Continued Fraction Interpolation Research Articles

Numerical Continued Fraction Interpolation

Numerical Continued Fraction Interpolation

Numerical continued fraction interpolation

UDC 517.524 We show that highly accurate approximations can often be obtained by constructing Thiele interpolating continued fractions by a Greedy selection of the interpolation points together with an early termination condition. The obtained results are comparable with the outcome of state-of-the-art rational interpolation techniques based on the barycentric form.

Continued Fraction Interpolation of Preserving Horizontal Asymptote

The classical Thiele‐type continued fraction interpolation is an important method of rational interpolation. However, the rational interpolation based on the classical Thiele‐type continued fractions cannot maintain the horizontal asymptote when the interpolated function is of a horizontal asymptote. By means of the relationship between the leading coefficients of the numerator and the denominator and the reciprocal differences of the continued fraction interpolation, a novel algorithm for the continued fraction interpolation is constructed in an effort to preserve the horizontal asymptote while approximating the given function with a horizontal asymptote. The uniqueness of the interpolation problem is proved, an error estimation is given, and numerical examples are provided to verify the effectiveness of the presented algorithm.

Bivariate Thiele-Like Rational Interpolation Continued Fractions with Parameters Based on Virtual Points

The interpolation of Thiele-type continued fractions is thought of as the traditional rational interpolation and plays a significant role in numerical analysis and image interpolation. Different to the classical method, a novel type of bivariate Thiele-like rational interpolation continued fractions with parameters is proposed to efficiently address the interpolation problem. Firstly, the multiplicity of the points is adjusted strategically. Secondly, bivariate Thiele-like rational interpolation continued fractions with parameters is developed. We also discuss the interpolant algorithm, theorem, and dual interpolation of the proposed interpolation method. Many interpolation functions can be gained through adjusting the parameter, which is flexible and convenient. We also demonstrate that the novel interpolation function can deal with the interpolation problems that inverse differences do not exist or that there are unattainable points appearing in classical Thiele-type continued fractions interpolation. Through the selection of proper parameters, the value of the interpolation function can be changed at any point in the interpolant region under unaltered interpolant data. Numerical examples are given to show that the developed methods achieve state-of-the-art performance.

Super-resolution reconstruction based on continued fractions interpolation kernel in the polar coordinates

Super-resolution reconstruction based on continued fractions interpolation kernel in the polar coordinates

An Adaptive Image Inpainting Method Based on Continued Fractions Interpolation

In view of the drawback of most image inpainting algorithms by which texture was not prominent, an adaptive inpainting algorithm based on continued fractions was proposed in this paper. In order to restore every damaged point, the information of known pixel points around the damaged point was used to interpolate the intensity of the damaged point. The proposed method included two steps; firstly, Thiele’s rational interpolation combined with the mask image was used to interpolate adaptively the intensities of damaged points to get an initial repaired image, and then Newton-Thiele’s rational interpolation was used to refine the initial repaired image to get a final result. In order to show the superiority of the proposed algorithm, plenty of experiments were tested on damaged images. Subjective evaluation and objective evaluation were used to evaluate the quality of repaired images, and the objective evaluation was comparison of Peak Signal to Noise Ratios (PSNRs). The experimental results showed that the proposed algorithm had better visual effect and higher Peak Signal to Noise Ratio compared with the state-of-the-art methods.

Barycentric-Thiele Type Blending Rational Interpolation

In this paper, we construct Barycentric-Thiele type rational interpolation, which is based on Thiele continued fraction interpolation and Barycentric rational interpolation. Compared with Thiele continued fraction interpolation, Barycentric-Thiele type rational interpolation is more accuracy, better numerical stability and smaller calculation cost. While constructing the corresponding Thiele continued fraction interpolation, we can choose the appropriate number of nodes to avoid poles. Finally, the numerical examples are given to verify the correctness and validity of our method.

