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

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## Abstract

**:**

## 1. Introduction

- How to construct a proper polynomial or rational interpolation with explicit mathematical expression and a simple calculation which make the function easy to use and convenient to study theoretically;
- For the given data, how to modify the curve or surface shape to enable the function to meet the actual requirements.

## 2. Univariate Thiele-Like Interpolation Continued Fractions with Parameters

_{0},y

_{0}),(x

_{1},y

_{1}),…(x

_{n},y

_{n})} on interval $\left[a,b\right]$, we can gain the following classical Thiele-like rational interpolation continued fractions [1]:

#### 2.1. Thiele-Like Rational Interpolation Continued Fractions with Parameter

#### 2.2. The Interpolation Problem with Unattainable Points

**Definition**

**1**

**.**Suppose the given point is $D=\{({x}_{i},{y}_{i})|i=0,1,\dots ,n\}$ and ${x}_{i}$ is diverse, $R(x)=\frac{N(x)}{D(x)}$ is the Thiele-type interpolation continued fractions in Formula (1) if $({x}_{k},{y}_{k})$ satisfies

**Theorem**

**1**

**.**Suppose the given point is $D=\{({x}_{i},{y}_{i})|i=0,1,\dots ,n\}$, and ${x}_{i}$ is diverse, the Thiele-type interpolation continued fraction is as shown in Formula (1), where ${b}_{k}\ne \infty $, $(k=0,1,\dots ,n-1)$, ${b}_{n}\ne 0$, and then the necessary and sufficient condition of $({x}_{k},{y}_{k})$ for an unattainable point is

**Theorem**

**2.**

**Proof.**

#### 2.3. The Thiele-Like Rational Interpolation Continued Fractions Problem with a Nonexistent Inverse Difference

## 3. Multivariate Thiele-Like Branched Rational Interpolation Continued Fractions with Parameters

**Theorem**

**3**

#### 3.1. Bivariate Thiele-Type Branched Rational Interpolation Continued Fractions with a Single Parameter

Algorithm 1 Algorithm of the bivariate Thiele-like branched rational interpolation continued fractions with a single parameter |

Step 1: Initialization.
$${f}_{i,j}^{\left(0,0\right)}=f({x}_{i},{y}_{j}),i=0,1,\dots ,m;\text{\hspace{0.17em}}j=0,1,\dots ,n.$$
Step 2: For $j=0\text{\hspace{0.17em}},\text{\hspace{0.17em}}1\text{\hspace{0.17em}},\dots ,n\text{\hspace{0.17em}};\text{\hspace{0.17em}}p=\text{\hspace{0.17em}}1\text{\hspace{0.17em}},2\text{\hspace{0.17em}},\dots ,m\text{\hspace{0.17em}};i=p\text{\hspace{0.17em}},p+\text{\hspace{0.17em}}1\text{\hspace{0.17em}},\dots ,m\text{\hspace{0.17em}}$,
$${f}_{i,j}^{\left(p,0\right)}=\frac{{x}_{i}-{x}_{p-1}}{{f}_{i,j}^{\left(p-1,0\right)}-{f}_{p-1,j}^{\left(p-1,0\right)}}.$$
