On the Exact Order of Convergence of Discrete Methods for Volterra-type Equations

1988 ◽  
Vol 8 (4) ◽  
pp. 511-515 ◽  
Author(s):  
JENNIFER SCOTT ◽  
SEAN MCKEE
Keyword(s):  
Order Of Convergence ◽  
Exact Order ◽  
Volterra Type ◽  
Discrete Methods

On exact order of convergence of random polynomials

2000 ◽  
Vol 99 (3) ◽  
pp. 1234-1243
Author(s):  
A. Basalykas ◽  
O. Yanushkevichiene
Keyword(s):  
Order Of Convergence ◽  
Exact Order ◽  
Random Polynomials

scholarly journals COLLOCATION APPROXIMATIONS FOR WEAKLY SINGULAR VOLTERRA INTEGRO‐DIFFERENTIAL EQUATIONS

2003 ◽  
Vol 8 (4) ◽  
pp. 315-328 ◽  
Author(s):  
I. Parts ◽  
A. Pedas
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Order Of Convergence ◽  
Piecewise Polynomial ◽  
Volterra Type ◽  
Weakly Singular ◽  
Uniform Grids ◽  
Attainable Order ◽  
Piecewise Polynomial Collocation Method ◽  
Polynomial Collocation

A piecewise polynomial collocation method for solving linear weakly singular integro‐differential equations of Volterra type is constructed. The attainable order of convergence of collocation approximations on arbitrary and quasi‐uniform grids is studied theoretically and numerically.

scholarly journals Spectral refinement using a new projection method

2004 ◽  
Vol 46 (2) ◽  
pp. 203-224 ◽  
Author(s):  
Rekha P. Kulkarni ◽  
N. Gnaneshwar
Keyword(s):  
Order Of Convergence ◽  
Projection Methods ◽  
Iteration Scheme ◽  
Spline Space ◽  
New Method ◽  
Galerkin Projection ◽  
Polynomial Space ◽  
Discrete Methods ◽  
Gauss Points ◽  
Better Than

AbstractIn this paper we consider two spectral refinement schemes, elementary and double iteration, for the approximation of eigenelements of a compact operator using a new approximating operator. We show that the new method performs better than the Galerkin, projection and Sloan methods. We obtain precise orders of convergence for the approximation of eigenelements of an integral operator with a smooth kernel using either the orthogonal projection onto a spline space or the interpolatory projection at Gauss points onto a discontinuous piecewise polynomial space. We show that in the double iteration scheme the error for the eigenvalue iterates using the new method is of the order of , where h is the mesh of the partition and k = 0, 1, 2,… denotes the step of the iteration. This order of convergence is to be compared with the orders in the Galerkin and projection methods and in the Sloan method. The error in eigenvector iterates is shown to be of the order of in the new method, in the Galerkin and projection methods and in the Sloan method. Similar improvement is observed in the case of the elementary iteration. We show that these orders of convergence are preserved in the corresponding discrete methods obtained by replacing the integration by a numerical quadrature formula. We illustrate this improvement in the order of convergence by numerical examples.

Discrete Methods and their Applications

1993 ◽  
Author(s):  
Peter L. Hammer ◽  
Fred S. Roberts ◽  
Gunzburger
Keyword(s):  
Discrete Methods

A Note on the Accuracy of Normal Approximation of Random Quantities

2021 ◽  
Vol 73 (1) ◽  
pp. 62-67
Author(s):  
Ibrahim A. Ahmad ◽  
A. R. Mugdadi
Keyword(s):  
Sample Size ◽  
Order Statistic ◽  
Normal Approximation ◽  
Order Approximation ◽  
Random Variables ◽  
Random Variable ◽  
Limiting Distribution ◽  
Exact Order ◽  
Fixed Sample ◽  
Independent Identically Distributed

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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