Irregular Bivariate Blending Continued Fractions Interpolation in Image Resizing

With the concept of information super-highway was proposed, internet is widely used, and the highdefinition (high resolution) media (panel TV DVD and projector) are popular, the human being have been in digital home hd era. The emerging industry-digital home should be free to timeliness, motility and cost performance, which makes it be a hot case in current manufacture as well as in research field. There are a lot of challenging works on hd media, which is composed by many frames of images, so we can classify video processing as image processing. Many digital image processing software generally resize image by polynomial interpolation, but the results are not very ideal. A bunch of researches show that we can get more ideal results via nonlinear rational function models. Bivariate blending continued fractions rational interpolation is not only a very efficient nonlinear approximate, but also can be calculated recurrently [4] [5], which means good feasible computability. This paper takes Bivariate blending continued fractions rational interpolation as theoretical basis, generalizes its definition on regular mesh to irregular mesh, and combines with digital image feature, applies it on resizing image to get better results.

Compression and Reconstruction of Time Series Data Based on Vector Valued Continued Fraction

In this paper we introduce a novel method to compress time series data based on interpolation of vector valued continued fraction. Compared with straight segments method, our method has a better compression. In our method, multiple attributes of an object can also be computed together by means of vector. Our method first get a raw series of compression data points by using top-down method, then the raw series is divided into some segments by using continued fraction interpolation. Finally, a discriminant criterion will help us to delete some compression points on segments. In reconstructing, piecewise interpolation of vector valued continued fraction is used to recover original data.

A novel algorithm for removal of salt and pepper noise using continued fractions interpolation

We propose a novel continued fractions interpolation filter (CFIF) to remove salt and pepper noise. The proposed algorithm replaces median filter with continued fractions interpolation filter. First, noise free pixels are left unchanged, then, the weighted mean of four-direction continued fractions interpolation is computed to estimate the corrupted pixel value. Our method requires neither a window nor any prior knowledge. The proposed method also preserves edges very well while removing impulse noise. Simulation results indicate that the CFIF outperforms other filters existing in the literature.

An Efficient Continued Fraction Algorithm for Nonlinear Optimization and Its Computer Implementation

Optimization has been a basic tool in all areas of applied mathematics, engineering, medicine, economics and other sciences.There has been much attention to develop iterative methods for solving nonlinear equations in these years.New algorithms and theoretical techniques have been developed, the diffusion into other disciplines has proceeded at a rapid pace. One of the most striking trends in optimization is the constantly increasing emphasis on the interdisciplinary nature of the field.Among wide various of papers have been published in the recent years, there are some progress about multi-step methods.These multi-step methods have been suggested by combining the well-known Newton's method with other methods. In this work, we develop a simple yet practical algorithm for solving nonlinear optimization problems by solving nonlinear equations with a good local convergence.The algorithm uses a continued fraction interpolation that can be easily implemented in software packages for achieving desired convergence orders.For the general $n$-point formula,the order of convergence rate of the presented algorithm is $\tau_n$, the unique positive root of the equation $x^n-x^{n-1}-\cdots-x-1=0$. Computational results ascertain that the developed algorithm is efficient and demonstrate equal or better performance as compared with other well known methods.

A New Algorithm to Approximate Bivariate Matrix Function via Newton-Thiele Type Formula

A new method for computing the approximation of bivariate matrix function is introduced. It uses the construction of bivariate Newton-Thiele type matrix rational interpolants on a rectangular grid. The rational interpolant is of the form motivated by Tan and Fang (2000), which is combined by Newton interpolant and branched continued fractions, with scalar denominator. The matrix quotients are based on the generalized inverse for a matrix which is introduced by C. Gu the author of this paper, and it is effective in continued fraction interpolation. The algorithm and some other important conclusions such as divisibility and characterization are given. In the end, two examples are also given to show the effectiveness of the algorithm. The numerical results of the second example show that the algorithm of this paper is better than the method of Thieletype matrix-valued rational interpolant in Gu (1997).

Fitting Algorithm of Parameterized Continued Fractions with Keeping Endpoints and its Application

In order to overcome fitting uncertainty, A nonlinear rational parameterized fitting algorithm with keeping endpoints was presented through inverse difference-continued fraction algorithm, the algorithm has property of keeping endpoints, parameterization of middle data points selected. The parameters obtained through minimum error function make fitting functions have much flexibility. Experimental results illustrate that the proposed algorithm may reduce degree of continued fraction interpolation function, numbers of iteration and avoiding non-existence on interpolation function of the continued fraction.The fitting values obtained are more accuracy, the computing efficiency is much more improved,the fitting errors are more stable, approximate effect is better, and the algorithm will be widely used in predict etc.

