scholarly journals A New Approach to General Interpolation Formulae for Bivariate Interpolation

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Le Zou ◽  
Shuo Tang
Keyword(s):  
Continued Fraction ◽  
Rational Interpolation ◽  
Interpolation Theorem ◽  
Interpolation Scheme ◽  
Bivariate Interpolation ◽  
New Approach ◽  
Multiple Parameters ◽  
Special Cases ◽  
Block Based ◽  
Branched Continued Fraction

General interpolation formulae for bivariate interpolation are established by introducing multiple parameters, which are extensions and improvements of those studied by Tan and Fang. The general interpolation formulae include general interpolation formulae of symmetric branched continued fraction, general interpolation formulae of univariate and bivariate interpolation, univariate block based blending rational interpolation, bivariate block based blending rational interpolation and their dual schemes, and some interpolation form studied by many scholars in recent years. We discuss the interpolation theorem, algorithms, dual interpolation, and special cases and give many kinds of interpolation scheme. Numerical examples are given to show the effectiveness of the method.

A New Approach to Trivariate Blending Rational Interpolation

2012 ◽  
Vol 546-547 ◽  
pp. 570-575 ◽  
Author(s):  
Le Zou ◽  
Jin Xie ◽  
Chang Wen Li
Keyword(s):  
Continued Fractions ◽  
Rational Interpolation ◽  
Interpolation Formula ◽  
Linear Interpolation ◽  
Interpolation Theorem ◽  
Newton Interpolation ◽  
New Approach ◽  
Barycentric Interpolation ◽  
Rational Interpolants ◽  
Data Pair

The advantages of barycentric interpolation formulations in computation are small number of floating points operations and good numerical stability. Adding a new data pair, the barycentric interpolation formula don’t require to renew computation of all basis functions. Thiele-type continued fractions interpolation and Newton interpolation may be the favoured nonlinear and linear interpolation. A new kind of trivariate blending rational interpolants were constructed by combining barycentric interpolation, Thiele continued fractions and Newton interpolation. We discussed the interpolation theorem, dual interpolation, no poles of the property and error estimation.

scholarly journals A New Approach for Analyzing the Reliability of the Repair Facility in a Series System with Vacations

2012 ◽  
Vol 2012 ◽  
pp. 1-16
Author(s):  
Renbin Liu ◽  
Yong Wu
Keyword(s):  
Renewal Process ◽  
Decomposition Method ◽  
Process Theory ◽  
Series System ◽  
New Approach ◽  
Numerical Examples ◽  
Special Cases ◽  
The Mean ◽  
Repair Facility

Based on the renewal process theory we develop a decomposition method to analyze the reliability of the repair facility in ann-unit series system with vacations. Using this approach, we study the unavailability and the mean replacement number during(0,t]of the repair facility. The method proposed in this work is novel and concise, which can make us see clearly the structures of the facility indices of a series system with an unreliable repair facility, two convolution relations. Special cases and numerical examples are given to show the validity of our method.

scholarly journals Continued Fractions Related to $(t,q)$-Tangents and Variants

10.37236/2014 ◽  
2011 ◽  
Vol 18 (2) ◽  
Author(s):  
Helmut Prodinger
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
General Version ◽  
Continued Fraction Expansion ◽  
Special Cases ◽  
Tangent Function

For the $q$-tangent function introduced by Foata and Han (this volume) we provide the continued fraction expansion, by creative guessing and a routine verification. Then an even more recent $q$-tangent function due to Cieslinski is also expanded. Lastly, a general version is considered that contains both versions as special cases.

scholarly journals On the convergence of multidimensional regular C-fractions with independent variables

2018 ◽  
Vol 26 (1) ◽  
pp. 18 ◽  
Author(s):  
R.I. Dmytryshyn
Keyword(s):  
Power Series ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Open Disk ◽  
Efficient Tool ◽  
Independent Variables ◽  
Multidimensional Generalization ◽  
Multiple Power ◽  
Multiple Power Series ◽  
Branched Continued Fraction

In this paper, we investigate the convergence of multidimensional regular С-fractions with independent variables, which are a multidimensional generalization of regular С-fractions. These branched continued fractions are an efficient tool for the approximation of multivariable functions, which are represented by formal multiple power series. We have shown that the intersection of the interior of the parabola and the open disk is the domain of convergence of a multidimensional regular С-fraction with independent variables. And, in addition, we have shown that the interior of the parabola is the domain of convergence of a branched continued fraction, which is reciprocal to the multidimensional regular С-fraction with independent variables.

scholarly journals A New Approach to Newton-Type Polynomial Interpolation with Parameters

2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Le Zou ◽  
Liangtu Song ◽  
Xiaofeng Wang ◽  
Thomas Weise ◽  
Yanping Chen ◽  
...  
Keyword(s):  
Polynomial Interpolation ◽  
Information Matrix ◽  
Divided Differences ◽  
Mathematical Representation ◽  
Interpolation Algorithm ◽  
New Approach ◽  
Multiple Parameters ◽  
Parameter Values ◽  
The Given ◽  
Interpolation Functions

