A New Approach to General Interpolation Formulae for Bivariate Interpolation

Abstract

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.

1. Introduction

Newton interpolation and Thiele-type continued fractions interpolation may be the favored linear interpolation and nonlinear interpolation [1]. Symmetric branched continued fraction is a bivariate continued fractions interpolation scheme discussed by Cuyt and Verdonk [2, 3], Kučminskaja [4], and Murphy and O’Donohoe [5]. In recent years, Kuchmins’ka and Vozna [6, 7], Pahirya [8], Zhao [9], and Wang [10] studied some new kinds of symmetric blending rational interpolation. Wang and Qian studied bivariate polynomial interpolation and continued fractions interpolation over ortho-triples [11]. Zhao and Tan studied block based Newton-like blending rational interpolation [12] and block based Thiele-like blending rational interpolation [13]. The general frames of interpolation problem have been widely studied. Kahng showed the generalizations of univariate Newton’s method and applied it to the approximation problems in 1967 [14]; Kahng described a class of interpolation functions and showed the explicit method of osculatory interpolation with a function in the class in 1969 [15]. In 1999, Tan and Fang [1] studied several general frames for bivariate interpolation which include many classical interpolation schemes; Tan also discussed the more general interpolation grids [16]. Recently, Tang and Zou [17] have improved and extended the general frames studied by Tan and Fang by introducing multiple parameters, so that the new frames can be used to deal with the interpolation problems where inverse differences are nonexistent or unattainable points occur. The general form of block based bivariate blending rational interpolation with the error estimation is established by introducing two parameters [18]; four different block based interpolations are included. Then an efficient algorithm for computing bivariate lacunary rational interpolation is constructed based on block based bivariate blending rational interpolation. One of authors constructs the frames of symmetry interpolation [19] and general structures of one and two variable interpolation function without depending on the existence of divided difference or inverse differences, and he also discusses the block based osculatory interpolation in one variable case [20].

Our contribution in this paper is to obtain a new type of general interpolation formulae for bivariate interpolation by introducing multiple parameters, which includes general interpolation formula 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 new interpolation scheme studied by many scholars in recent years. The organization of the paper is as follows. In Section 2 we discuss the interpolation theorem, algorithms, dual interpolation, and special cases of general interpolation formulae of symmetric interpolation. The interpolation theorem, algorithms, dual interpolation, and special cases of the general interpolation formulae of block based univariate and bivariate interpolation are discussed in Section 3. Numerical examples are given to show the effectiveness of the method in Section 4.

2. General Interpolation Formulae of Symmetric Interpolation

Given a set of real points Π_𝑛,𝑚={(𝑥_𝑖,𝑦_𝑗)∣𝑖=0,1,…,𝑛,𝑗=0,1,…,𝑚}⊂[𝑎,𝑏]×[𝑐,𝑑]⊂𝑅² and a bivariate function 𝑓(𝑥,𝑦) defined in a domain [𝑎,𝑏]×[𝑐,𝑑].

Notation 1. Let 𝑁=max𝑖∣𝑥_𝑖,𝑦_𝑖∈Π_𝑛,𝑚. (1)
Now we construct a function 𝑄𝑥,𝑦=𝑓₀𝐴₀𝑥,𝑦+𝑆₀𝑥,𝑦×𝑓₁𝐴₁𝑥,𝑦+𝑆₁𝑥,𝑦×𝑓₂𝐴₂𝑥,𝑦+⋯+𝑆_𝑁−1𝑥,𝑦×𝑓_𝑁𝐴_𝑁𝑥,𝑦⋯, (2) by constructing different 𝐴_𝑖(𝑥,𝑦); then, 𝑄(𝑥,𝑦) can be changed into general frame of symmetric interpolation [19], general frame of block based univariate interpolation [17, 20], general frame of block based bivariate interpolation [17, 20], and so on.
If we choose 𝐴_𝑖(𝑥,𝑦) as follows in formula (2): 𝐴_𝑖𝑥,𝑦=𝑓_𝑖,𝑖𝑎_𝑖,𝑖𝑥,𝑦+𝑔_𝑖(𝑥)×𝑓_𝑖+1,𝑖𝑎_𝑖+1,𝑖𝑥,𝑦+⋯+𝑔_𝑛−1(𝑥)×𝑓_𝑛,𝑖𝑎_𝑛,𝑖𝑥,𝑦⋯+ℎ_𝑖𝑦𝑓_𝑖,𝑖+1𝑎_𝑖,𝑖+1𝑥,𝑦+⋯+ℎ_𝑚−1𝑦𝑓_𝑖,𝑚𝑎_𝑖,𝑚𝑥,𝑦⋯, (3) where 𝑆_𝑖(𝑥,𝑦)=(𝑥−𝑥_𝑖)(𝑦−𝑦_𝑖), 𝑎_𝑖,𝑗(𝑥,𝑦) are constants, 𝑔_𝑖(𝑥)=𝑥−𝑥_𝑖, ℎ_𝑖(𝑦)=𝑦−𝑦_𝑖, then 𝑄(𝑥,𝑦) is a general interpolation formula of symmetric interpolation.
We cite a theorem and one can prove (2), (3) are a general interpolation formula of symmetric interpolation and satisfy interpolation conditions.

Notation 2. Let ℎ(𝐴)={ℎ(𝑥)},𝑅(ℎ)(𝑥).∣𝑥∈𝐴∶rangeofℎ (4)

Theorem 1 (see [15]). Given a function 𝑦(𝑥) continuous in a finite interval [𝑎,𝑏] and 𝑛+1 points 𝑥_𝑖 with 𝑎⩽𝑥₀<𝑥₁<⋯<𝑥_𝑛⩽𝑏, there exists a unique set of parameters 𝑎₀,𝑎₁,…,𝑎_𝑛 such that the interpolation function 𝑄(𝑥)=𝑓₀𝑎₀+𝑔₀(𝑥)𝑓₁𝑎₁+⋯+𝑔_𝑛−1(𝑥)𝑓_𝑛𝑎_𝑛,…, (5) satisfying 𝑄𝑥_𝑖=𝑦𝑥_𝑖,𝑖=0,1…,𝑛, (6) and 𝑄(𝑥) is continuous if(a)𝑓_𝑖 is continuous and strictly monotone in (−∞,+∞) and the range of 𝑓_𝑖(𝑥) covers (−∞,+∞),𝑖=1,2,…,𝑛;(b)𝑓₀ is continuous and its inverse function 𝑓₀⁻¹ exists in 𝑅(𝑓₀) and 𝑅(𝑓₀)⊃𝑦([𝑎,𝑏]);(c)the functions 𝑔_𝑗(𝑥),𝑗=0,1,…,𝑛−1, are continuous in [𝑎,𝑏] and 𝑔_𝑗(𝑥)=0,𝑥=𝑥_𝑗;≠0,𝑥>𝑥_𝑗. (7)
When the above conditions are satisfied, the parameters are determined from the following equations in sequence: 𝑎₀=𝑓₀⁻¹(𝑄(𝑥₀)) is determined from 𝑄(𝑥₀)=𝑓₀(𝑎₀), 𝑎₁=𝑓₁⁻¹((𝑓₀⁻¹(𝑄(𝑥₁))−𝑎₀)/𝑔₀(𝑥₁)) is found from 𝑄(𝑥₁)=𝑓₀(𝑎₀+𝑔₀(𝑥₁)𝑓₁(𝑎₁)), and so on.
The conditions on the functions 𝑓_𝑖,𝑓_𝑖,𝑗,𝑔_𝑖,ℎ_𝑖,𝑗 for the existence of unique parameters 𝑎_𝑖,𝑗(𝑖=0,1,…,𝑛,𝑗=0,1,…,𝑚) are given next using the following notations.