Step 3: For $i=0\text{\hspace{0.17em}},\text{\hspace{0.17em}}1\text{\hspace{0.17em}},\dots ,k-1\text{\hspace{0.17em}},\text{\hspace{0.17em}}k+1,\dots ,m\text{\hspace{0.17em}};\text{\hspace{0.17em}}q=\text{\hspace{0.17em}}1\text{\hspace{0.17em}},\text{\hspace{0.17em}}2,\dots ,n\text{\hspace{0.17em}};j=q\text{\hspace{0.17em}},q+\text{\hspace{0.17em}}1\text{\hspace{0.17em}},\dots ,n\text{\hspace{0.17em}}$,
$${f}_{i,j}^{(i,q)}=\frac{{y}_{j}-{y}_{q-1}}{{f}_{i,j}^{(i,q-1)}-{f}_{i,q-1}^{(i,q-1)}}.$$
Step 4: By introducing parameter $\lambda $ into the formula ${f}_{k,j}^{\left(k,0\right)}(j=0,1,\dots ,n)$, then one can calculate them with Formulas (2)–(5), and mark the final results as
$${({a}_{k,0},\text{\hspace{0.17em}}{a}_{k,1}\text{\hspace{0.17em}},\dots \text{\hspace{0.17em}},\text{\hspace{0.17em}}{a}_{k,l},\text{\hspace{0.17em}}{a}_{k,l+1\text{\hspace{0.17em}}}^{0},\text{\hspace{0.17em}}{a}_{k,l+1},\text{\hspace{0.17em}}{a}_{k,l+2}\text{\hspace{0.17em}},\text{\hspace{0.17em}}\dots \text{\hspace{0.17em}},\text{\hspace{0.17em}}{a}_{k,n})}^{T}.$$
Step 5: Using the elements in Formulas (19) and (20), the Thiele-like interpolation continued fractions with a single parameter with respect to $y$ can be constructed:
$${A}_{i}(y)=\{\begin{array}{l}{f}_{i,0}^{(i,0)}+\frac{y-{y}_{0}}{{f}_{i,1}^{(i,1)}}\text{\hspace{0.17em}}\begin{array}{c}\\ +\end{array}\frac{y-{y}_{1}}{{f}_{i,2}^{(i,2)}}\text{\hspace{0.17em}}\begin{array}{c}\\ +\end{array}\begin{array}{c}\\ \cdots \end{array}\text{\hspace{0.17em}}\begin{array}{c}\\ +\end{array}\frac{y-{y}_{n-1}}{{f}_{i,n}^{(i,n)}},\text{\hspace{0.17em}}\begin{array}{cc}& \end{array}i=0,1,\dots ,k-1,k+1,\dots ,m,\\ {a}_{k,0}+\frac{y-{y}_{0}}{{a}_{k,1}}\begin{array}{c}\\ +\end{array}\begin{array}{c}\\ \cdots \end{array}\begin{array}{c}\\ +\end{array}\frac{y-{y}_{l-1}}{{a}_{k,l}}\begin{array}{c}\\ +\end{array}\frac{y-{y}_{l}}{{a}_{k,l+1}^{0}}\begin{array}{c}\\ +\end{array}\frac{y-{y}_{l}}{{a}_{k,l+1}}\begin{array}{c}\\ +\end{array}\frac{y-{y}_{l+1}}{{a}_{k,l+2}}\begin{array}{c}\\ +\end{array}\begin{array}{c}\\ \cdots \end{array}\begin{array}{c}\\ +\end{array}\frac{y-{y}_{n-1}}{{a}_{k,n}},\begin{array}{cc}& i=k,\end{array}\end{array}.$$
Step 6: Let
$${R}_{m,n}^{0}(x,y)={A}_{0}(y)+\frac{x-{x}_{0}}{{A}_{1}(y)}\begin{array}{c}\\ +\end{array}\frac{x-{x}_{1}}{{A}_{2}(y)}\begin{array}{c}\\ +\cdots +\end{array}\frac{x-{x}_{m-1}}{{A}_{m}(y)}.$$
Then, ${R}_{m,n}^{0}(x,y)$ is a bivariate Thiele-like branched rational interpolation continued fractions with a single parameter. |

