scholarly journals On Hermite-Fejér type interpolation on the Chebyshev nodes

1993 ◽  
Vol 47 (1) ◽  
pp. 13-24 ◽  
Author(s):  
Graeme J. Byrne ◽  
T.M. Mills ◽  
Simon J. Smith
Keyword(s):  
Approximation Error ◽  
Interpolation Polynomial ◽  
Precise Estimate ◽  
Type Interpolation ◽  
Interpolation Methods ◽  
Chebyshev Nodes ◽  
Interpolatory Process

Given f ∈ C [−1, 1], let Hn, 3(f, x) denote the (0,1,2) Hermite-Fejér interpolation polynomial of f based on the Chebyshev nodes. In this paper we develop a precise estimate for the magnitude of the approximation error |Hn, 3(f, x) − f(x)|. Further, we demonstrate a method of combining the divergent Lagrange and (0,1,2) interpolation methods on the Chebyshev nodes to obtain a convergent rational interpolatory process.

scholarly journals The theorem about the transformer excitation current waveform mapping into the dynamic hysteresis loop branch for the sinusoidal magnetic flux case

2015 ◽  
Vol 12 (1) ◽  
pp. 33-52
Author(s):  
Nenad Petrovic ◽  
Velibor Pjevalica ◽  
Vladimir Vujicic
Keyword(s):  
Magnetic Flux ◽  
Hysteresis Loop ◽  
Hysteresis Loops ◽  
Stationary Regime ◽  
Interpolation Polynomial ◽  
Magnetic Core ◽  
Excitation Current ◽  
Approximation Accuracy ◽  
Dynamic Hysteresis ◽  
Chebyshev Nodes

This paper analyses aspects of the approximation theory application on the certain subsets of the measured samples of the transformer excitation current and the sinusoidal magnetic flux. The presented analysis is performed for single-phase transformer case, Epstein frame case and toroidal core case. In the paper the theorem of direct mapping the transformer excitation current in the stationary regime is proposed. The excitation current is mapped to the dynamic hysteresis loop branch (in further text DHLB) by an appropriate cosine transformation. This theorem provides the necessary and satisfactory conditions for above described mapping. The theorem highlights that the transformer excitation current under the sinusoidal magnetic flux has qualitatively equivalent information about magnetic core properties as the DHLB. Furthermore, the theorem establishes direct relationship between the number of the transformer excitation current harmonics and their coefficients with the degree of the DHLB interpolation polynomial and its coefficients. The DHLB interpolation polynomial is calculated over the measured subsets of samples representing Chebyshev nodes of the first and the second kind. These nonequidistant Chebyshev nodes provides uniform convergence of the interpolation polynomial to the experimentally obtained DHLB with an excellent approximation accuracy and are applicable on the approximation of the static hysteresis loops and the DC magnetization curves as well.

scholarly journals The Lebesgue function for Hermite-Fejér interpolation on the extended Chebyshev nodes

2002 ◽  
Vol 66 (1) ◽  
pp. 151-162
Author(s):  
Simon J. Smith
Keyword(s):  
Chebyshev Polynomial ◽  
Minimum Degree ◽  
Convergence Property ◽  
Lebesgue Constant ◽  
Interpolation Polynomial ◽  
Lebesgue Function ◽  
Surprising Result ◽  
End Points ◽  
Chebyshev Nodes ◽  
Interpolation Polynomials

Given f ∈ C[−1, 1] and n point (nodes) in [−1, 1], the Hermite-Fejér interpolation polynomial is the polynomial of minimum degree which agrees with f and has zero derivative at each of the nodes. In 1916, L. Fejér showed that if the nodes are chosen to be zeros of Tn (x), the nth Chebyshev polynomial of the first kind, then the interpolation polynomials converge to f uniformly as n → ∞. Later, D.L. Berman demonstrated the rather surprising result that this convergence property no longer holds true if the Chebyshev nodes are extended by the inclusion of the end points −1 and 1 in the interpolation process. The aim of this paper is to discuss the Lebesgue function and Lebesgue constant for Hermite-Fejér interpolation on the extended Chebyshev nodes. In particular, it is shown that the inclusion of the two endpoints causes the Lebesgue function to change markedly, from being identically equal to 1 for the Chebyshev nodes, to having the form 2n2(1 − x2)(Tn (x))2 + O (1) for the extended Chebyshev nodes.

scholarly journals On the divergence of Hermite-Fejér type interpolation with equidistant nodes

1994 ◽  
Vol 49 (1) ◽  
pp. 101-110 ◽  
Author(s):  
T.M. Mills ◽  
Simon J. Smith
Keyword(s):  
Lagrange Interpolation ◽  
Interpolation Polynomial ◽  
Simple Function ◽  
Convergence Properties ◽  
Lagrange Interpolation Polynomial ◽  
Type Interpolation ◽  
General Terms ◽  
Equidistant Nodes ◽  
Unique Polynomial

If f(x) is defined on [−1, 1], let H¯1 n(f, x) denote the Lagrange interpolation polynomial of degree n (or less) for f which agrees with f at the n+1 equally spaced points xk, n = −1 + (2k)/n (0 ≤ k ≤ n). A famous example due to S. Bernstein shows that even for the simple function h(x) = │x│, the sequence H¯1 n (h, x) diverges as n → ∞ for each x in 0 < │x│ < 1. A generalisation of Lagrange interpolation is the Hermite-Fejér interpolation polynomial H¯mn (f, x), which is the unique polynomial of degree no greater than m(n + 1) – 1 which satisfies (f, Xk, n) = δo, pf(xk, n) (0 ≤ p ≤ m − 1, 0 ≤ k ≤ n). In general terms, if m is an even number, the polynomials H¯mn(f, x) seem to possess better convergence properties than the H¯1 n (f, x). Nevertheless, D.L. Berman was able to show that for g(x) ≡ x, the sequence H¯2n(g, x) diverges as n → ∞ for each x in 0 < │x│. In this paper we extend Berman's result by showing that for any even m, H¯mn(g, x) diverges as n → ∞ for each x in 0 < │x│ < 1. Further, we are able to obtain an estimate for the error │H¯mn(g, x) – g(x)│.