Notation 3. Let ℎ()=ℎ∣∈[]×[𝑆(ℎ).𝐴,𝐵𝑥,𝑦𝑥,𝑦𝑎,𝑏𝑐,𝑑]∶rangeofℎ𝑥,𝑦 (8)

Theorem 2. Given a function 𝑓(𝑥,𝑦) continuous in [𝑎,𝑏]×[𝑐,𝑑] and (𝑛+1)×(𝑚+1) points (𝑥_𝑖,𝑦_𝑗) such that 𝑎⩽𝑥₀<𝑥₁<⋯<𝑥_𝑛⩽𝑏;𝑐⩽𝑦₀<𝑦₁<⋯<𝑦_𝑚⩽𝑑, (9) then there exists a unique set of parameters 𝑎_0,0,𝑎_0,1,…,𝑎_𝑁,𝑚 for the interpolation function 𝑄𝑥,𝑦=𝑓₀𝐴₀𝑥,𝑦+𝑆₀𝑥,𝑦×𝑓₁𝐴₁𝑥,𝑦+𝑆₁𝑥,𝑦×𝑓₂𝐴₂𝑥,𝑦+⋯+𝑆_𝑁−1𝑥,𝑦×𝑓_𝑁𝐴_𝑁⋯,𝐴𝑥,𝑦_𝑖𝑥,𝑦=𝑓_𝑖,𝑖𝑎_𝑖,𝑖𝑥,𝑦+𝑔_𝑖(𝑥)×𝑓_𝑖+1,𝑖𝑎_𝑖+1,𝑖𝑥,𝑦+⋯+𝑔_𝑛−1(𝑥)×𝑓_𝑛,𝑖𝑎_𝑛,𝑖𝑥,𝑦⋯+ℎ_𝑖𝑦𝑓_𝑖,𝑖+1𝑎_𝑖,𝑖+1𝑥,𝑦+⋯+ℎ_𝑚−1𝑦×𝑓_𝑖,𝑚𝑎_𝑖,𝑚𝑥,𝑦⋯, (10) satisfying 𝑄𝑥_𝑖,𝑦_𝑗=𝑓𝑥_𝑖,𝑦_𝑗,𝑖=0,1,…,𝑛,𝑗=0,1,…,𝑚, (11) if(a)𝑓_𝑖, 𝑓_𝑖,𝑗 are continuous and strictly monotone in their domain of definitions and their ranges are (−∞,+∞),𝑖=1,2,…,𝑛,𝑗=0,1,…,𝑚,(b)𝑓₀,𝑓_𝑖,0 are continuous and their inverse functions 𝑓₀⁻¹, 𝑓⁻¹_𝑖,0 exist in 𝑆(𝑓₀), 𝑆(𝑓_𝑖,0), respectively, and 𝑆(𝑓₀)⊃𝑓([𝑎,𝑏],𝑦₀), 𝑆(𝑓_𝑖,0)⊃𝑓(𝑥_𝑖,[𝑐,𝑑]),(c)functions 𝑔_𝑖(𝑥),𝑖=0,1,…,𝑛−1;ℎ_𝑖,𝑗(𝑦),𝑖=0,1,…,𝑛,𝑗=0,1,…,𝑚−1, are continuous in [𝑎,𝑏], [𝑐,𝑑], respectively, and 𝑔_𝑖(𝑥)==0,𝑥=𝑥_𝑖;≠0,𝑥>𝑥_𝑖;ℎ_𝑖,𝑗𝑦=0,𝑦=𝑦_𝑗;≠0,𝑦>𝑦_𝑗. (12)

Proof. If 𝑥=𝑥₀, then 𝑄𝑥₀,𝑦=𝑓₀𝑓_0,0𝑎_0,0+ℎ₀𝑦𝑓_0,1𝑎_0,1+⋯+ℎ_𝑚−1𝑦×𝑓_0,𝑚𝑎_0,𝑚⋯; (13) this is just the univariate structure; from Theorem 1, we can get 𝑄(𝑥₀,𝑦_𝑗)=𝑓(𝑥₀,𝑦_𝑗), and 𝑄(𝑥₀,𝑦) is continuous.
Similarly, if 𝑦=𝑦₀, 𝑄𝑥,𝑦₀=𝑓₀𝑓_0,0𝑎_0,0+𝑔₀(𝑥)𝑓_1,0𝑎_1,0+⋯+𝑔_𝑛−1(𝑥)×𝑓_𝑛,0𝑎_𝑛,0⋯; (14) we can get 𝑄(𝑥,𝑦₀)=𝑓(𝑥,𝑦₀) easily, and 𝑄(𝑥,𝑦₀) is continuous. Similarly, if 𝑥=𝑥₁, 𝑄𝑥₁,𝑦=𝑓₀𝐴₀𝑥₁,𝑦+𝑆₀𝑥₁,𝑦𝑓₁𝐴₁𝑥₁,𝐴,𝑦₀𝑥₁,𝑦=𝑓_0,0𝑎_0,0+𝑔₀𝑥₁𝑓_1,0𝑎_1,0+ℎ₀𝑦×𝑓_0,1𝑎_0,1+⋯+ℎ_𝑚−1𝑦𝑓_0,𝑚𝑎_0,𝑚⋯,𝐴₁𝑥₁,𝑦=𝑓_1,1𝑎_1,1+ℎ₁𝑦×𝑓_1,2𝑎_1,2+⋯+ℎ_𝑚−1𝑦𝑓_1,𝑚𝑎_1,𝑚⋯, (15) and if 𝑦=𝑦₁, 𝑄𝑥,𝑦₁=𝑓₀𝐴₀𝑥,𝑦₁+𝑆₀𝑥,𝑦₁𝑓₁𝐴₁𝑥,𝑦₁,𝐴₀𝑥,𝑦₁=𝑓_0,0𝑎_0,0+𝑔₀(𝑥)𝑓_1,0𝑎_1,0+⋯+𝑔_𝑛−1(𝑥)𝑓_𝑛,0𝑎_𝑛,0+ℎ₀𝑦₁𝑓_0,1𝑎_0,1,𝐴₁𝑥,𝑦₁=𝑓_1,1𝑎_1,1+𝑔₁(𝑥)𝑓_2,1𝑎_2,1+⋯+𝑔_𝑛−1(𝑥)𝑓_𝑛,1𝑎_𝑛,1. (16)
We repeat the above process similarly, and finally we can obtain 𝑄(𝑥_𝑛,𝑦_𝑗)=𝑓(𝑥_𝑛,𝑦_𝑗), and 𝑄(𝑥_𝑛,𝑦) is continuous. When the above conditions are satisfied, the parameters 𝑎_0,0,𝑎_0,1,…,𝑎_𝑛,𝑚 are determined from the following equations in sequence.
From 𝑄(𝑥₀,𝑦₀)=𝑓₀(𝑓_0,0(𝑎_0,0)), we can get 𝑎_0,0=𝑓⁻¹_0,0𝑓₀⁻¹𝑄𝑥₀,𝑦₀; (17) from 𝑄(𝑥₀,𝑦₁)=𝑓₀(𝑓_0,0(𝑎_0,0+ℎ₀(𝑦₁)𝑓_0,1(𝑎_0,1))), we can get 𝑎_0,1=𝑓⁻¹_0,1𝑓⁻¹_0,0𝑓₀⁻¹𝑄𝑥₀,𝑦₁−𝑎_0,0ℎ₀𝑦₁; (18) from 𝑄(𝑥₀,𝑦₂)=𝑓₀(𝑓_0,0(𝑎_0,0+ℎ₀(𝑦₂)𝑓_0,1(𝑎_0,1+ℎ₁(𝑦₂)𝑓_0,2(𝑎_0,2)))), we can get 𝑎_0,2=𝑓⁻¹_0,2𝑓⁻¹_0,1𝑓⁻¹_0,0𝑓₀⁻¹𝑄𝑥₀,𝑦₂−𝑎_0,0ℎ₀𝑦₂−𝑎_1,1×ℎ₁𝑦₂⁻¹; (19) finally, we can obtain the parameters 𝑎_0,𝑗,𝑗=0,1,…,𝑚; (20) from 𝑄(𝑥₁,𝑦₀)=𝑓₀(𝑓_0,0(𝑎_0,0+𝑔₀(𝑥₁)𝑓_1,0(𝑎_1,0))), we can get 𝑎_1,0=𝑓⁻¹_1,0𝑓⁻¹_0,0𝑓₀⁻¹𝑄𝑥₁,𝑦₀−𝑎_0,0𝑔₀𝑥₁; (21) from 𝑄(𝑥₂,𝑦₀)=𝑓₀(𝑓_0,0(𝑎_0,0+𝑔₀(𝑥₂)𝑓_1,0(𝑎_1,0+𝑔₁(𝑥₂)𝑓_2,0(𝑎_2,0)))), we can get 𝑎_2,0=𝑓⁻¹_1,0𝑓⁻¹_1,0𝑓⁻¹_0,0𝑓₀⁻¹𝑄𝑥₂,𝑦₀−𝑎_0,0𝑔₀𝑥₂−𝑎_1,0×𝑔₁𝑥₂⁻¹. (22)
Using the induction method, finally, we can obtain all the parameters 𝑎_𝑖,𝑗,𝑖=0,1,…,𝑛,𝑗=0,1,…,𝑚. (23) Thus, this proves the theorem.