**Theorem**

**4.**

**Proof.**

#### 3.2. Bivariate Thiele-Like Branched Rational Interpolation Continued Fractions with Multiple Parameters

#### 3.2.1. Bivariate Thiele-Like Branched Rational Interpolation Continued Fractions with Two Parameters Based on a Virtual Treble Point

Algorithm 2 Algorithm of the bivariate Thiele-like rational interpolation continued fractions with two parameters |

Step 1: Initialization: Step 2: If $j=0\text{\hspace{0.17em}},\text{\hspace{0.17em}}1\text{\hspace{0.17em}},\dots ,n\text{\hspace{0.17em}};\text{\hspace{0.17em}}p=\text{\hspace{0.17em}}1\text{\hspace{0.17em}},2\text{\hspace{0.17em}},\dots ,m\text{\hspace{0.17em}};i=p\text{\hspace{0.17em}},p+\text{\hspace{0.17em}}1\text{\hspace{0.17em}},\dots ,m\text{\hspace{0.17em}}$, |

Step 3: For $i=0\text{\hspace{0.17em}},\text{\hspace{0.17em}}1\text{\hspace{0.17em}},\dots ,k-1\text{\hspace{0.17em}},\text{\hspace{0.17em}}k+1,\dots ,m\text{\hspace{0.17em}};\text{\hspace{0.17em}}q=\text{\hspace{0.17em}}1\text{\hspace{0.17em}},\text{\hspace{0.17em}}2,\dots ,n\text{\hspace{0.17em}};j=q\text{\hspace{0.17em}},q+\text{\hspace{0.17em}}1\text{\hspace{0.17em}},\dots ,n\text{\hspace{0.17em}}$,
$${f}_{i,j}^{(i,q)}=\frac{{y}_{j}-{y}_{q-1}}{{f}_{i,j}^{(i,q-1)}-{f}_{i,q-1}^{(i,q-1)}}.$$
By introducing parameters $\alpha ,\beta $ into the formula ${f}_{k,j}^{\left(k,0\right)}(j=0,1,\dots ,n)$, then one can calculate the final result as
$${({a}_{k,0},\text{\hspace{0.17em}}{a}_{k,1},\text{\hspace{0.17em}}\dots ,\text{\hspace{0.17em}}{a}_{k,l},\text{\hspace{0.17em}}{a}_{k,l+1}^{0},\text{\hspace{0.17em}}{a}_{k,l+1}^{1}\text{\hspace{0.17em}},\text{\hspace{0.17em}}{a}_{k,l+1},\text{\hspace{0.17em}}{a}_{k,l+2},\text{\hspace{0.17em}}\dots ,\text{\hspace{0.17em}}{a}_{k,n})}^{T}.$$
Step 4: By using the elements in Formulas (26) and (27), the Thiele-like interpolation continued fractions with a single parameter with respect to $y$ can be constructed:
$${A}_{i}(y)=\{\begin{array}{l}{f}_{i,0}^{(i,0)}+\frac{y-{y}_{0}}{{f}_{i,1}^{(i,1)}}\text{\hspace{0.17em}}\begin{array}{c}\\ +\end{array}\frac{y-{y}_{1}}{{f}_{i,2}^{(i,2)}}\text{\hspace{0.17em}}\begin{array}{c}\\ +\end{array}\begin{array}{c}\\ \cdots \end{array}\text{\hspace{0.17em}}\begin{array}{c}\\ +\end{array}\frac{y-{y}_{n-1}}{{f}_{i,n}^{(i,n)}},\text{\hspace{0.17em}}\begin{array}{cc}& \end{array}i=0,1,\dots ,k-1,k+1,\dots ,m,\\ {a}_{k,0}+\frac{y-{y}_{0}}{{a}_{k,1}}\begin{array}{c}\\ +\end{array}\begin{array}{c}\\ \cdots \end{array}\begin{array}{c}\\ +\end{array}\frac{y-{y}_{l-1}}{{a}_{k,l}}\begin{array}{c}\\ +\end{array}\frac{y-{y}_{l}}{{a}_{k,l+1}^{0}}\begin{array}{c}\\ +\end{array}\frac{y-{y}_{l}}{{a}_{k,l+1}^{1}}\begin{array}{c}\\ +\end{array}\frac{y-{y}_{l}}{{a}_{k,l+1}^{}}\begin{array}{c}\\ +\end{array}\frac{y-{y}_{l+1}}{{a}_{k,l+2}}\begin{array}{c}\\ +\end{array}\begin{array}{c}\\ \cdots \end{array}\begin{array}{c}\\ +\end{array}\frac{y-{y}_{n-1}}{{a}_{k,n}},\begin{array}{cc}& i=k,\end{array}\end{array}$$
Step 5: Let
$${R}_{m,n}^{1}(x,y)={A}_{0}(y)+\frac{x-{x}_{0}}{{A}_{1}(y)}\begin{array}{c}\\ +\end{array}\frac{x-{x}_{1}}{{A}_{2}(y)}\begin{array}{c}\\ +\cdots +\end{array}\frac{x-{x}_{m-1}}{{A}_{m}(y)}.$$
Then, ${R}_{m,n}^{1}(x,y)$ is a bivariate Thiele-like branched rational interpolation continued fractions with two parameters based on a treble virtual point. |

**Theorem**

**5.**

#### 3.2.2. Bivariate Thiele-Like Branched Rational Interpolation Continued Fractions with Two Parameters Based on Two Virtual Double Points in the Same Column

Algorithm 3 Algorithm of the Thiele-like branched rational interpolation continued fractions with two parameters in the same column |

