scholarly journals Convex interpolation by rational spline functions of class $ C ^ 2 $

2018 ◽  
pp. 62-67
Author(s):  
◽  
Abdul-Rashid Ramazanov ◽  
Vazipat Magomedova
Keyword(s):  
Spline Functions ◽  
Rational Spline ◽  
Convex Interpolation ◽  
Class C

On the approximation of $ \ exp (-x) $ on the half-axis by spline functions in three-point rational interpolants

2019 ◽  
pp. 32-37
Author(s):  
Abdul-Rashid Ramazanov ◽  
V.G. Magomedova
Keyword(s):  
Convergence Rate ◽  
Spline Functions ◽  
Rational Spline ◽  
Rational Interpolants ◽  
Half Axis

For the function $f(x)=\exp(-x)$, $x\in [0,+\infty)$ on grids of nodes $\Delta: 0=x_0<x_1<\dots $ with $x_n\to +\infty$ we construct rational spline-functions such that $R_k(x,f, \Delta)=R_i(x,f)A_{i,k}(x)\linebreak+R_{i-1}(x, f)B_{i,k}(x)$ for $x\in[x_{i-1}, x_i]$ $(i=1,2,\dots)$ and $k=1,2,\dots$ Here $A_{i,k}(x)=(x-x_{i-1})^k/((x-x_{i-1})^k+(x_i-x)^k)$, $B_{i,k}(x)=1-A_{i,k}(x)$, $R_j(x,f)=\alpha_j+\beta_j(x-x_j)+\gamma_j/(x+1)$ $(j=1,2,\dots)$, $R_j(x_m,f)=f(x_m)$ при $m=j-1,j,j+1$; we take $R_0(x,f)\equiv R_1(x,f)$. Bounds for the convergence rate of $R_k(x,f, \Delta)$ with $f(x)=\exp(-x)$, $x\in [0,+\infty)$, are found.

scholarly journals LINEAR/LINEAR RATIONAL SPLINE INTERPOLATION

2010 ◽  
Vol 15 (4) ◽  
pp. 447-455 ◽  
Author(s):  
Erge Ideon ◽  
Peeter Oja
Keyword(s):  
Spline Interpolation ◽  
Monotone Function ◽  
Uniform Mesh ◽  
Second Derivative ◽  
Numerical Examples ◽  
First Derivative ◽  
Rational Spline ◽  
Smoothness Class ◽  
Class C ◽  
Theoretical Results

For a strictly monotone function y on [a, b] we describe the construction of an interpolating linear/linear rational spline S of smoothness class C 1. We show that for the linear/linear rational splines we obtain ¦S(xi ) − y(xi )¦8 = O(h 4) on uniform mesh xi = a + ih, i = 0,…, n. We prove also the superconvergence of order h3 for the first derivative and of order h2 for the second derivative of S in certain points. Numerical examples support the obtained theoretical results. This work was supported by the Estonian Science Foundation grant 8313.

Approximation by rational spline functions

CALCOLO ◽  
2006 ◽  
Vol 43 (4) ◽  
pp. 279-286 ◽  
Author(s):  
Gancho T. Tachev
Keyword(s):  
Spline Functions ◽  
Rational Spline

Environmental Quality Assessment Based on a New Kind of Neural Network Using Rational Spline Weight Functions

2015 ◽  
Vol 713-715 ◽  
pp. 1708-1711
Author(s):  
Dai Yuan Zhang ◽  
Shan Jiang Hou
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Quality Assessment ◽  
Environmental Quality ◽  
Weight Functions ◽  
Spline Functions ◽  
Environmental Quality Assessment ◽  
Rational Spline ◽  
The Neural Network ◽  
Very High

As we all known, artificial neural network can be used in the process of environmental quality assessment. To improve the accuracy and science of assessment, a method of environmental quality assessment is presented in this paper, which is based on spline weight function (SWF) neural networks. The weigh functions of the neural network are composed of rational spline functions with cubic numerator and linear denominator (3/1 rational SWF). The simulation results show that, compared with the conventional BP neural networks, this method can get very high precision and accuracy. This case demonstrates that SWF neural networks can offer a very prospective tool for environmental quality assessment.

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