2.1. Special Cases

Some of the special cases of the above general interpolation formula of bivariate symmetry interpolation function are shown below. (1)If 𝑓_𝑖(𝑥)=𝑥, 𝑖=0,1,…,𝑁, 𝑓_𝑖,𝑗(𝑥)=𝑥,𝑆_𝑖(𝑥,𝑦)=(𝑥−𝑥_𝑖)(𝑦−𝑦_𝑖),𝑎_𝑖,𝑗(𝑥,𝑦) are constants, 𝑔_𝑖(𝑥)=𝑥−𝑥_𝑖, ℎ_𝑖(𝑦)=𝑦−𝑦_𝑖, 𝑖=0,1,…,𝑛,𝑗=0,1,…,𝑚, then 𝑄(𝑥,𝑦) is bivariate Newton interpolation polynomial [16]. (2)If 𝑓₀(𝑥)=𝑥, 𝑓_𝑖(𝑥)=1/𝑥, 𝑓_𝑖,𝑖(𝑥)=𝑥, 𝑓_𝑖,𝑗(𝑥)=1/𝑥, 𝑖≠𝑗, 𝑆_𝑖(𝑥,𝑦)=(𝑥−𝑥_𝑖)(𝑦−𝑦_𝑖), 𝑎_𝑖,𝑗(𝑥,𝑦) are constants, 𝑔_𝑖(𝑥)=𝑥−𝑥_𝑖, ℎ_𝑖(𝑦)=𝑦−𝑦_𝑖, 𝑖=0,1,…,𝑛, 𝑗=0,1,…,𝑚, then 𝑄(𝑥,𝑦) is bivariate symmetric continued fractions interpolation studied by many authors [2–5, 8]. (3)If 𝑓₀(𝑥)=𝑥, 𝑓_𝑖(𝑥)=1/𝑥, 𝑓_𝑖,𝑖(𝑥)=𝑥, 𝑓_𝑖,𝑗(𝑥)=1/𝑥, 𝑖≠𝑗, 𝑆_𝑖(𝑥,𝑦)=(𝑥−𝑥_𝑖)(𝑦−𝑦_𝑖), 𝑎_𝑖,𝑗(𝑥,𝑦) are constants, 𝑔_𝑖(𝑥)=𝑥−𝑥_𝑖, ℎ_𝑖(𝑦)=𝑦−𝑦_𝑖, 𝑖=0,1,…,𝑛, 𝑗=0,1,…,𝑚, then 𝑄(𝑥,𝑦) is Stieltjes-Newton rational interpolation studied by Wang [10]; Zhao and Tan [9] also studied it and its limiting case. (4)If 𝑓₀(𝑥)=𝑥, 𝑓_𝑖(𝑥)=1/𝑥, 𝑓_𝑖,𝑖(𝑥)=𝑥, 𝑓_𝑖,𝑗(𝑥)=𝑥^(−1)𝑖+1, 𝑖>𝑗, 𝑓_𝑖,,𝑗(𝑥)=𝑥^(−1)𝑗+1, 𝑖<𝑗, 𝑆_𝑖(𝑥,𝑦)=(𝑥−𝑥_𝑖)(𝑦−𝑦_𝑖), 𝑎_𝑖,𝑗(𝑥,𝑦) are constants, 𝑔_𝑖(𝑥)=𝑥−𝑥_𝑖, ℎ_𝑖(𝑦)=𝑦−𝑦_𝑖, 𝑖=0,1,…,𝑛, 𝑗=0,1,…,𝑚, then 𝑄(𝑥,𝑦) is Newton-Thiele-like interpolation formula studied by Kuchmins’ka and Vozna [6, 7]. (5)If 𝑓_𝑖(𝑥)=𝑥, 𝑓_𝑖,𝑖(𝑥)=𝑥, 𝑓_𝑖,𝑗(𝑥)=𝑥^(−1)𝑖+1,𝑖>𝑗, 𝑓_𝑖,𝑗(𝑥)=𝑥^(−1)𝑗+1, 𝑖<𝑗,𝑆_𝑖(𝑥,𝑦)=(𝑥−𝑥_𝑖)(𝑦−𝑦_𝑖), 𝑎_𝑖,𝑗(𝑥,𝑦) are constants, 𝑔_𝑖(𝑥)=𝑥−𝑥_𝑖, ℎ_𝑖(𝑦)=𝑦−𝑦_𝑖,  𝑖=0,1,…,𝑛, 𝑗=0,1,…,𝑚, then 𝑄(𝑥,𝑦) is symmetric Newton associated continued fraction blending rational interpolation. (6)If 𝑓₀(𝑥)=𝑥, 𝑓_𝑖(𝑥)=𝑥^𝛿, 𝑓_𝑖,𝑖(𝑥)=𝑥, 𝑓_𝑖,𝑗(𝑥)=𝑥^𝛿, 𝛿=1or−1, 𝑔_𝑖(𝑥)=𝑥−𝑥_𝑖, ℎ_𝑖(𝑦)=𝑦−𝑦_𝑖, 𝑖=0,1,…,𝑛, 𝑗=0,1,…,𝑚, then 𝑄(𝑥,𝑦) is the general frame of symmetry interpolation studied by Tan and Fang [1]. (7)If 𝑓₀(𝑥)=𝑥, 𝑓_𝑖(𝑥)=𝑥^(−1)𝑖+1, 𝑓_𝑖,𝑗(𝑥)=𝑥^(−1)𝑖+1, 𝑔_𝑖(𝑥)=𝑥−𝑥_𝑖, ℎ_𝑖(𝑦)=𝑦−𝑦_𝑖,𝑖=0,1,…,𝑛, 𝑗=0,1,…,𝑚, then 𝑄(𝑥,𝑦) is the general frame of symmetry interpolation studied by Zou and Tang [19]. (8)Suppose that the fixed points are arranged in groups of threes, which form 𝑛+1 L-like configurations. If 𝑓₀(𝑥)=𝑥, 𝑓_𝑖(𝑥)=𝑥, 1≤𝑖≤𝑁−1, 𝑓_𝑖,𝑖(𝑥)=𝑥, 𝑓_𝑖,𝑖+1(𝑥)=𝑓_𝑖+1,𝑖(𝑥)=𝑥, 𝑓_𝑖,𝑗(𝑥)=𝑓_𝑗,𝑖(𝑥)=𝑥, 𝑗≥𝑖+2, 𝑔_𝑖(𝑥)=𝑥−𝑥_2𝑖, ℎ_𝑖(𝑦)=𝑦−𝑦_2𝑖, 𝑎_{𝑖,𝑖+𝑠}(𝑥)=𝑎_{2𝑖,2𝑖+𝑠}, 1≤𝑠≤𝑛−𝑖, 𝑎_{𝑖+𝑡,𝑖}(𝑥)=𝑎_{2𝑖+𝑡,2𝑖}, 1≤𝑡≤𝑚−𝑖, 𝑖=0,1,…,𝑛; 𝑗=0,1,…,𝑚, then 𝑄(𝑥,𝑦) is bivariate polynomial interpolation over ortho-triples studied by Salzer [21]. (9)Suppose that the fixed points are arranged in groups of threes, which form 𝑛+1 L-like configurations. If 𝑓₀(𝑥)=𝑥, 𝑓_𝑖(𝑥)=1/𝑥, 1≤𝑖≤𝑁−1, 𝑓_𝑖,𝑖(𝑥)=𝑥, 𝑓_𝑖,𝑖+1(𝑥)=𝑓_𝑖+1,𝑖(𝑥)=1/𝑥, 𝑓_𝑖,𝑗(𝑥)=𝑓_𝑗,𝑖(𝑥)=𝑥, 𝑗≥𝑖+2, 𝑔_𝑖(𝑥)=𝑥−𝑥_2𝑖, ℎ_𝑖(𝑦)=𝑦−𝑦_2𝑖, 𝑎_{𝑖,𝑖+𝑠}(𝑥)=𝑎_{2𝑖,2𝑖+𝑠}, 1≤𝑠≤𝑛−𝑖, 𝑎_{𝑖+𝑡,𝑖}(𝑥)=𝑎_{2𝑖+𝑡,2𝑖}, 1≤𝑡≤𝑚−𝑖, 𝑖=0,1,…,𝑛, 𝑗=0,1,…,𝑚, then 𝑄(𝑥,𝑦) is bivariate continued fraction interpolation over ortho-triples studied by Wang and Qian [11]. (10)If 𝑓₀(𝑥)=𝑥, 𝑓_𝑖(𝑥)=𝑓_𝑖,𝑖(𝑥)=𝑥^(−1)𝑖+1, 𝑓_𝑖,𝑗(𝑥)=𝑥^(−1)𝑖+1, 𝑖>𝑗, 𝑓_𝑖,𝑗(𝑥)=𝑥^(−1)𝑗+1, 𝑖<𝑗, 𝑆_𝑖(𝑥,𝑦)=(𝑥−𝑥_𝑖)(𝑦−𝑦_𝑖),𝑎_𝑖,𝑗(𝑥,𝑦) are constants, 𝑔_𝑖(𝑥)=𝑥−𝑥_𝑖, ℎ_𝑖(𝑦)=𝑦−𝑦_𝑖, 𝑖=0,1,…,𝑛,𝑗=0,1,…,𝑚, then 𝑄(𝑥,𝑦) is a new type of symmetric blending rational interpolation. (11)If 𝑓₀(𝑥)=𝑥, 𝑓_𝑖(𝑥)=𝑓_𝑖,𝑖(𝑥)=𝑥, 𝑓_𝑖,𝑗(𝑥)=𝑥,𝑖>𝑗, 𝑓_𝑖,𝑗(𝑥)=𝑥^(−1)𝑗+1, 𝑖<𝑗, 𝑆_𝑖(𝑥,𝑦)=(𝑥−𝑥_𝑖)(𝑦−𝑦_𝑖), 𝑎_𝑖,𝑗(𝑥,𝑦) are constants, 𝑔_𝑖(𝑥)=𝑥−𝑥_𝑖, ℎ_𝑖(𝑦)=𝑦−𝑦_𝑖,  𝑖=0,1,…,𝑛, 𝑗=0,1,…,𝑚, then 𝑄(𝑥,𝑦) is a new type of symmetric blending rational interpolation. (12)If 𝑓_𝑖(𝑥)=𝑥, 𝑓_𝑖,𝑖(𝑥)=𝑥, 𝑓_𝑖,𝑗(𝑥)=𝑥,𝑖>𝑗, 𝑓_𝑖,𝑗(𝑥)=𝑥⁽⁻¹⁾,𝑖<𝑗,𝑆_𝑖(𝑥,𝑦)=(𝑥−𝑥_𝑖)(𝑦−𝑦_𝑖), 𝑎_𝑖,𝑗(𝑥,𝑦) are constants, 𝑔_𝑖(𝑥)=𝑥−𝑥_𝑖, ℎ_𝑖(𝑦)=𝑦−𝑦_𝑖, 𝑖=0,1,…,𝑛, 𝑗=0,1,…,𝑚, then 𝑄(𝑥,𝑦) is a new type of symmetric blending rational interpolation. (13)If 𝑓₀(𝑥)=𝑥, 𝑓_𝑖(𝑥)=𝑥⁻¹, 𝑓_𝑖,𝑖(𝑥)=𝑥, 𝑓_𝑖,𝑗(𝑥)=𝑥,𝑖>𝑗, 𝑓_𝑖,𝑗(𝑥)=𝑥⁽⁻¹⁾, 𝑖<𝑗, 𝑆_𝑖(𝑥,𝑦)=(𝑥−𝑥_𝑖)(𝑦−𝑦_𝑖), 𝑎_𝑖,𝑗(𝑥,𝑦) are constants, 𝑔_𝑖(𝑥)=𝑥−𝑥_𝑖, ℎ_𝑖(𝑦)=𝑦−𝑦_𝑖, 𝑖=0,1,…,𝑛, 𝑗=0,1,…,𝑚, then 𝑄(𝑥,𝑦) is a new type of symmetric blending rational interpolation. (14)If 𝑓₀(𝑥)=𝑥, 𝑓_𝑖(𝑥)=𝑥^(−1)𝑖+1, 𝑓_𝑖,𝑖(𝑥)=𝑥, 𝑓_𝑖,𝑗(𝑥)=𝑥, 𝑖>𝑗, 𝑓_𝑖,𝑗(𝑥)=𝑥⁽⁻¹⁾, 𝑖<𝑗, 𝑆_𝑖(𝑥,𝑦)=(𝑥−𝑥_𝑖)(𝑦−𝑦_𝑖), 𝑎_𝑖,𝑗(𝑥,𝑦) are constants, 𝑔_𝑖(𝑥)=𝑥−𝑥_𝑖, ℎ_𝑖(𝑦)=𝑦−𝑦_𝑖, 𝑖=0,1,…,𝑛, 𝑗=0,1,…,𝑚, then 𝑄(𝑥,𝑦) is a new type of symmetric blending rational interpolation. (15)If 𝑖(mod3)=0, 𝑓_𝑖(𝑥)=1/𝑥, otherwise, 𝑓_𝑖(𝑥)=𝑥, 𝑓_𝑖,𝑖(𝑥)=𝑥, 𝑓_𝑖,𝑗(𝑥)=𝑥, 𝑖>𝑗, 𝑓_𝑖,𝑗(𝑥)=𝑥⁽⁻¹⁾, 𝑖<𝑗, 𝑆_𝑖(𝑥,𝑦)=(𝑥−𝑥_𝑖)(𝑦−𝑦_𝑖), 𝑎_𝑖,𝑗(𝑥,𝑦) are constants, 𝑔_𝑖(𝑥)=𝑥−𝑥_𝑖, ℎ_𝑖(𝑦)=𝑦−𝑦_𝑖, 𝑖=0,1,…,𝑛,𝑗=0,1,…,𝑚, then 𝑄(𝑥,𝑦) is a new type of symmetric blending rational interpolation.