Step 1: Initialization: Step 2: If $j=0\text{\hspace{0.17em}},\text{\hspace{0.17em}}1\text{\hspace{0.17em}},\dots ,n\text{\hspace{0.17em}};\text{\hspace{0.17em}}p=\text{\hspace{0.17em}}1\text{\hspace{0.17em}},2\text{\hspace{0.17em}},\dots ,m\text{\hspace{0.17em}};i=p\text{\hspace{0.17em}},p+\text{\hspace{0.17em}}1\text{\hspace{0.17em}},\dots ,m\text{\hspace{0.17em}}$, Step 3: For $i=0\text{\hspace{0.17em}},\text{\hspace{0.17em}}1\text{\hspace{0.17em}},\dots ,k-1\text{\hspace{0.17em}},\text{\hspace{0.17em}}k+1,\dots ,m\text{\hspace{0.17em}};\text{\hspace{0.17em}}q=\text{\hspace{0.17em}}1\text{\hspace{0.17em}},\text{\hspace{0.17em}}2,\dots ,n\text{\hspace{0.17em}};j=q\text{\hspace{0.17em}},q+\text{\hspace{0.17em}}1\text{\hspace{0.17em}},\dots ,n\text{\hspace{0.17em}}$,
$${f}_{i,j}^{(i,q)}=\frac{{y}_{j}-{y}_{q-1}}{{f}_{i,j}^{(i,q-1)}-{f}_{i,q-1}^{(i,q-1)}}.$$
Step 4: By introducing parameters $\varphi ,\delta $ into ${f}_{k,j}^{\left(k,0\right)}(j=0,1,\dots ,n)$, then one can calculate them with inverse differences similar to Equation (2)–(5) , and mark the final results as
$${({a}_{k,0}\text{\hspace{0.17em}},\text{\hspace{0.17em}}{a}_{k,1}\text{\hspace{0.17em}},\text{\hspace{0.17em}}\dots ,\text{\hspace{0.17em}}{a}_{k,l}\text{\hspace{0.17em}},\text{\hspace{0.17em}}{a}_{k,l+1\text{\hspace{0.17em}}}^{0}\text{\hspace{0.17em}},\text{\hspace{0.17em}}{a}_{k,l+1}\text{\hspace{0.17em}},\text{\hspace{0.17em}}{a}_{k,l+2}\text{\hspace{0.17em}},\text{\hspace{0.17em}}\dots \text{\hspace{0.17em}},\text{\hspace{0.17em}}{a}_{k,s}\text{\hspace{0.17em}},\text{\hspace{0.17em}}{a}_{k,s+1\text{\hspace{0.17em}}}^{0}\text{\hspace{0.17em}},\text{\hspace{0.17em}}{a}_{k,s+1}\text{\hspace{0.17em}},\text{\hspace{0.17em}}{a}_{k,s+2}\text{\hspace{0.17em}},\text{\hspace{0.17em}}\dots ,\text{\hspace{0.17em}}{a}_{k,n})}^{T}.$$
Step 5: Using the elements in Formulas (52) and (53), the univariate Thiele-like interpolation continued fractions with two parameters with respect to $y$ can be constructed:
$${A}_{i}(y)=\{\begin{array}{l}{f}_{i,0}^{(i,0)}+\frac{y-{y}_{0}}{{f}_{i,1}^{(i,1)}}\text{\hspace{0.17em}}\begin{array}{c}\\ +\end{array}\frac{y-{y}_{1}}{{f}_{i,2}^{(i,2)}}\text{\hspace{0.17em}}\begin{array}{c}\\ +\end{array}\begin{array}{c}\\ \cdots \end{array}\text{\hspace{0.17em}}\begin{array}{c}\\ +\end{array}\frac{y-{y}_{n-1}}{{f}_{i,n}^{(i,n)}},\text{\hspace{0.17em}}\begin{array}{cc}& \end{array}i=0,1,\dots ,k-1,k+1,\dots ,m,\\ {a}_{k,0}+\frac{y-{y}_{0}}{{a}_{k,1}}\begin{array}{c}\\ +\end{array}\begin{array}{c}\\ \cdots \end{array}\begin{array}{c}\\ +\end{array}\frac{y-{y}_{l-1}}{{a}_{k,l}}\begin{array}{c}\\ +\end{array}\frac{y-{y}_{l}}{{a}_{k,l+1}^{0}}\begin{array}{c}\\ +\end{array}\frac{y-{y}_{l}}{{a}_{k,l+1}}\begin{array}{c}\\ +\end{array}\frac{y-{y}_{l+1}}{{a}_{k,l+2}}\begin{array}{c}\\ +\end{array}\begin{array}{c}\\ \cdots \end{array}\begin{array}{c}\\ +\end{array}\\ \frac{y-{y}_{s-1}}{{a}_{k,s}}\begin{array}{c}\\ +\end{array}\frac{y-{y}_{s}}{{a}_{k,s+1}^{0}}\begin{array}{c}\\ +\end{array}\frac{y-{y}_{s}}{{a}_{k,s+1}}\begin{array}{c}\\ +\end{array}\frac{y-{y}_{s+1}}{{a}_{k,s+2}}\begin{array}{c}\\ +\end{array}\begin{array}{c}\\ \cdots \end{array}\begin{array}{c}\\ +\end{array}\frac{y-{y}_{n-1}}{{a}_{k,n}},\begin{array}{cc}& i=k,\end{array}\end{array}.$$
Step 6: Let
$${R}_{m,n}^{2}(x,y)={A}_{0}(y)+\frac{x-{x}_{0}}{{A}_{1}(y)}\begin{array}{c}\\ +\end{array}\frac{x-{x}_{1}}{{A}_{2}(y)}\begin{array}{c}\\ +\cdots +\end{array}\frac{x-{x}_{m-1}}{{A}_{m}(y)}.$$
Then, ${R}_{m,n}^{2}(x,y)$ is a bivariate Thiele-type branched interpolation continued fraction with two parameters based on two virtual double nodes in the same column. |

**Theorem**

**6.**

#### 3.2.3. Bivariate Thiele-Like Branched Rational Interpolation Continued Fractions with Two Parameters Based on Two Virtual Double Points in Different Columns

Algorithm 4 Algorithm of the Thiele-like rational interpolation continued fractions with two parameters on the different columns |