A New Approach to Trivariate Blending Rational Interpolation

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.

A tractable approximation approach to improving hotel service quality

Typically, the service quality in the hotel industry is analyzed using cross-sectional empirical studies. For example, to examine current causal relationships among various latent and manifest variables using structural equation modeling plays a key role in understanding how services are provided at a given hotel. Note that accurate prediction of service quality certainly helps decision-makers better manage their services operations to sustain quality services to meet the ever-changing needs of customers. In this paper, by integrating continued fraction interpolation theory and genetic algorithms, an innovative service quality prediction model in the hotel industry is presented. An illustrative example is provided, which essentially shows the applicability of this proposed innovative methodology.

Bivariate polynomial and continued fraction interpolation over ortho-triples

By means of the barycentric coordinates expression of the interpolating polynomial over each ortho-triple, some properties are obtained. Moreover, the explicit coefficients in terms of B-net for one ortho-triple, and two ortho-triples are worked out, respectively. Thus the computation of multiple integrals can be converted into the sum of the coefficients in terms of the B-net over triangular domain much effectively and conveniently. Based on a new symmetrical algorithm of partial inverse differences, a novel continued fractions interpolation scheme is presented over arbitrary ortho-triples in R 2, which is a bivariate osculatory interpolation formula with one-order partial derivatives at all corner points in the ortho-triples. Furthermore, its characterization theorem is presented by three-term recurrence relations. The new scheme is advantageous over the polynomial one with some numerical examples.

Block-based Thiele-like blending rational interpolation

Thiele-type continued fraction interpolation may be the favoured nonlinear interpolation in the sense that it is constructed by means of the inverse differences which can be calculated recursively and produce useful intermediate results. However, Thiele's interpolation is in fact a point-based interpolation by which we mean that a new interpolating continued fraction with one more degree of its numerator or denominator polynomial is obtained by adding a tail to the current one, or in other words, this can be reshaped simply by adding a new support point to the current set of support points once at a time. In this paper we introduce so-called block-based inverse differences to extend the point-based Thiele-type interpolation to the block-based Thiele-like blending rational interpolation. The construction process may be outlined as follows: first of all, divide the original set of support points into some subsets (blocks), then construct each block by using whatever interpolation means, linear or rational, and finally assemble these blocks by Thiele's method to shape the whole interpolation scheme. Clearly, many flexible rational interpolation schemes can be obtained in this case including the classical Thiele's continued fraction interpolation as its special case. As an extension, a bivariate analogy is also discussed and numerical examples are given to show the effectiveness of our method.

Two dual expansions for generalized bivariate thiele-type matrix valued interpolating continued fractions

A new method for the construction of bivariate matrix valued rational interpolants on a rectangular grid is introduced. The rational interpolants are expressed in the continued fraction form with scalar denominator. The matrix quotients are based on the generalized inverse for a matrix, which is found to be effective in continued fraction interpolation. In this paper, two dual expansions for bivariate matrix valued Thiele-type interpolating continued fractions are presented, then, two dual rational interpolants are defined out of them.

Bivariate Thiele-type matrix-valued rational interpolants

A new method for the construction of bivariate matrix-valued rational interpolants on a rectangular grid is introduced in this paper. The rational interpolants are of the continued fraction form, with scalar denominator. In this respect the approach is essentially different from that of Bose and Basu (1980) where a rational matrix-valued approximant with matrix-valued numerator and denominator is used for the approximation of a bivariate matrix power series. The matrix quotients are based on the generalized inverse for a matrix introduced by Gu Chuanqing and Chen Zhibing (1995) which is found to be effective in continued fraction interpolation. A sufficient condition of existence is obtained. Some important conclusions such as characterisation and uniqueness are proven respectfully. The inner connection between two type interpolating functions is investigated. Some examples are given so as to illustrate the results in the paper.

Extrapolation to the limit by using continued fraction interpolation

Extrapolation to the limit by using continued fraction interpolation