Furthermore, one can get more symmetric blending rational interpolations via choosing 𝑓_𝑖(𝑥),𝑓_𝑖,𝑗(𝑥) appropriately, for example, some new schemes given in the paper [19]. It is not difficult to generalize the general structure in this paper to higher dimensions or a vector-valued case or a matrix-valued case [16, 22, 23].

3. General Interpolation Formulae for Block Based Bivariate Interpolation

Now we consider the general interpolation formulae of the following scheme; we divide ∏_𝑛,𝑚={(𝑥_𝑖,𝑦_𝑗)∣𝑖=0,1,…,𝑛,𝑗=0,1,…,𝑚} into (𝑢+1)×(𝜈+1) subsets; namely, ^𝑠,𝑡_𝑛,𝑚=𝑥_𝑖,𝑦_𝑗∣𝑐_𝑠≤𝑖≤𝑑_𝑠,ℎ_𝑡≤𝑗≤𝑟_𝑡,𝑠=0,1,…,𝑢,𝑡=0,1,…,𝑣. (24)

The subsets may be achieved by reordering the interpolation points if necessary.

If we choose 𝐴_𝑖(𝑥,𝑦) as follows in formula (2): 𝐴_𝑖𝑥,𝑦=𝑓_𝑖,0𝑎_𝑖,0𝑥,𝑦+𝑔₀(𝑥)×𝑓_1,𝑖𝑎_1,𝑖𝑥,𝑦+⋯+𝑔_𝑢−1(𝑥)×𝑓_𝑢,𝑖𝑎_𝑢,𝑖𝑥,𝑦…+ℎ₀𝑦𝑓_𝑖,1𝑎_𝑖,1𝑥,𝑦+⋯+ℎ_𝑣−1𝑦×𝑓_𝑖,𝑣𝑎_𝑖,𝑣𝑥,𝑦…, (25) then 𝑄^′(𝑥,𝑦) is a general interpolation formula of block based bivariate interpolation.