Step 1: Initialization. Step 2: If $j=0\text{\hspace{0.17em}},\text{\hspace{0.17em}}1\text{\hspace{0.17em}},\dots ,n\text{\hspace{0.17em}};\text{\hspace{0.17em}}p=\text{\hspace{0.17em}}1\text{\hspace{0.17em}},2\text{\hspace{0.17em}},\dots ,m\text{\hspace{0.17em}};i=p\text{\hspace{0.17em}},p+\text{\hspace{0.17em}}1\text{\hspace{0.17em}},\dots ,m\text{\hspace{0.17em}}$, Step 3: For $i=0\text{\hspace{0.17em}},\text{\hspace{0.17em}}1\text{\hspace{0.17em}},\dots ,k-1\text{\hspace{0.17em}},\text{\hspace{0.17em}}k+1,\dots ,s-1\text{\hspace{0.17em}},\text{\hspace{0.17em}}s+1,\dots ,m\text{\hspace{0.17em}};\text{\hspace{0.17em}}q=\text{\hspace{0.17em}}1\text{\hspace{0.17em}},\text{\hspace{0.17em}}2,\dots ,n\text{\hspace{0.17em}};j=q\text{\hspace{0.17em}},q+\text{\hspace{0.17em}}1\text{\hspace{0.17em}},\dots ,n\text{\hspace{0.17em}}$,
$${f}_{i,j}^{(i,q)}=\frac{{y}_{j}-{y}_{q-1}}{{f}_{i,j}^{(i,q-1)}-{f}_{i,q-1}^{(i,q-1)}}.$$
Step 4: By introducing parameter $\phi $ into ${f}_{k,j}^{\left(k,0\right)}\text{\hspace{0.17em}}(j=0,1,\dots ,n)$, we can calculate them by using Formulas (2)–(5) and mark the final results as
$${({a}_{k,0},\text{\hspace{0.17em}}{a}_{k,1}\text{\hspace{0.17em}},\text{\hspace{0.17em}}\dots \text{\hspace{0.17em}},\text{\hspace{0.17em}}{a}_{k,l},\text{\hspace{0.17em}}{a}_{k,l+1\text{\hspace{0.17em}}}^{0}\text{\hspace{0.17em}}{a}_{k,l+1}\text{\hspace{0.17em}},\text{\hspace{0.17em}}{a}_{k,l+2},\text{\hspace{0.17em}}\dots \text{\hspace{0.17em}},\text{\hspace{0.17em}}{a}_{k,n})}^{T}.$$
Step 5: By introducing parameter $\varpi $ into ${f}_{s,j}^{\left(s,0\right)}\text{\hspace{0.17em}}(j=0,1,\dots ,n)$, we can calculate them by using Formulas (2)–(5), and mark the final results as
$${({a}_{s,0},{a}_{s,1},\dots ,{a}_{s,t},{a}_{s,t+1}^{0},{a}_{s,t+1},{a}_{s,t+2},\dots ,{a}_{s,n})}^{T}.$$
Step 6: By using the elements in Formulas (40)–(42), the Thiele-like interpolation continued fractions with a single parameter with respect to $y$ was constructed:
$${A}_{i}(y)=\{\begin{array}{l}{f}_{i,0}^{(i,0)}+\frac{y-{y}_{0}}{{f}_{i,1}^{(i,1)}}\text{\hspace{0.17em}}\begin{array}{c}\\ +\end{array}\frac{y-{y}_{1}}{{f}_{i,2}^{(i,2)}}\text{\hspace{0.17em}}\begin{array}{c}\\ +\end{array}\begin{array}{c}\\ \cdots \end{array}\text{\hspace{0.17em}}\begin{array}{c}\\ +\end{array}\frac{y-{y}_{n-1}}{{f}_{i,n}^{(i,n)}},\text{\hspace{0.17em}}\begin{array}{cc}& \end{array}i=0,1,\dots ,k-1,k+1,\dots ,s-1,s+1,\dots ,m,\\ {a}_{k,0}+\frac{y-{y}_{0}}{{a}_{k,1}}\begin{array}{c}\\ +\end{array}\begin{array}{c}\\ \cdots \end{array}\begin{array}{c}\\ +\end{array}\frac{y-{y}_{l-1}}{{a}_{k,l}}\begin{array}{c}\\ +\end{array}\frac{y-{y}_{l}}{{a}_{k,l+1}^{0}}\begin{array}{c}\\ +\end{array}\frac{y-{y}_{l}}{{a}_{k,l+1}}\begin{array}{c}\\ +\end{array}\frac{y-{y}_{l+1}}{{a}_{k,l+2}}\begin{array}{c}\\ +\end{array}\begin{array}{c}\\ \cdots \end{array}\begin{array}{c}\\ +\end{array}\frac{y-{y}_{n-1}}{{a}_{k,n}},\begin{array}{cc}& i=k,\end{array}\\ {a}_{s,0}+\frac{y-{y}_{0}}{{a}_{s,1}}\begin{array}{c}\\ +\end{array}\begin{array}{c}\\ \cdots \end{array}\begin{array}{c}\\ +\end{array}\frac{y-{y}_{t-1}}{{a}_{s,t}}\begin{array}{c}\\ +\end{array}\frac{y-{y}_{t}}{{a}_{s,t+1}^{0}}\begin{array}{c}\\ +\end{array}\frac{y-{y}_{t}}{{a}_{s,t+1}}\begin{array}{c}\\ +\end{array}\frac{y-{y}_{t+1}}{{a}_{s,t+2}}\begin{array}{c}\\ +\end{array}\begin{array}{c}\\ \cdots \end{array}\begin{array}{c}\\ +\end{array}\frac{y-{y}_{n-1}}{{a}_{s,n}},\begin{array}{cc}& \begin{array}{l}i=s,\\ \end{array}\end{array}\end{array}.$$
Step 7: Let
$${R}_{m,n}^{3}(x,y)={A}_{0}(y)+\frac{x-{x}_{0}}{{A}_{1}(y)}\begin{array}{c}\\ +\end{array}\frac{x-{x}_{1}}{{A}_{2}(y)}\begin{array}{c}\\ +\cdots +\end{array}\frac{x-{x}_{m-1}}{{A}_{m}(y)}.$$
Then, ${R}_{m,n}^{3}(x,y)$ is a bivariate Thiele-like branched rational interpolation continued fractions with two parameters based on two virtual double points in the different columns. |

