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.


Introduction e classical
iele-type continued fraction interpolation [1][2][3][4][5][6][7][8][9] is an important method of rational interpolation. Suppose y f(x) is the interpolated function and y j f(x j ), j 0, 1, . . . , n where x 0 , x 1 , . . . , x n are n + 1 di erent interpolating nodes. e classical iele-type continued fraction has the form as follows: where b j φ x 0 , x 1 , x 2 , . . . , x j , j 0, 1, . . . , n, is the jth inverse di erence of the function f(x) with respect to x 0 , x 1 , . . . , x j , which can be calculated recursively as follows: φ x j f x j , j 0, 1, . . . , n, It is not di cult to show that R n (x) is a rational function whose numerator and denominator are polynomials of degrees not exceeding [(n + 1)/2] and [n/2], respectively, where [u] denotes the largest integer not exceeding u, and R n (x) satis es R n x j y j , j 0, 1, 2, . . . , n. (4) e nth reciprocal di erence ρ[x 0 , x 1 , . . . , x n ] of the function f(x) with respect to x 0 , x 1 , . . . , x n is de ned recursively as follows: ρ x j � f x j , j � 0, 1, . . . , n, e inverse differences can be calculated via the reciprocal differences as follows [10,11]: Denote by L(P n (x)) the leading coefficient of the polynomial P n (x), then when n 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 n is even, L Q n (x) � 1.
e classical iele-type continued fraction interpolation may not necessarily maintain the original horizontal asymptote of the interpolated function when the interpolated function has a horizontal asymptote.
is paper presents an algorithm to construct the continued fraction interpolation preserving the horizontal asymptote that the interpolated function possesses. e 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.

The Algorithm for Continued Fraction
Interpolation of Preserving Horizontal Asymptote e problem for continued fraction interpolation of preserving horizontal asymptote: Let y � f(x) be defined in I and x 0 , x 1 , . . . , x 2m− 1 ⊂ I be 2m distinct interpolation nodes such that y j � f(x j ), j � 0, 1, . . . , 2m − 1. Suppose y � f(x) has a horizontal asymptote y � A, i.e., lim x⟶∞ f(x) � A, where A is a constant. Our purpose is to seek for a rational function of the following form: such that Since x 2m is unknown, the formula of inverse differences cannot be used to calculate It is not difficult to show R 2m (x) 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 have i.e., 2

Journal of Mathematics
As a result, the continued fraction interpolation with the preserved horizontal asymptote is given by (20) [13,14])

The Uniqueness of Interpolant
then, we have (26) e following can be obtained with the formulas (16), (17), and (19): lim Both meet the interpolation conditions in formula (12), we have at is, turns out to be a polynomial of degree not exceeding 2m − 1, which has 2m distinct zeros. erefore, namely,

The Error Estimation
and where Using the Lagrange interpolation formula with remainder term yields (see [15]), erefore, and □

Numerical Examples
.50837752, f (x 5 ) � 1.52083793 and lim x⟶∞ |arctanx| � 1.57079633. We want to construct the continued fraction interpolant R 6 (x) such that it meets the interpolation conditions and lim x⟶∞ R 6 (x) � 1.57079633 (keep eight decimal places). According to what is known, the involved inverse differences can be calculated as shown in Table 1.
Using the classical iele-type continued fraction interpolation, one can get A comparison is made between the curves y � f(x) and y � R 6 (x) as shown in Figure 1. e values of |f(x) − R 6 (x)| and |f(x) − R 5 (x)| at certain points are calculated as shown in Table 2. e errors |f(x) − R 6 (x)| and |f(x) − R 5 (x)| are illustrated in Figure 2.

Example 2. Given six interpolation nodes
)/x) � 1. We want to construct the continued fraction interpolant R 6 (x) such that it meets the interpolation conditions and lim x⟶∞ R 6 (x) � 1.
According to what is known, the involved inverse differences can be calculated as shown in Table 3.
From equation (19), it follows: Substituting b 6 into R 6 (x) gives which can be simplified as      Journal of Mathematics So, Clearly, Using the classical iele-type continued fraction interpolation, one can get which can be simplified as A comparison is made between the curves y � f(x) and y � R 6 (x), as shown in Figure 3. e values of |f(x) − R 6 (x)| and |f(x) − R 5 (x)| at certain points are calculated, as shown in Table 4. e errors |f(x) − R 6 (x)| and |f(x) − R 5 (x)| are shown in Figure 4.
According to what is known, the involved inverse differences can be calculated and listed in the following Table 5. From (19), it follows: Substituting b 6 into R 6 (x) gives So, Obviously, lim x⟶∞ R 6 (x) � 1.
Using the classical iele-type continued fraction interpolation, we have

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which can be simplified as A comparison is conducted between the curves y � f(x) and y � R 6 (x) as shown in Figure 5. e values of |f(x) − R 6 (x)| and |f(x) − R 5 (x)| at certain points are calculated and listed in Table 6. e errors |f(x) − R 6 (x)| and |f(x) − R 5 (x)| are shown in Figure 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 e data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.