Theorem 3. Given a function 𝑓(𝑥,𝑦) continuous in [𝑎,𝑏]×[𝑐,𝑑] and (𝑛+1)×(𝑚+1) points (𝑥_𝑖,𝑦_𝑗)(𝑖=0,1,⋯<𝑛,𝑗=0,1,…,𝑚) such that 𝑎⩽𝑥₀<𝑥₁<⋯<𝑥_𝑛⩽𝑏;𝑐⩽𝑦₀<𝑦₁<⋯<𝑦_𝑚⩽𝑑, (26) then there exists a unique set of parameters 𝑎_0,0,𝑎_0,1,…,𝑎_𝑛,𝑚 for the interpolation function 𝑄^′𝑥,𝑦=𝑓₀𝐴₀𝑥,𝑦+𝑆₀𝑥,𝑦×𝑓₁𝐴₁𝑥,𝑦+𝑆₁𝑥,𝑦×𝑓₂𝐴₂𝑥,𝑦+⋯+𝑆_𝑁−1𝑥,𝑦×𝑓_𝑁𝐴_𝑁𝑥,𝑦⋯,𝐴_𝑖𝑥,𝑦=𝑓_𝑖,0𝑎_𝑖,0𝑥,𝑦+𝑔₀(𝑥)×𝑓_1,𝑖𝑎_1,𝑖𝑥,𝑦+⋯+𝑔_𝑢−1(𝑥)×𝑓_𝑢,𝑖𝑎_𝑢,𝑖𝑥,𝑦⋯+ℎ₀𝑦𝑓_𝑖,1𝑎_𝑖,1𝑥,𝑦+⋯+ℎ_𝑣−1𝑦×𝑓_𝑖,𝑣𝑎_𝑖,𝑣𝑥,𝑦⋯, (27)(28) satisfying 𝑄^′𝑥_𝑖,𝑦_𝑗=𝑓𝑥_𝑖,𝑦_𝑗,𝑖=0,1,…,𝑛,𝑗=0,1,…,𝑚, (29) if(a)𝑓_𝑖,𝑓_𝑖,𝑗 are continuous and strictly monotone in their domain of definitions and their ranges are (−∞,+∞), 𝑖=1,2,…,𝑛; 𝑗=0,1,…,𝑚;(b)𝑓₀,𝑓_𝑖,0 are continuous and their inverse functions 𝑓₀⁻¹, 𝑓⁻¹_𝑖,0 exist in 𝑆(𝑓₀), 𝑆(𝑓_𝑖,0), respectively, and 𝑆(𝑓₀)⊃𝑓([𝑎,𝑏],𝑦₀), 𝑆(𝑓_𝑖,0)⊃𝑓(𝑥_𝑖,[𝑐,𝑑]);(c)functions 𝑔_𝑖(𝑥),𝑖=0,1,…,𝑛−1; ℎ_𝑖,𝑗(𝑦), 𝑖=0,1,…,𝑛, 𝑗=0,1,…,𝑚−1, are continuous in [𝑎,𝑏], [𝑐,𝑑], respectively, and 𝑔_𝑖(𝑥)=0,𝑥=𝑥_𝑖;≠0,𝑥>𝑥_𝑖;ℎ_𝑖,𝑗𝑦=0𝑦=𝑦_𝑗;≠0𝑦>𝑦_𝑗. (30)
We can prove the previous theorem similarly.

3.1. General Interpolation Formulae for Block Based Univariate Interpolation

If we choose the parameters in formulae (2), (25) as follows, we can get general interpolation formulae for block based univariate interpolation.(1)If we choose 𝑓_𝑖,1(𝑥)≡0, 𝑓₁(𝑥)≡0, 𝑔_𝑠(𝑥)=∏^𝑑𝑠_{𝑖=𝑐𝑠}(𝑥−𝑥_𝑖),𝑖=0,1,…,𝑛, 𝑠=0,1,…,𝑢−1, we can get 𝑄𝑥,𝑦=𝑓₀𝑓_0,0𝑎_0,0𝑥,𝑦+𝑔₀(𝑥)×𝑓_1,0𝑎_1,0𝑥,𝑦+⋯+𝑔_𝑢−1(𝑥)×𝑓_𝑢,0𝑎_𝑢,0𝑥,𝑦⋯, (31) where 𝑎_𝑖,𝑗(𝑥,𝑦)=𝑎_𝑖(𝑥) are univariate interpolating polynomial, rational interpolation, the Hermite interpolating polynomial, or Salzer-type osculatory rational interpolation. Then 𝑄(𝑥,𝑦) is a general interpolation formula of block based univariate interpolation.(2)If we choose 𝑓_1,𝑖(𝑥)≡0, 𝑓₁(𝑥)≡0, ℎ_𝑡(𝑦)=𝑔_𝑠(𝑥)=∏^𝑑𝑠_{𝑖=𝑐𝑠}(𝑥−𝑥_𝑖), 𝑠=0,1,…,𝑢−1, we can get 𝑄𝑥,𝑦=𝑓₀𝑓_0,0𝑎_0,0𝑥,𝑦+ℎ₀𝑦×𝑓_0,1𝑎_0,1𝑥,𝑦+⋯+ℎ_𝑣−1(𝑥)×𝑓_0,𝑣𝑎_0,𝑣𝑥,𝑦⋯, (32) where 𝑎_𝑖,𝑗(𝑥,𝑦)=𝑎_𝑖(𝑥) are univariate interpolating polynomial, rational interpolation, the Hermite interpolating polynomial, or Salzer-type osculatory rational interpolation. Then 𝑄(𝑥,𝑦) is a general interpolation formula of block based univariate interpolation.

3.1.1. Special Case

We discuss the case that we choose the parameters in (2), 𝑄𝑥,𝑦=𝑓₀𝑓_0,0𝑎_0,0𝑥,𝑦+𝑔₀(𝑥)×𝑓_1,0𝑎_1,0𝑥,𝑦+⋯+𝑔_𝑢−1(𝑥)×𝑓_𝑢,0𝑎_𝑢,0𝑥,𝑦⋯. (33) Some of the special cases of the above general interpolation formula of interpolation functions are shown below. (1)If 𝑓₀(𝑥)=𝑥, 𝑓_𝑖,0(𝑥)=𝑥, 𝑔_𝑠(𝑥)=∏^𝑑𝑠_{𝑖=𝑐𝑠}(𝑥−𝑥_𝑖), 𝑖=0,1,…,𝑛, 𝑠=0,1,…,𝑢−1, then 𝑄(𝑥,𝑦) is univariate block based Newton-like interpolation polynomial [12]. (2)If 𝑓₀(𝑥)=𝑥, 𝑓_0,0(𝑥)=𝑥, 𝑓_𝑖,0(𝑥)=1/𝑥, 𝑔_𝑠(𝑥)=∏^𝑑𝑠_{𝑖=𝑐𝑠}(𝑥−𝑥_𝑖), 𝑖=0,1,…,𝑛, 𝑠=0,1,…,𝑢−1, then 𝑄(𝑥,𝑦) is univariate block based Thiele-like continued fractions interpolation [13]. (3)If 𝑓₀(𝑥)=𝑥,  𝑓_0,0(𝑥)=𝑥, 𝑓_𝑖,0(𝑥)=𝑥^(−1)𝑖+1, 𝑔_𝑠(𝑥)=∏^𝑑𝑠_{𝑖=𝑐𝑠}(𝑥−𝑥_𝑖), 𝑖=0,1,…,𝑛, 𝑠=0,1,…,𝑢−1, then 𝑄(𝑥,𝑦) is univariate block based associated continued fractions interpolation [17]. (4)If 𝑓₀(𝑥)=𝑥, 𝑓_0,0(𝑥)=𝑥, 𝑓_𝑖,0(𝑥)=𝑥 or 1/𝑥, 𝑔_𝑠(𝑥)=𝑥−𝑥_𝑠, 𝑖=0,1,…,𝑛, 𝑠=0,1,…,𝑢−1, then 𝑄(𝑥,𝑦) is the general frame of interpolation scheme studied by Tan and Fang [1]. (5)If 𝑓₀(𝑥)=𝑥, 𝑓_0,0(𝑥)=𝑥, 𝑓_𝑖,0(𝑥)=𝑥^(−1)𝑖+1, 𝑔_𝑠(𝑥)=∏^𝑑𝑠_{𝑖=𝑐𝑠}(𝑥−𝑥_𝑖), 𝑖=0,1,…,𝑛, 𝑠=0,1,…,𝑢−1, then 𝑄(𝑥,𝑦) is univariate block based associated continued fractions interpolation [17]. (6)If 𝑓₀(𝑥)=𝑥, 𝑓_0,0(𝑥)=𝑥, 𝑓_𝑖,0(𝑥)=𝑥 or 1/𝑥, 𝑖=0,1,…,𝑛, then 𝑄(𝑥,𝑦) is the general frame of interpolation scheme studied by Tang and Zou [17]. (7)If 𝑓₀(𝑥)=𝑥, then 𝑄(𝑥,𝑦) is the general frames of interpolation scheme studied by Zou and Tang [20].

If we choose 𝑑_𝑠=𝑐_𝑠+1 in scheme as shown above, that is to say, every block only includes one point, then 𝑄(𝑥,𝑦) is changed into univariate Newton interpolation polynomial, Thiele continued fractions interpolation, and associated continued fractions interpolation. Furthermore, one can get some blending rational interpolations or osculatory interpolation via choosing 𝑓_𝑖,𝑓_𝑖,𝑗 appropriately; for example, one can get modified Thiele continued fractions blending rational interpolation, three associated continued fractions interpolation, block based Newton-Werner blending osculatory rational interpolation, Thiele-Werner blending osculatory rational interpolation, and so on [16, 24].

