On exact order of convergence of random polynomials

For a sequence of independent, identically distributed random variable (iid rv's) [Formula: see text] and a sequence of integer-valued random variables [Formula: see text], define the random quantiles as [Formula: see text], where [Formula: see text] denote the largest integer less than or equal to [Formula: see text], and [Formula: see text] the [Formula: see text]th order statistic in a sample [Formula: see text] and [Formula: see text]. In this note, the limiting distribution and its exact order approximation are obtained for [Formula: see text]. The limiting distribution result we obtain extends the work of several including Wretman[Formula: see text]. The exact order of normal approximation generalizes the fixed sample size results of Reiss[Formula: see text]. AMS 2000 subject classification: 60F12; 60F05; 62G30.

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Zeros of derivatives of Gaussian random polynomials on plane domains

Journal of Mathematical Analysis and Applications ◽

10.1016/j.jmaa.2021.125229 ◽

2021 ◽

pp. 125229

Author(s):

Yu Wang ◽

Tao Jiang

Keyword(s):

Random Polynomials ◽

Derivatives Of ◽

Zeros Of Derivatives

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A necessary and sufficient condition for convergence of the zeros of random polynomials

Advances in Mathematics ◽

10.1016/j.aim.2021.107691 ◽

2021 ◽

Vol 384 ◽

pp. 107691

Author(s):

Duncan Dauvergne

Keyword(s):

Sufficient Condition ◽

Random Polynomials ◽

Necessary And Sufficient Condition ◽

Necessary And Sufficient

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Fractional Newton–Raphson Method Accelerated with Aitken’s Method

Axioms ◽

10.3390/axioms10020047 ◽

2021 ◽

Vol 10 (2) ◽

pp. 47

Author(s):

A. Torres-Hernandez ◽

F. Brambila-Paz ◽

U. Iturrarán-Viveros ◽

R. Caballero-Cruz

Keyword(s):

Fractional Derivative ◽

Iterative Methods ◽

Fractional Integral ◽

Order Of Convergence ◽

Fractional Operators ◽

Speed Of Convergence ◽

Newton Raphson ◽

Raphson Method

In the following paper, we present a way to accelerate the speed of convergence of the fractional Newton–Raphson (F N–R) method, which seems to have an order of convergence at least linearly for the case in which the order α of the derivative is different from one. A simplified way of constructing the Riemann–Liouville (R–L) fractional operators, fractional integral and fractional derivative is presented along with examples of its application on different functions. Furthermore, an introduction to Aitken’s method is made and it is explained why it has the ability to accelerate the convergence of the iterative methods, in order to finally present the results that were obtained when implementing Aitken’s method in the F N–R method, where it is shown that F N–R with Aitken’s method converges faster than the simple F N–R.

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Derivative-Free King’s Scheme for Multiple Zeros of Nonlinear Functions

Mathematics ◽

10.3390/math9111242 ◽

2021 ◽

Vol 9 (11) ◽

pp. 1242

Author(s):

Ramandeep Behl ◽

Sonia Bhalla ◽

Eulalia Martínez ◽

Majed Aali Alsulami

Keyword(s):

Iterative Methods ◽

Fourth Order ◽

Order Of Convergence ◽

Real Life ◽

Multiple Roots ◽

Computational Results ◽

Nonlinear Functions ◽

First Order ◽

Life Problems ◽

Derivative Free

There is no doubt that the fourth-order King’s family is one of the important ones among its counterparts. However, it has two major problems: the first one is the calculation of the first-order derivative; secondly, it has a linear order of convergence in the case of multiple roots. In order to improve these complications, we suggested a new King’s family of iterative methods. The main features of our scheme are the optimal convergence order, being free from derivatives, and working for multiple roots (m≥2). In addition, we proposed a main theorem that illustrated the fourth order of convergence. It also satisfied the optimal Kung–Traub conjecture of iterative methods without memory. We compared our scheme with the latest iterative methods of the same order of convergence on several real-life problems. In accordance with the computational results, we concluded that our method showed superior behavior compared to the existing methods.

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Time and space meshes adaptivity for the resolution of a semi-linear heat equation

Asymptotic Analysis ◽

10.3233/asy-201663 ◽

2021 ◽

pp. 1-10

Author(s):

Nejmeddine Chorfi

Keyword(s):

Parabolic Equation ◽

Heat Equation ◽

Order Of Convergence ◽

Linear Parabolic Equation ◽

Spectral Elements ◽

Euler Scheme ◽

Time Step ◽

Implicit Euler Scheme ◽

Linear Heat ◽

Error Indicators

The aim of this work is to highlight that the adaptivity of the time step when combined with the adaptivity of the spectral mesh is optimal for a semi-linear parabolic equation discretized by an implicit Euler scheme in time and spectral elements method in space. The numerical results confirm the optimality of the order of convergence. The later is similar to the order of the error indicators.

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