Abstract

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.

1. Introduction

The classical Thiele-type continued fraction interpolation [1–9] is an important method of rational interpolation. Suppose is the interpolated function and where are different interpolating nodes. The classical Thiele-type continued fraction has the form as follows:whereis the th inverse difference of the function with respect to , which can be calculated recursively as follows:

It is not difficult to show that is a rational function whose numerator and denominator are polynomials of degrees not exceeding and , respectively, where denotes the largest integer not exceeding , and satisfies

The th reciprocal difference of the function with respect to is defined recursively as follows:

The inverse differences can be calculated via the reciprocal differences as follows [10, 11]:

Let

Denote by the leading coefficient of the polynomial , then when is odd, the reciprocal differences and the leading coefficients of the numerator polynomial and denominator polynomial of continued fraction interpolation have the following identity relationship [12]:when is even,

The classical Thiele-type continued fraction interpolation may not necessarily maintain the original horizontal asymptote of the interpolated function when the interpolated function has a horizontal asymptote. This paper presents an algorithm to construct the continued fraction interpolation preserving the horizontal asymptote that the interpolated function possesses. The uniqueness of solution of the numerical problem is proved, an error estimation is worked out, and numerical examples are provided to show the effectiveness of the new algorithm.

2. The Algorithm for Continued Fraction Interpolation of Preserving Horizontal Asymptote

The problem for continued fraction interpolation of preserving horizontal asymptote: Let be defined in and be distinct interpolation nodes such that . Suppose has a horizontal asymptote , i.e., , where is a constant. Our purpose is to seek for a rational function of the following form:such thatwhere

Since is unknown, the formula of inverse differences cannot be used to calculate directly.

It is not difficult to show

can be written as in the following form:

It follows from (9) and (10),

Using the relationship between the inverse differences and the reciprocal differences gives

With (12), (16), and (17) in mind, we havei.e.,

As a result, the continued fraction interpolation with the preserved horizontal asymptote is given by

3. The Uniqueness of Interpolant

Theorem 1. Let

If all the th inverse differences exist and , then set (see [13, 14])

Ifthen, we have

Proof. , using (20), (22)–(24) givesThe following can be obtained with the formulas (16), (17), and (19):The proof is completed.

Theorem 2. If a rational interpolation function of type with the preserved horizontal asymptote exists, it must be the unique one.

Proof. Suppose the two rational functions of type .Both meet the interpolation conditions in formula (12), we haveThat is,Since the leading term of is , turns out to be a polynomial of degree not exceeding , which has distinct zeros. Therefore,namely,

4. The Error Estimation

Theorem 3. Suppose is the smallest interval containing and is times differentiable in . Letsatisfy . Then, for each , there exists a point such thatandwhere .

Proof. Let . Then, from , it follows,Using the Lagrange interpolation formula with remainder term yields (see [15]),Therefore,and

5. Numerical Examples

Example 1. Given six interpolation nodes . Suppose , then, and . We want to construct the continued fraction interpolant such that it meets the interpolation conditions and (keep eight decimal places).
According to what is known, the involved inverse differences can be calculated as shown in Table 1.
By equation (19), we haveSubstituting into giveswhich can be simplified asso,It is obvious thatUsing the classical Thiele-type continued fraction interpolation, one can getwhich can be simplified asA comparison is made between the curves and as shown in Figure 1. The values of and at certain points are calculated as shown in Table 2. The errors and are illustrated in Figure 2.

Table 1 
Table of inverse differences.

Figure 1 
Function plots of and .
Table 2 
Numerical experiments about the error.

Figure 2 
Function plots of and .

Example 2. Given six interpolation nodes . Suppose , then, , and . We want to construct the continued fraction interpolant such that it meets the interpolation conditions and .
According to what is known, the involved inverse differences can be calculated as shown in Table 3.
From equation (19), it follows:Substituting into giveswhich can be simplified asSo,Clearly,Using the classical Thiele-type continued fraction interpolation, one can getwhich can be simplified asA comparison is made between the curves and , as shown in Figure 3. The values of and at certain points are calculated, as shown in Table 4. The errors and are shown in Figure 4.

Table 3 
Table of inverse differences.

Figure 3 
Function plots of and .
Table 4 
Numerical experiments about the error.

Figure 4 
Function plots of and .

Example 3. Given six interpolation nodes . Suppose , then, and . We try to construct a continued fraction interpolant such that it meets the interpolation conditions and .
According to what is known, the involved inverse differences can be calculated and listed in the following Table 5. From (19), it follows:Substituting into giveswhich can be simplified asSo,Obviously,Using the classical Thiele-type continued fraction interpolation, we havewhich can be simplified asA comparison is conducted between the curves and as shown in Figure 5. The values of and at certain points are calculated and listed in Table 6. The errors and are shown in Figure 6.

Table 5 
Table of inverse differences.

Figure 5 
Function plots of and .
Table 6 
Numerical experiments about the error.

Figure 6 
Function plots of and .

6. Conclusion

As classical approximation tool, continued fractions have been playing an important role in numerical rational approximation. However, continued fractions are rarely involved in shape-preserving design which is an interesting research topic in geometric modeling. In this paper, we construct an interpolating rational function based on the continued fractions, which serves to approximate the functions with the horizontal asymptotes. An algorithm is presented for the interpolating rational function to preserve the horizontal asymptote, the uniqueness of the interpolating rational function is proved and the error is analyzed. Numerical examples are given to verify the effectiveness of the new method.

Data Availability

The data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by the Anhui University Postgraduate Scientific Research Project under Grant No. YJS20210368 and the National Natural Science Foundation of China under Grant No. 62172135.