3.2. General Interpolation Formulae for Block Based Bivariate Blending Rational Interpolation

If we choose the parameters in formulae (2) and (25) as follows, 𝑓_1,𝑖(𝑥)𝑆≡0,𝑖=1,2,…,𝑛,_𝑠𝑥,𝑦=^𝑑𝑠_{𝑘=𝑐𝑠}𝑥−𝑥_𝑘,ℎ_𝑡𝑦=^𝑟𝑡_𝑗=ℎ𝑡𝑦−𝑦_𝑗,𝑠=0,1,…,𝑢,𝑡=0,1,…𝑣, (34) we can get 𝑄𝑥,𝑦=𝑓₀𝐴₀𝑥,𝑦+𝑆₀𝑥,𝑦×𝑓₁𝐴₁𝑥,𝑦+𝑆₁𝑥,𝑦×𝑓₂𝐴₂𝑥,𝑦+⋯+𝑆_𝑢−1𝑥,𝑦×𝑓_𝑢𝐴_𝑢⋯𝐴𝑥,𝑦_𝑖𝑥,𝑦=𝑓_𝑖,0𝑎_𝑖,0𝑥,𝑦+ℎ₀𝑦×𝑓_𝑖,1𝑎_𝑖,1𝑥,𝑦+⋯+ℎ_𝑣−1𝑦×𝑓_𝑖,𝑣𝑎_𝑖,𝑣𝑥,𝑦⋯. (35) then 𝑄(𝑥,𝑦) is a general interpolation formula of block based bivariate blending rational interpolation.

3.2.1. Special Case

Some of the special cases of the general interpolation formula of block based bivariate interpolation are shown below. (1)If 𝑓_𝑖(𝑥)=𝑥, 𝑓_𝑖,𝑗(𝑥)=𝑥, 𝑖=0,1,…,𝑛, 𝑗=0,1,…,𝑚, 𝑆_𝑠(𝑥,𝑦)=∏^𝑑𝑠_{𝑘=𝑐𝑠}(𝑥−𝑥_𝑘), ℎ_𝑡(𝑦)=∏^𝑟𝑡_𝑗=ℎ𝑡(𝑦−𝑦_𝑗),  𝑠=0,1,…,𝑢, 𝑡=0,1,…𝑣, then 𝑄(𝑥,𝑦) is block based bivariate Newton-like blending rational interpolation [12]; especially, let 𝑑_𝑠=𝑐_𝑠+1; that is to say, every block only includes one point; then, 𝑄(𝑥,𝑦) is bivariate Newton interpolation polynomial. (2)If 𝑓₀(𝑥)=𝑥, 𝑓_𝑖,0(𝑥)=𝑥, 𝑓_𝑖(𝑥)=1/𝑥, 𝑓_𝑖,𝑗(𝑥)=1/𝑥, 𝑖=0,1,…,𝑛, 𝑗=0,1,…,𝑚,𝑆_𝑠(𝑥,𝑦)=∏^𝑑𝑠_{𝑘=𝑐𝑠}(𝑥−𝑥_𝑘), ℎ_𝑡(𝑦)=∏^𝑟𝑡_𝑗=ℎ𝑡(𝑦−𝑦_𝑗), 𝑠=0,1,…,𝑢, 𝑡=0,1,…𝑣, then 𝑄(𝑥,𝑦) is block based bivariate Thiele-like blending rational interpolation [13]; especially, let 𝑑_𝑠=𝑐_𝑠+1; that is to say, every block only includes one point; then, 𝑄(𝑥,𝑦) is bivariate Thiele-type branched continued fractions interpolation [11, 25]. (3)If 𝑓₀(𝑥)=𝑥, 𝑓_𝑖,0(𝑥)=𝑥, 𝑓_𝑖,𝑗(𝑥)=𝑥, 𝑓_𝑖(𝑥)=1/𝑥, 𝑖=0,1,…,𝑛, 𝑗=0,1,…,𝑚, 𝑆_𝑠(𝑥,𝑦)=∏^𝑑𝑠_{𝑘=𝑐𝑠}(𝑥−𝑥_𝑘), ℎ_𝑡(𝑦)=∏^𝑟𝑡_𝑗=ℎ𝑡(𝑦−𝑦_𝑗), 𝑠=0,1,…,𝑢, 𝑡=0,1,…𝑣, then 𝑄(𝑥,𝑦) is block based Newton-Thiele-like blending rational interpolation [17]; especially, let 𝑑_𝑠=𝑐_𝑠+1; that is to say, every block only includes one point; then, 𝑄(𝑥,𝑦) is bivariate Newton-Thiele blending rational interpolation [11, 20]. (4)If 𝑓₀(𝑥)=𝑥,𝑓_𝑖,0(𝑥)=𝑥,𝑓_𝑖(𝑥)=𝑥,𝑓_𝑖,𝑗(𝑥)=1/𝑥,𝑖=0,1,…,𝑛,𝑗=0,1,…,𝑚,  𝑆_𝑠(𝑥,𝑦)=∏^𝑑𝑠_{𝑘=𝑐𝑠}(𝑥−𝑥_𝑘),ℎ_𝑡(𝑦)=∏^𝑟𝑡_𝑗=ℎ𝑡(𝑦−𝑦_𝑗),𝑠=0,1,…,𝑢,𝑡=0,1,…𝑣, then 𝑄(𝑥,𝑦) is block based Thiele-Newton-like blending rational interpolation [17, 18]; especially, let 𝑑_𝑠=𝑐_𝑠+1; that is to say, every block only includes one point; then, 𝑄(𝑥,𝑦) is bivariate Thiele-Newton blending rational interpolation [1, 17, 20]. (5)If 𝑓₀(𝑥)=𝑥,𝑓_𝑖(𝑥)=𝑥^(−1)𝑖+1, 𝑓_𝑖,𝑗(𝑥)=𝑥, 𝑖=0,1,…,𝑛, 𝑗=0,1,…,𝑚, 𝑆_𝑠(𝑥,𝑦)=∏^𝑑𝑠_{𝑘=𝑐𝑠}(𝑥−𝑥_𝑘), ℎ_𝑡(𝑦)=∏^𝑟𝑡_𝑗=ℎ𝑡(𝑦−𝑦_𝑗), 𝑠=0,1,…,𝑢,𝑡=0,1,…𝑣, then 𝑄(𝑥,𝑦) is block based bivariate Newton associated continued fractions blending rational interpolation [17, 20]; especially, let 𝑑_𝑠=𝑐_𝑠+1; that is to say, every block only includes one point; then, 𝑄(𝑥,𝑦) is bivariate Newton associated continued fractions blending rational interpolation [17, 20, 26]. (6)If 𝑓_𝑖,0(𝑥)=𝑥, 𝑓_𝑖(𝑥)=𝑥,𝑓_𝑖,𝑗(𝑥)=𝑥^(−1)𝑖+1,𝑖=0,1,…,𝑛,𝑗=0,1,…,𝑚, 𝑆_𝑠(𝑥,𝑦)=∏^𝑑𝑠_{𝑘=𝑐𝑠}(𝑥−𝑥_𝑘), ℎ_𝑡(𝑦)=∏^𝑟𝑡_𝑗=ℎ𝑡(𝑦−𝑦_𝑗), 𝑠=0,1,…,𝑢,𝑡=0,1,…𝑣, then 𝑄(𝑥,𝑦) is block based bivariate associated continued fractions Newton blending rational interpolation [17, 20, 26]; especially, let 𝑑_𝑠=𝑐_𝑠+1; that is to say, every block only includes one point; then, 𝑄(𝑥,𝑦) is bivariate associated continued fractions Newton blending rational interpolation [17, 20, 26]. (7)If 𝑓₀(𝑥)=𝑥,𝑓_𝑖,0(𝑥)=𝑥, 𝑓_𝑖(𝑥)=1/𝑥, 𝑓_𝑖,𝑗(𝑥)=𝑥^(−1)𝑖+1, 𝑖=0,1,…,𝑛, 𝑗=0,1,…,𝑚, 𝑆_𝑠(𝑥,𝑦)=∏^𝑑𝑠_{𝑘=𝑐𝑠}(𝑥−𝑥_𝑘), ℎ_𝑡(𝑦)=∏^𝑟𝑡_𝑗=ℎ𝑡(𝑦−𝑦_𝑗), 𝑠=0,1,…,𝑢,𝑡=0,1,…𝑣, then 𝑄(𝑥,𝑦) is block based bivariate Thiele associated continued fractions blending rational interpolation [17, 20, 26]; especially, let 𝑑_𝑠=𝑐_𝑠+1; that is to say, every block only includes one point; then, 𝑄(𝑥,𝑦) is bivariate Thiele associated continued fractions blending rational interpolation [17, 20, 26]. (8)If 𝑓₀(𝑥)=𝑥,𝑓_𝑖,0(𝑥)=𝑥, 𝑓_𝑖(𝑥)=𝑥^(−1)𝑖+1, 𝑓_𝑖,𝑗(𝑥)=1/𝑥,𝑖=0,1,…,𝑛,𝑗=0,1,…,𝑚, 𝑆_𝑠(𝑥,𝑦)=∏^𝑑𝑠_{𝑘=𝑐𝑠}(𝑥−𝑥_𝑘), ℎ_𝑡(𝑦)=∏^𝑟𝑡_𝑗=ℎ𝑡(𝑦−𝑦_𝑗),  𝑠=0,1,…,𝑢, 𝑡=0,1,…𝑣, then 𝑄(𝑥,𝑦) is block based bivariate associated continued fractions Thiele blending rational interpolation [17, 20, 26]; especially, let 𝑑_𝑠=𝑐_𝑠+1; that is to say, every block only includes one point; then, 𝑄(𝑥,𝑦) is bivariate associated continued fractions Thiele blending rational interpolation [17, 20, 26]. (9)If 𝑓₀(𝑥)=𝑥,𝑓_𝑖,0(𝑥)=𝑥, 𝑓_𝑖(𝑥)=𝑥^(−1)𝑖+1, 𝑓_𝑖,𝑗(𝑥)=𝑥^(−1)𝑖+1, 𝑖=0,1,…,𝑛,𝑗=0,1,…,𝑚, 𝑆_𝑠(𝑥,𝑦)=∏^𝑑𝑠_{𝑘=𝑐𝑠}(𝑥−𝑥_𝑘), ℎ_𝑡(𝑦)=∏^𝑟𝑡_𝑗=ℎ𝑡(𝑦−𝑦_𝑗), 𝑠=0,1,…,𝑢, 𝑡=0,1,…𝑣, then 𝑄(𝑥,𝑦) is block based bivariate associated continued fractions blending rational interpolation [17, 20, 26]; especially, let 𝑑_𝑠=𝑐_𝑠+1; that is to say, every block only includes one point; then, 𝑄(𝑥,𝑦) is bivariate branched associated continued fractions blending rational interpolation [17, 20, 26, 27]. (10)If 𝑓_𝑖(𝑥)=𝑥, 𝑓_𝑖,𝑗(𝑥)=𝑥, 𝑢=𝑛, 𝑆_𝑠(𝑥)=sin𝑥−sin𝑥_𝑖, ℎ_𝑗(𝑦)=sin𝑦−sin𝑦_𝑗, 𝑖=0,1,…,𝑛; 𝑗=0,1,…,𝑚, then 𝑄(𝑥,𝑦) is a bivariate trigonometric function and may be expanded to a finite Fourier series. (11)If we set 𝑢=𝑛,𝑆_𝑠(𝑥)=𝑠(𝑥)−𝑠(𝑥_𝑖), ℎ_𝑗(𝑦)=𝑠(𝑦)−𝑠(𝑦_𝑗) and choose 𝑓_𝑖,𝑗(𝑥), 𝑓_𝑖(𝑥), 𝑖=0,1,…,𝑛,𝑗=0,1,…,𝑚, from 𝑥,1/𝑥; 𝑠(𝑥),𝑠(𝑦) are chosen from 𝑒^𝑥,𝑥²,cos𝑥,𝑒^−𝑥,𝑠ℎ𝑥, and so forth, then we have a class of interpolation functions. (12)If 𝑓₀(𝑥)=𝑥,𝑓_𝑖(𝑥)=𝑥, or 𝑓_𝑖(𝑥)=1/𝑥, 𝑓_𝑖,𝑗(𝑥)=𝑥, or 𝑓_𝑖,𝑗(𝑥)=1/𝑥, 𝑢=𝑛, 𝑣=𝑚, 𝑆_𝑠(𝑥,𝑦)=(𝑥−𝑥_𝑠), ℎ_𝑡(𝑦)=(𝑦−𝑦_𝑗), 𝑖=0,1,…,𝑛,𝑗=0,1,…,𝑚, then 𝑄(𝑥,𝑦) is the general frame of interpolation scheme studied by Tan and Fang [1]. (13)If 𝑓₀(𝑥)=𝑥, 𝑓_𝑖(𝑥)=𝑥^(−1)𝑖+1, 𝑓_𝑖,𝑗(𝑥)=𝑥^(−1)𝑖+1, 𝑢=𝑛, 𝑣=𝑚, 𝑆_𝑠(𝑥,𝑦)=(𝑥−𝑥_𝑠), ℎ_𝑡(𝑦)=(𝑦−𝑦_𝑗),  𝑖=0,1,…,𝑛,𝑗=0,1,…,𝑚, then 𝑄(𝑥,𝑦) is the general frame of interpolation scheme studied by Tang and Zou [17]. (14)If 𝑓₀(𝑥)=𝑥, 𝑓_𝑖(𝑥)=𝑥^(−1)𝑖+1, 𝑓_𝑖,𝑗(𝑥)=𝑥^(−1)𝑖+1, 𝑖=0,1,…,𝑛, 𝑗=0,1,…,𝑚, then 𝑄(𝑥,𝑦) is the general frame of interpolation scheme studied by Tang and Zou [17]. (15)If 𝑆_𝑠(𝑥,𝑦), ℎ_𝑡(𝑦), 𝑠=0,1,…,𝑢,𝑡=0,1,…,𝑣, are the same as 𝑔_𝑖(𝑥),ℎ_𝑖,𝑗(𝑦),𝑖=0,1,…,𝑛, 𝑗=0,1,…,𝑚, of paper [20], then 𝑄(𝑥,𝑦) is the general frame of interpolation scheme studied by Zou and Tang [20].