**Theorem**

**7.**

#### 3.3. Dual Bivariate Thiele-Like Branched Rational Interpolation Continued Fractions with a Single Parameter

Algorithm 5 Algorithm of the dual bivariate Thiele-like branched rational interpolation continued fractions with a single parameter |

Step 1: Initialization: Step 2: If $i=0,1,\dots ,m;q=1,2,\dots ,n;j=q,q+1,\dots ,n$,
$${f}_{i,j}^{(0,q)}=\frac{{y}_{j}-{y}_{q-1}}{{f}_{i,j}^{(0,q-1)}-{f}_{i,q-1}^{(0,q-1)}}.$$
Step 3: For $j=0,1,\dots ,l-1,l+1,\dots ,n;p=1,2,\dots ,m;i=p,p+1,\dots ,m$,
$${f}_{i,j}^{\left(p,j\right)}=\frac{{x}_{i}-{x}_{p-1}}{{f}_{i,j}^{\left(p-1,j\right)}-{f}_{p-1,j}^{\left(p-1,j\right)}}.$$
Step 4: By introducing a parameter $\theta $ into ${f}_{i,l}^{\left(0,l\right)}(i=0,1,\dots ,m)$, we can calculate them using Formulas (2)–(5) and mark the final results as
$${a}_{0,l},{a}_{1,l},\dots ,{a}_{k,l},{a}_{k+1,l}^{0},{a}_{k+1,l},{a}_{k+2,l},\dots ,{a}_{m,l}.$$
Step 5: By using the elements in Formulas (48) and (49), the Thiele-type interpolation continued fractions with a single parameter regarded to $x$ can be constructed:
$${A}_{j}(x)=\{\begin{array}{l}{f}_{0,j}^{(0,j)}+\frac{x-{x}_{0}}{{f}_{1,j}^{(1,j)}}\begin{array}{c}\\ +\end{array}\frac{x-{x}_{1}}{{f}_{2,j}^{(2,j)}}\begin{array}{c}\\ +\end{array}\begin{array}{c}\\ \cdots \end{array}\begin{array}{c}\\ +\end{array}\frac{x-{x}_{m-1}}{{f}_{m,j}^{(m,j)}},\begin{array}{cc}& \end{array}j=0,1,\dots ,l-1,l+1,\dots ,n,\\ {a}_{0,l}+\frac{x-{x}_{0}}{{a}_{1,l}}\begin{array}{c}\\ +\end{array}\begin{array}{c}\\ \cdots \end{array}\begin{array}{c}\\ +\end{array}\frac{x-{x}_{k-1}}{{a}_{k,l}}\begin{array}{c}\\ +\end{array}\frac{x-{x}_{k}}{{a}_{k+1,l}^{0}}\begin{array}{c}\\ +\end{array}\frac{x-{x}_{k}}{{a}_{k+1,l}}\begin{array}{c}\\ +\end{array}\frac{x-{x}_{k+1}}{{a}_{k+2,l}}\begin{array}{c}\\ +\end{array}\begin{array}{c}\\ \cdots \end{array}\begin{array}{c}\\ +\end{array}\frac{x-{x}_{m-1}}{{a}_{m,l}},\begin{array}{cc}& j=l,\end{array}\end{array}.$$
Step 6: Let
$${R}_{m,n}^{4}(x,y)={A}_{0}(x)+\frac{y-{y}_{0}}{{A}_{1}(x)}\begin{array}{c}\\ +\end{array}\frac{y-{y}_{1}}{{A}_{2}(x)}\begin{array}{c}\\ +\cdots +\end{array}\frac{y-{y}_{n-1}}{{A}_{n}(x)}.$$
Then, ${R}_{m,n}^{4}(x,y)$ is a dual bivariate Thiele-like branched rational interpolation continued fractions with a single parameter. |

