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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|数值分析代写numerical analysis代考|Inverse Newton’s Divided Difference Interpolation

In this section, we introduce interpolations that are a special case of fractional and recursive interpolation. Among these interpolations are inverse differences, which are defined recursively for interpolation points $\left(x_{\mathrm{i}}, f_{i}\right), i=0,1, \ldots$ as follows:
\begin{aligned} &\varphi\left(x_{i}, x_{j}\right)=\frac{x_{i}-x_{i}}{f_{i}-f_{j}} \ &\varphi\left(x_{i}, x_{j}, x_{k}\right)=\frac{x_{j}-x_{k}}{\varphi\left(x_{i}, x_{j}\right)-\varphi\left(x_{i}, x_{k}\right)} \ &\varphi\left(x_{i}, \ldots, x_{l}, x_{m}, x_{n}\right)=\frac{x_{m}-x_{n}}{\varphi\left(x_{i}, \ldots, x_{l}, x_{m}\right)-\varphi\left(x_{i}, \ldots, x_{l}, x_{n}\right)} \end{aligned}
The above process is an Aitken recursive interpolation process. Now using the equations of (4.7), we introduce a fractional expression that interpolates points $\left(x_{i}, f_{i}\right), i=0,1, \ldots$ as follows:

$\phi^{n . n}(x)=f_{0}+\underline{x-x_{0}} \sqrt{\varphi\left(x_{0}, x_{1}\right)}+\underline{x-x_{1}} \sqrt{\varphi\left(x_{0}, x_{1}, x_{2}\right)}$
$+x-x_{2} \sqrt{\varphi\left(x_{0}, x_{1}, x_{2}, x_{3}\right)}+\cdots+x-x_{2 n-1} \sqrt{\varphi\left(x_{0}, \ldots, x_{2 n}\right)}$
This fractional expression is called inverse difference interpolator. This method moves in the data table according to a right triangle and has no symmetry. Therefore, we introduce another method called reciprocal differences, which is defined recursively and is symmetrical.

## 数学代写|数值分析代写numerical analysis代考|Trigonometric Interpolation

When the function $f$ is a periodic function, we approximate it with periodic functions that have a definite periodicity. Therefore, in this section, we examine the trigonometric interpolating functions. This type of interpolation is a special mode of linear and non-recursive interpolation. The structure of a trigonometric interpolator is a combination of functions $\sin l x$ and $\cos l x$ where $l \in \mathbb{Z}$.

Suppose that for $k=0, \ldots, N-1,\left(x_{k}, f_{k}\right)$ s are interpolation points. We consider two modes for the trigonometric interpolating function:
$$\psi(x)=\left{\begin{array}{cc} \frac{A_{0}}{2}+\sum_{l=1}^{M}\left(A_{l} \cos l x+B_{l} \sin l x\right), & N=2 M+1 \ \frac{A_{0}}{2}+\sum_{I=1}^{M-1}\left(A_{l} \cos l x+B_{l} \sin l x\right)+\frac{A_{M}}{2} \cos M x, & N=2 M \end{array}\right.$$
In both cases, $\psi(x)$ is a periodic function in terms of $x$ with periodicity of $2 \pi$. Suppose that the interpolation points are as $0=x_{0}<x_{1}<\cdots<x_{N-1}=2 \pi$ where
$$x_{k}=\frac{2 k \pi}{N}, \quad k=0, \ldots, N-1$$

# 数值分析代考

## 数学代写|数值分析代写numerical analysis代考|Inverse Newton’s Divided Difference Interpolation

$$\varphi\left(x_{i}, x_{j}\right)=\frac{x_{i}-x_{i}}{f_{i}-f_{j}} \quad \varphi\left(x_{i}, x_{j}, x_{k}\right)=\frac{x_{j}-x_{k}}{\varphi\left(x_{i}, x_{j}\right)-\varphi\left(x_{i}, x_{k}\right)} \varphi\left(x_{i}, \ldots, x_{l}, x_{m}, x_{n}\right)=\frac{x_{m}-x_{n}}{\varphi\left(x_{i}, \ldots, x_{l}, x_{m}\right)-\varphi\left(x_{i}, \ldots, x_{l}, x_{n}\right)}$$

\begin{aligned} &\phi^{n . n}(x)=f_{0}+x-x_{0} \sqrt{\varphi\left(x_{0}, x_{1}\right)}+x-x_{1} \sqrt{\varphi\left(x_{0}, x_{1}, x_{2}\right)} \ &+x-x_{2} \sqrt{\varphi\left(x_{0}, x_{1}, x_{2}, x_{3}\right)}+\cdots+x-x_{2 n-1} \sqrt{\varphi\left(x_{0}, \ldots, x_{2 n}\right)} \end{aligned}

## 数学代写|数值分析代写numerical analysis代考|Trigonometric Interpolation

$\$ \$$\backslash \mathrm{psi}(\mathrm{x})=\backslash left {$$
\frac{A_{0}}{2}+\sum_{l=1}^{M}\left(A_{l} \cos l x+B_{l} \sin l x\right), \quad N=2 M+1 \frac{A_{0}}{2}+\sum_{I=1}^{M-1}\left(A_{l} \cos l x+B_{l} \sin l x\right)+\frac{A M}{2} \cos M x, \quad N=2 M
$$正确的。 Inbothcases, \ \psi(x) \ i saperiodicfunctionintermsof \ \$$ withperiodicityof $\$ 2 \pi \. Supposethattheinterpolationpointsareas $\$ 0=x_{0}<x_{1}<\cdots<x_{N}x_{{}{k}=\backslash$frac${2 k \backslash$pi}$\left.} N\right}, \backslash$quad$k=0, \backslash$dots,$N-1\$

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## MATLAB代写

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