Special cases (10) and (11) are interesting and useful; we will investigate them in depth in the future. Furthermore, one can get more blending rational interpolations via choosing 𝑓_𝑖(𝑥),𝑓_𝑖,𝑗(𝑥) appropriately. It could be used to deal with the interpolation problems where inverse differences are nonexistent or unattainable points occur via choosing 𝑓_𝑖(𝑥),𝑓_𝑖,𝑗(𝑥) appropriately [17].

3.3. Algorithm of General Interpolation Formulae of Block Based Bivariate Interpolation

In this section, we give the algorithm of general interpolation formula of block based bivariate interpolation.

𝑓_𝑖,𝑓_𝑖,𝑗 given is initialized.

Step  1. Let 𝑓𝑥,𝑦=𝑓^0,0_𝑖,𝑗,𝑖=0,1,…,𝑛,𝑗=0,1,…,𝑚. (36)

Step  2. For 𝑡=1,2,…,𝑣, 𝑓^0,𝑡_𝑖,𝑗=𝑓⁻¹_0,𝑡𝑓^0,𝑡−1_𝑖,𝑗−𝑎_0,𝑡−1𝑥_𝑖,𝑦_𝑗ℎ_𝑡−1𝑦_𝑗,𝑖=0,1,…,𝑛,𝑗=ℎ_𝑡,ℎ_𝑡+1,…,𝑚. (37)

Step  3. For𝑠=1,2,…,𝑢, 𝑓^𝑠,0_𝑖,𝑗=𝑓_𝑠⁻¹𝑓^𝑠−1,0_𝑖,𝑗−𝑍_𝑠−1𝑥_𝑖,𝑦_𝑗𝑆_𝑠−1𝑥_𝑖,𝑖=𝑐_𝑠,𝑐_𝑠+1,…,𝑛;𝑗=0,1,…,𝑚. (38)

Step  4. For 𝑠=1,2,…,𝑢, 𝑡=1,2,…,𝑣, 𝑓^𝑠,𝑡_𝑖,𝑗=𝑓⁻¹_𝑠,𝑡𝑓^{𝑠,𝑡−1}_𝑖,𝑗−𝑎_{𝑠,𝑡−1}𝑥_𝑖,𝑦_𝑗ℎ_𝑡−1𝑦_𝑗,𝑖=𝑐_𝑠,𝑐_𝑠+1,…,𝑛;𝑗=ℎ_𝑡,ℎ_𝑡+1,…,𝑚. (39)

Step  5. For𝑠=1,2,…,𝑢, 𝑡=1,2,…,𝑣, 𝑎_𝑠,𝑡𝑥,𝑦=𝑓^𝑠,𝑡_𝑖,𝑗,𝑐_𝑠≤𝑖≤𝑑_𝑠,ℎ_𝑡≤𝑗≤𝑟_𝑡,𝑠=0,1,…,𝑢,𝑡=0,1,…,𝑣, (40) where 𝑎_𝑠,𝑡(𝑥,𝑦)(𝑡=0,1,…,𝑣) are bivariate polynomials or rational interpolations on the subsets ∏^𝑠,𝑡_𝑛,𝑚.

