Iterative Method ◽  
Fourth Order ◽  
Newton’S Method ◽  
Newton's Method ◽  
New Method ◽  
Order Convergence ◽  
Third Order ◽  
Numerical Examples ◽  
Modified Newton’S Method ◽  
Jarratt Method

We present a new modified Newton's method with third-order convergence and compare it with the Jarratt method, which is of fourth-order. Based on this new method, we obtain a family of Newton-type methods, which converge cubically. Numerical examples show that the presented method can compete with Newton's method and other known third-order modifications of Newton's method.

scholarly journals Image denoising using anisotropic second and fourth order diffusions based on gradient vector convolution

2012 ◽  
Vol 9 (4) ◽  
pp. 1493-1511 ◽  
Author(s):  
Huaibin Wang ◽  
Yuanquan Wang ◽  
Wenqi Ren
Keyword(s):  
Image Denoising ◽  
Fourth Order ◽  
Numerical Stability ◽  
Anisotropic Diffusion ◽  
Computational Cost ◽  
Gradient Vector ◽  
Order Model ◽  
Robustness To Noise ◽  
Fourth Order Model ◽  
Low Computational Cost

In this paper, novel second order and fourth order diffusion models are proposed for image denoising. Both models are based on the gradient vector convolution (GVC) model. The second model is coined by incorporating the GVC model into the anisotropic diffusion model and the fourth order one is by introducing the GVC to the You-Kaveh fourth order model. Since the GVC model can be implemented in real time using the FFT and possesses high robustness to noise, both proposed models have many advantages over traditional ones, such as low computational cost, high numerical stability and remarkable denoising effect. Moreover, the proposed fourth order model is an anisotropic filter, so it can obviously improve the ability of edge and texture preserving except for further improvement of denoising. Some experiments are presented to demonstrate the effectiveness of the proposed models.

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