Newton’s interpolation is a classical polynomial interpolation approach and plays a significant role in numerical analysis and image processing. The interpolation function of most classical approaches is unique to the given data. In this paper, univariate and bivariate parameterized Newton-type polynomial interpolation methods are introduced. In order to express the divided differences tables neatly, the multiplicity of the points can be adjusted by introducing new parameters. Our new polynomial interpolation can be constructed only based on divided differences with one or multiple parameters which satisfy the interpolation conditions. We discuss the interpolation algorithm, theorem, dual interpolation, and information matrix algorithm. Since the proposed novel interpolation functions are parametric, they are not unique to the interpolation data. Therefore, its value in the interpolant region can be adjusted under unaltered interpolant data through the parameter values. Our parameterized Newton-type polynomial interpolating functions have a simple and explicit mathematical representation, and the proposed algorithms are simple and easy to calculate. Various numerical examples are given to demonstrate the efficiency of our method.

scholarly journals Fractal Behavior of a Ternary 4-Point Rational Interpolation Subdivision Scheme

2018 ◽  
Vol 23 (4) ◽  
pp. 65 ◽  
Author(s):  
Kaijun Peng ◽  
Jieqing Tan ◽  
Zhiming Li ◽  
Li Zhang
Keyword(s):  
Singular Point ◽  
Sufficient Conditions ◽  
Rational Interpolation ◽  
Subdivision Scheme ◽  
Fractal Curve ◽  
Rational Scheme ◽  
Special Cases ◽  
Necessary And Sufficient ◽  
Parameter Values ◽  
Selection Of

In this paper, a ternary 4-point rational interpolation subdivision scheme is presented, and the necessary and sufficient conditions of the continuity are analyzed. The generalization incorporates existing schemes as special cases: Hassan–Ivrissimtzis’s scheme, Siddiqi–Rehan’s scheme, and Siddiqi–Ahmad’s scheme. Furthermore, the fractal behavior of the scheme is investigated and analyzed, and the range of the parameter of the fractal curve is the neighborhood of the singular point of the rational scheme. When the fractal curve and surface are reconstructed, it is convenient for the selection of parameter values.

scholarly journals Plotting positions via maximum-likelihood for a non-standard situation

1997 ◽  
Vol 1 (2) ◽  
pp. 357-366 ◽  
Author(s):  
D. A. Jones
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Frequency Analysis ◽  
Likelihood Estimation ◽  
Similar Data ◽  
Statistical Distributions ◽  
Annual Maximum ◽  
New Approach ◽  
Special Cases ◽  
Standard Situation

Abstract. A new approach is developed for the specification of the plotting positions used in the frequency analysis of extreme flows, rainfalls or similar data. The approach is based on the concept of maximum likelihood estimation and it is applied here to provide plotting positions for a range of problems which concern non-standard versions of annual-maximum data. This range covers the inclusion of incomplete years of data and also the treatment of cases involving regional maxima, where the number of sites considered varies from year to year. These problems, together with a not-to-be-recommended approach to using historical information, can be treated as special cases of a non-standard situation in which observations arise from different statistical distributions which vary in a simple, known, way.

Structuralism? Functionalism? Behaviorism? Or mechanism? Looking for a better approach to AI

2008 ◽  
Vol 1 (3) ◽  
pp. 325-336 ◽  
Author(s):  
Yixin Zhong
Keyword(s):  
Intelligent System ◽  
New Approach ◽  
Content Type ◽  
Advantages And Disadvantages ◽  
Special Cases ◽  
Powerful Approach ◽  
Reasonable Approach ◽  
The Common ◽  
To Come ◽  
The One

PurposeResearch of artificial intelligence (AI), has aimed at making machines intelligent via the simulation of natural intelligence, particularly human intelligence. During the past decades, there have been three major approaches aimed at achieving this goal, namely structuralism, functionalism and behaviorism. Unfortunately, they work separately and contradictorily to a large extent. The purpose of this paper is to present a better and more unified approach.Design/methodology/approachThe paper analyses each of the three major approaches to AI, describing their advantages and disadvantages. There then follows an attempt to explore a new and more reasonable approach to AI. The new approach should be able to solve all the problems that the existing approaches can solve on one hand and can solve the problems that the existing approaches cannot solve on the other hand.FindingsIt was found that the more reasonable and more powerful approach is the one that directly touches the common and core mechanism of intelligence formation. This is due to the fact that the mechanism of intelligence formation is much more essential than other windows of an intelligent system, such as structure, function, or behavior. It was also found that the common and core mechanism of intelligence formation can be implemented through the information‐knowledge‐intelligence transformation. The third finding is that the three existing approaches are special cases of the mechanism approach under different conditions and can thus be harmoniously unified within the frame of the mechanism approach.Originality/valueThe three findings in the paper: the mechanism approach, the implementation of the mechanism approach, and the unification of the existed three major approaches, are important laws never found before in the literature. The breakthrough of the mechanism approach to AI will be of great significance to both theoretical and practical research in AI in the years to come.

Sign in / Sign up

Export Citation Format

Share Document