3.4. Dual Scheme for General Interpolation Formulae for Block Based Bivariate Interpolation

If we choose the parameters in formulae (2) and (25) as follows, 𝑓_𝑖,1(𝑥)≡0,𝑆_𝑡𝑥,𝑦=^𝑟𝑡_𝑗=ℎ𝑡𝑦−𝑦_𝑗,𝑔_𝑠(𝑥)=^𝑑𝑠_{𝑘=𝑐𝑠}𝑥−𝑥_𝑘,𝑠=0,1,…,𝑢,𝑡=0,1,…𝑣, (41) then 𝑄𝑥,𝑦=𝑓₀𝐴₀𝑥,𝑦+𝑆₀𝑥,𝑦×𝑓₁𝐴₁𝑥,𝑦+𝑆₁𝑥,𝑦×𝑓₂𝐴₂𝑥,𝑦+⋯+𝑆_𝑣−1𝑥,𝑦×𝑓_𝑣𝐴_𝑣⋯,𝐴𝑥,𝑦_𝑖𝑥,𝑦=𝑓_0,𝑖𝑎_0,𝑖𝑥,𝑦+𝑔₀(𝑥)×𝑓_1,𝑖𝑎_1,𝑖𝑥,𝑦+⋯+𝑔_𝑢−1(𝑥)×𝑓_𝑢,𝑖𝑎_𝑢,𝑖𝑥,𝑦⋯. (42)

We call the scheme defined by formulae (41)-(42) dual scheme of general interpolation formula for block based bivariate interpolation of (34)-(35). We can discuss this frame similarly, and the above dual interpolation function also includes many kinds of the interpolation schemes which does not as the same as the schemes we have discussed in Section 3.2.

4. Numerical Examples

In this section, we take two simple examples to show the effectiveness of the result in this paper. Example 4 is to show how the proposed construction takes out under different choice of 𝑓_𝑖’s, 𝑓_𝑖,𝑗’s and 𝑔’s, ℎ’s. Example 10 is given to solve the interpolation problem where inverse differences are nonexistent.

Example 4. Let (𝑥_𝑖,𝑦_𝑗) and 𝑓(𝑥_𝑖,𝑦_𝑗)(𝑖=0,1,2,𝑗=0,1,2) be given in Table 1.
Using the frame in the paper, one can get many special interpolations; some of them are as follows.

Table 1 

Interpolation data.

Table 2 

Interpolation data.

Scheme 1. Symmetric continued fractions interpolation is 𝑄₁𝑥𝑥,𝑦=1+1+𝑥−11+𝑦−1+𝑦−12+𝑥𝑦/()+()1/2+𝑦−1−2/5𝑥−1𝑦−1.1/3 (43)

Scheme 2. Bivariate Newton interpolation polynomial is 𝑄₂1𝑥,𝑦=1−𝑦−2𝑦+1𝑦−1+𝑥2𝑥()()+9𝑥−1+2𝑥𝑦−3𝑥𝑦𝑦−1−𝑥𝑦𝑥−14𝑥(𝑥−1)𝑦𝑦−1. (44)

Scheme 3. Bivariate Thiele branched continued fractions interpolation is 𝑄₃𝑦𝑥,𝑦=1+−1+𝑦−13+𝑥()++1+𝑦/−3/2𝑦−1/2𝑥−1()+/().−3+𝑦/1/18𝑦−1198/97 (45)

Scheme 4. Bivariate Newton-Thiele blending rational interpolation is 𝑄₄1𝑥,𝑦=1−𝑦−2𝑦+𝑥𝑦−1−𝑦+1−2𝑦/3𝑦−1/3𝑥−1()𝑦.−3+18𝑦−97/6𝑦−1 (46)

Scheme 5. Bivariate Thiele-Newton blending rational interpolation is 𝑄₅𝑦𝑥,𝑦=1+−1+𝑦−13+𝑦1+1/2+𝑦−1𝑥+1−2/32+𝑦−1+𝑦−1𝑥().5/9𝑥−1 (47)
If 𝑐₀=0,𝑑₀=1,𝑐₁=𝑑₁=2;ℎ₀=0,𝑟₀=1,ℎ₁=𝑟₁=2, namely, we divided ∏_2,2 into the following four subsets ∏^0,0_2,2,∏^0,1_2,2,∏^1,0_2,2, and ∏^1,1_2,2: ()()()0,0)(0,1)(0,21,0)(1,1)(1,22,0)(2,1)(2,2 (48)
Let 𝑎_0,0 be bivariate Newton interpolating polynomial ∏^0,0_2,2, 𝑎_0,1 bivariate Newton interpolating polynomial ∏^0,1_2,2, 𝑎_1,0 bivariate Newton interpolating polynomial ∏^1,0_2,2, and 𝑎_1,1 bivariate Newton interpolating polynomial ∏^1,1_2,2.

Scheme 6. Block based bivariate Thiele-Newton blending rational interpolation is 𝑄₆𝑦𝑥,𝑦=1+𝑥−𝑦+2𝑥𝑦+𝑦−1+1−2+12𝑥/72𝑦−𝑦+𝑦−1𝑥().−20/27𝑥−1 (49)

Scheme 7. Block based bivariate Newton-Thiele blending rational interpolation is 𝑄₇−1𝑥,𝑦=1+𝑥−𝑦+2𝑥𝑦+2𝑦+𝑥()−3𝑥𝑦−1𝑥−1()𝑦.2−4𝑦+19/6𝑦−1 (50)

Scheme 8. Block based bivariate Newton-like blending rational interpolation is 𝑄₈−1𝑥,𝑦=1+𝑥−𝑦+2𝑥𝑦+2𝑦+1−3𝑥𝑦−129−𝑦+4𝑦𝑦−1𝑥(𝑥−1). (51)

Scheme 9. Block based bivariate Thiele-like blending rational interpolation is 𝑄₉+𝑦𝑥,𝑦=1+𝑥−𝑦+2𝑥𝑦𝑦−1()+𝑥()−2+12𝑥/7𝑥−1𝑦/().2−4𝑦+𝑦−142/121 (52) It is easy to verify 𝑄_𝑠𝑥_𝑖,𝑦_𝑗=𝑓𝑥_𝑖,𝑦_𝑗,𝑖=0,1,2;𝑗=0,1,2,𝑠=1,2,…,9. (53)

Example 10. Let (𝑥_𝑖,𝑦_𝑗) and 𝑓(𝑥_𝑖,𝑦_𝑗) be given in Table 2.
Newton-Thiele blending rational interpolation fails in this case, since calculating inverse differences leads to that one of denominator is zero. From the general frame (34)-(35), by choosing 𝑓_𝑖(𝑥), 𝑓_𝑖,𝑗(𝑥) appropriately, we can get (54) and we can get the following interpolation function: 𝑄_2,2𝑥,𝑦=𝐿₁+𝑥𝐿₂+𝑥(𝑥−1)𝐿₃=8𝑦−8+3𝑥𝑦−20𝑥+7𝑥²𝑦+4𝑥²+2𝑥²𝑦³−10𝑥²𝑦²−2𝑥𝑦³+10𝑥𝑦²×2𝑦−8⁻¹. (55)
It is easy to verify 𝑄_2,2𝑥_𝑖,𝑦_𝑗=𝑓𝑥_𝑖,𝑦_𝑗,𝑖=0,1,2;𝑗=0,1,2. (56)

5. Conclusion

The general interpolation formulae of bivariate interpolation function are more general than the general frames studied by many scholars [1, 14–20]; it could be used to deal with the interpolation problems where inverse differences are nonexistent or unattainable points occur via choosing 𝑓_𝑖, 𝑓_𝑖,𝑗 appropriately [17]. Another question is coming; there are so many schemes we can use; how to choose formula appropriately is our further work. In practical applications, the choice of 𝑓_𝑖’s, 𝑓_𝑖,𝑗’s and 𝑔’s, ℎ’s may be determined by the desired form of interpolation, for example, polynomial, rational function of given degree of the numerator and the denominator, or certain transformation of a rational function. If there is no restriction as to the form of 𝑄(𝑥,𝑦), the best choice may be the interpolation function that gives the smallest error term among the functions certain complexity. However, it is not easy to determine such a function without the process of trial and comparison.

We conclude this paper by pointing out that it is not difficult to generalize the general interpolation formulae in this paper to rational interpolation for higher dimensions, vector-valued case, or matrix-valued case [16, 17, 22, 23].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work is supported by the Grants of the National Natural Science Foundation of China, nos. 61005010 and 61272024, and the Anhui Provincial Natural Science Foundation, nos. 1308085MF84 and 1308085QF115.