Rational spline functions

2001 ◽  
pp. 247-300 ◽  
Author(s):  
Ren-Hong Wang
Keyword(s):  
Spline Functions ◽  
Rational Spline

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.

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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