**Theorem**

**8.**

**Proof.**

#### 3.4. Bivariate Thiele-Like Branched Rational Interpolation Continued Fractions with Unattainable Points

**Definition**

**2.**

**Theorem**

**9**

**.**Suppose the bivariate Thiele-like branched rational interpolation continued fractions which is diverse for the given point set $D=\{({x}_{i},{y}_{j},{f}_{i,j})|i=0,1,\dots ,m,j=0,1,\dots ,n\}$, shown in Formula (15), satisfies

**Theorem**

**10.**

#### 3.5. Bivariate Thiele-Like Branched Rational Interpolation Continued Fractions for Nonexistent Partial Inverse Difference

## 4. Numerical Examples

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

## 5. Conclusions and Future Work

- How to select appropriate parameters and suitably alter the shape of the curves or surfaces according to actual requirements.
- The geometric properties of curves/surfaces based on the Thiele-like rational interpolation continued fractions function with parameters.
- How to design geometric modeling using Thiele-like rational interpolation continued fractions functions with parameters.
- The proposed Thiele-like rational interpolation continued fractions with parameters algorithm can be implemented in other pixel level image processing, such as image inpainting, removal of salt and pepper noise, image rotation, image super-resolution reconstruction, image metamorphosis and image upscaling.
- How to generalize the proposed algorithms to lacunary rational interpolants, rational interpolants over triangular grids.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Tan, J. Theory of Continued Fractions and Its Applications; Science Publishers: Beijing, China, 2007. [Google Scholar]
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**Figure 1.**(

**a**) Graph of ${R}_{2}(x)$ with $\lambda =-3$, (

**b**) graph of ${R}_{2}(x)$ with $\lambda =80$.

**Figure 3.**The zoomed images by factor 2 based on four different interpolant methods. (

**a**) The zoomed images by factor 2 based on the nearest-neighbor interpolation method, (

**b**) the zoomed images by factor 2 based on the bilinear interpolation, (

**c**) the zoomed images by factor 2 based on the bicubic interpolation, (

**d**) the zoomed images by factor 2 based on the proposed method.

**Figure 4.**Comparison of eye effects of the nearest neighbor interpolation and the proposed rational interpolation. (

**a**) The zoomed images by factor 4 based on the nearest-neighbor interpolation method, (

**b**) the zoomed images by factor 4 based on the proposed method.

Nodes | 0 Order Inverse Differences | 1 Order Inverse Differences | $\mathit{k}$ | $\mathit{k}+1\text{}\mathbf{Order}\text{}\mathbf{Inverse}\text{}\mathbf{Differences}$ | $\mathit{k}+2\text{}\mathbf{Order}\text{}\mathbf{Inverse}\text{}\mathbf{Differences}$ | $\mathit{n}+1\text{}\mathbf{Order}\text{}\mathbf{Inverse}\text{}\mathbf{Differences}$ | ||
---|---|---|---|---|---|---|---|---|

${x}_{0}$ | ${y}_{0}^{0}$ | |||||||

${x}_{1}$ | ${y}_{1}^{0}$ | ${y}_{1}^{1}$ | ||||||

$\begin{array}{c}\vdots \end{array}$ | $\begin{array}{c}\vdots \end{array}$ | $\begin{array}{c}\vdots \end{array}$ | $\ddots $ | |||||

${x}_{k}$ | ${y}_{k}^{0}$ | ${y}_{k}^{1}$ | $\cdots $ | ${y}_{k}^{k}$ | ||||

${x}_{k}$ | ${y}_{k}^{0}$ | ${y}_{k}^{1}$ | $\cdots $ | ${y}_{k}^{k}$ | $\frac{1}{\eta}$ | |||

${x}_{k+1}$ | ${y}_{k+1}^{0}$ | ${y}_{k+1}^{1}$ | $\cdots $ | ${y}_{k+1}^{k}$ | ${y}_{k+1}^{k+1}$ | ${z}_{k+1}^{k+1}$ | ||

$\begin{array}{c}\vdots \end{array}$ | $\begin{array}{c}\vdots \end{array}$ | $\begin{array}{c}\vdots \end{array}$ | $\cdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\ddots $ | |

${x}_{n}$ | ${y}_{n}^{0}$ | ${y}_{n}^{1}$ | $\cdots $ | $\begin{array}{c}{y}_{n}^{k}\end{array}$ | ${y}_{n}^{k+1}$ | ${z}_{n}^{k+1}$ | $\cdots $ | ${z}_{n}^{n}$ |

${\mathit{x}}_{\mathit{i}}$ | $\mathit{f}({\mathit{x}}_{\mathit{i}})=\frac{1}{1+25{\mathit{x}}_{\mathit{i}}^{2}}$ | $\mathbf{Newton}\text{}\mathbf{Interpolation}\text{}{\mathit{p}}_{6}(\mathit{x})$ | Thiele Continued Fractions Interpolation | Thiele Continued Fractions Interpolation with c = 1 | Thiele Continued Fractions Interpolation with c = −10 |
---|---|---|---|---|---|