scholarly journals Technology for restoring functional dependencies to determine reliability parameters

2021 ◽  
Vol 101 (1) ◽  
pp. 78-86
Author(s):  
V.P. Kvasnikov ◽  
◽  
S.V. Yehorov ◽  
T.Yu. Shkvarnytska ◽  
◽  
...  
Keyword(s):  
Interpolation Polynomial ◽  
Electronic Equipment ◽  
Optimal Interpolation ◽  
Electronic Systems ◽  
Functional Dependencies ◽  
Interpolation Methods ◽  
Reliability Parameters ◽  
Discrete Sample ◽  
Failure Statistics ◽  
Qualitative Characteristics

The problem of determining the properties of the object by analyzing the numerical and qualitative characteristics of a discrete sample is considered. A method has been developed to determine the probability of trouble-free operation of electronic systems for the case if the interpolation fields are different between several interpolation nodes. A method has been developed to determine the probability of trouble-free operation if the interpolation polynomial is the same for the entire interpolation domain. It is shown that local interpolation methods give more accurate results, in contrast to global interpolation methods. It is shown that in the case of global interpolation it is possible to determine the value of the function outside the given values by extrapolation methods, which makes it possible to predict the probability of failure. It is shown that the use of approximation methods to determine the probability of trouble-free operation reduces the error of the second kind. A method for analyzing the qualitative characteristics of functional dependences has been developed, which allows us to choose the optimal interpolation polynomial. With sufficient statistics, using the criteria of consent, it is possible to build mathematical models for the analysis of failure statistics of electronic equipment. Provided that the volume of statistics is not large, such statistics may not be sufficient and the application of consent criteria will lead to unsatisfactory results. Another approach is to use an approximation method that is applied to statistical material that was collected during testing or controlled operation. In this regard, it is extremely important to develop a method for determining the reliability of electronic systems in case of insufficiency of the collected statistics of failures of electronic equipment.

Effects of Sampling and Interpolation Methods on Accuracy of Extracted Watershed Features

2021 ◽  
Vol 26 (3) ◽  
pp. 05020053
Author(s):  
Jingwei Hou ◽  
Meiyan Zheng ◽  
Moyan Zhu ◽  
Yanjuan Wang
Keyword(s):  
Interpolation Methods

THE MODEL FOR MANAGING INVESTMENTS IN THE MOSCOW ENVIRONMENT BASED ON A POWER REGRESSION EQUATION

2020 ◽  
Vol 2 (7) ◽  
pp. 91-99
Author(s):  
E. V. KOSTYRIN ◽  
◽  
M. S. SINODSKAYA ◽  
Keyword(s):  
Solid Waste ◽  
Influencing Factors ◽  
Pair Correlation ◽  
Approximation Error ◽  
Regression Equation ◽  
Regression Equations ◽  
The Impact ◽  
The Relationship ◽  
Average Approximation ◽  
Average Approximation Error

The article analyzes the impact of certain factors on the volume of investments in the environment. Regression equations describing the relationship between the volume of investment in the environment and each of the influencing factors are constructed, the coefficients of the Pearson pair correlation between the dependent variable and the influencing factors, as well as pairwise between the influencing factors, are calculated. The average approximation error for each regression equation is determined. A correlation matrix is constructed and a conclusion is made. The developed econometric model is implemented in the program of separate collection of municipal solid waste (MSW) in Moscow. The efficiency of the model of investment management in the environment is evaluated on the example of the growth of planned investments in the activities of companies specializing in the export and processing of solid waste.

scholarly journals Spatio-Temporal Analysis of Water Quality Parameters in Machángara River with Nonuniform Interpolation Methods

Water ◽  
2016 ◽  
Vol 8 (11) ◽  
pp. 507 ◽  
Author(s):  
Iván Vizcaíno ◽  
Enrique Carrera ◽  
Margarita Sanromán-Junquera ◽  
Sergio Muñoz-Romero ◽  
José Luis Rojo-Álvarez ◽  
...  
Keyword(s):  
Water Quality ◽  
Temporal Analysis ◽  
Quality Parameters ◽  
Water Quality Parameters ◽  
Interpolation Methods ◽  
Spatio Temporal

POLYGONAL APPROXIMATION OF DIGITAL CURVE BASED ON REVERSE ENGINEERING CONCEPT

2013 ◽  
Vol 13 (04) ◽  
pp. 1350017 ◽  
Author(s):  
KUMAR S. RAY ◽  
BIMAL KUMAR RAY
Keyword(s):  
Reverse Engineering ◽  
Linear Time ◽  
Line Drawing ◽  
Approximation Error ◽  
Zero Approximation ◽  
Polygonal Approximation ◽  
Digital Curve ◽  
Symmetric Approximation ◽  
Engineering Concept ◽  
Automatic Algorithm

This paper applies reverse engineering on the Bresenham's line drawing algorithm [J. E. Bresenham, IBM System Journal, 4, 106–111 (1965)] for polygonal approximation of digital curve. The proposed method has a number of features, namely, it is sequential and runs in linear time, produces symmetric approximation from symmetric digital curve, is an automatic algorithm and the approximating polygon has the least non-zero approximation error as compared to other algorithms.

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