−1.00 | 0.03846 | 0.03846 | 0.03846000000 | 0.03846000000 | 0.03846000000 |

−0.96 | 0.04160 | 0.03298 | 0.04159488287 | 0.04159595868 | 0.04159595865 |

−0.90 | 0.04706 | 0.03770 | 0.04705547745 | 0.04705768836 | 0.04705768826 |

−0.86 | 0.05131 | 0.04534 | 0.05130475006 | 0.05130739205 | 0.05130739185 |

−0.80 | 0.05882 | 0.05882 | 0.05882000000 | 0.05882000000 | 0.05882000000 |

−0.76 | 0.06477 | 0.06763 | 0.06476367286 | 0.06476628712 | 0.06476628741 |

−0.70 | 0.07547 | 0.07971 | 0.07546947663 | 0.07547137553 | 0.07547137561 |

−0.66 | 0.08410 | 0.08737 | 0.08410290267 | 0.08410410847 | 0.08410410852 |

−0.60 | 0.10000 | 0.10000 | 0.09999999823 | 0.10000000000 | 0.10000000000 |

−0.56 | 0.11312 | 0.11071 | 0.11312302677 | 0.11312226113 | 0.11312226113 |

−0.50 | 0.13793 | 0.13338 | 0.13793269272 | 0.13793118870 | 0.13793118867 |

−0.46 | 0.15898 | 0.15487 | 0.15898409183 | 0.15898264914 | 0.15898264914 |

−0.40 | 0.2000 | 0.20000 | 0.19999998642 | 0.20000000000 | 0.20000000000 |

−0.36 | 0.23585 | 0.24042 | 0.23584678820 | 0.23584889034 | 0.23584889036 |

−0.30 | 0.30769 | 0.31883 | 0.30768550203 | 0.30769184706 | 0.30769184712 |

−0.26 | 0.37175 | 0.38372 | 0.37173857114 | 0.37174665711 | 0.37174665718 |

−0.20 | 0.50000 | 0.50000 | 0.49999990866 | 0.50000000000 | 0.50000000000 |

−0.16 | 0.60976 | 0.73720 | 0.60977966699 | 0.60975745619 | 0.60975745603 |

−0.10 | 0.80000 | 0.73720 | 0.80009647464 | 0.80000519086 | 0.80000519027 |

−0.06 | 0.91743 | 0.84193 | 0.91756721089 | 0.91743821176 | 0.91743821096 |

0.00 | 1.0000 | 1.00000 | 0.99999979851 | 1.00000000000 | 1.00000000000 |

$i$ | 0 | 1 | 2 |

${x}_{i}$ | 2 | 1 | 0 |

${f}_{i}$ | 1 | 0 | 0 |

Nodes | 0 Order Inverse Differences | 1 Order Inverse Differences | 2 Order Inverse Differences |
---|---|---|---|

2 | 1 | ||

1 | 0 | 1 | |

0 | 0 | 2 | −1 |

Nodes | 0 Order Inverse Differences | 1 Order Inverse Differences | 2 Order Inverse Differences | 3 Order Inverse Differences |
---|---|---|---|---|

2 | 1 | |||

2 | 1 | $\lambda $ | ||

1 | 0 | 1 | $\frac{1}{\lambda -1}$ | |

0 | 0 | 2 | $\frac{2}{\lambda -2}$ | $3-\frac{2}{\lambda}-\lambda $ |

$({\mathit{x}}_{\mathit{i}},{\mathit{y}}_{\mathit{j}})$ | ${\mathit{x}}_{0}=0$ | ${\mathit{x}}_{1}=0.5$ | ${\mathit{x}}_{2}=1$ |
---|---|---|---|

${y}_{0}=0$ | 2 | 2.3 | 2.5 |

${y}_{1}=0.5$ | 1.8 | 2 | 2.1 |

${y}_{1}=1$ | 1.5 | 1.55 | 1.5 |

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## Share and Cite

**MDPI and ACS Style**

Zou, L.; Song, L.; Wang, X.; Chen, Y.; Zhang, C.; Tang, C.
Bivariate Thiele-Like Rational Interpolation Continued Fractions with Parameters Based on Virtual Points. *Mathematics* **2020**, *8*, 71.
https://doi.org/10.3390/math8010071

**AMA Style**

Zou L, Song L, Wang X, Chen Y, Zhang C, Tang C.
Bivariate Thiele-Like Rational Interpolation Continued Fractions with Parameters Based on Virtual Points. *Mathematics*. 2020; 8(1):71.
https://doi.org/10.3390/math8010071

**Chicago/Turabian Style**

Zou, Le, Liangtu Song, Xiaofeng Wang, Yanping Chen, Chen Zhang, and Chao Tang.
2020. "Bivariate Thiele-Like Rational Interpolation Continued Fractions with Parameters Based on Virtual Points" *Mathematics* 8, no. 1: 71.
https://doi.org/10.3390/math8010071