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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|Iterative Interpolation

These types of interpolations are also a special case of type (1). Suppose that there is a sequence of points $\left{\left(x_{i_{i},}, f_{i_{j}}\right)\right}_{j=0}^{n}$. It should be noted that the reason for the presence of two indices is that the order of the points is not important. A polynomial $P_{i 0, \ldots, i_{n}}(x)$ of at most degree $n$ is called a Neville interpolator for the above points, where
$$P_{i_{0}, \ldots, i_{n}}(x)=\frac{\left(x-x_{i_{0}}\right) P_{i_{1}, \ldots, i_{n}}(x)-\left(x-x_{i_{n}}\right) P_{i_{1}, \ldots, i_{y,-1}}(x)}{x_{i_{q}}-x_{i 0}}$$
It is clear that Neville’s method works with the two first and last indices of the data and is also invariant under permutation of indices. This method moves symmetrically and in the form of an isosceles triangle in the data table.

A polynomial $P_{i 0, \ldots, i k, i, i, i t}(x)$ of at most degree $n$ is called Aitken interpolator in which
$$P_{i_{6}, \ldots, i_{k}, i_{i}, i_{i}}(x)=\frac{\left(x-x_{i j}\right) P_{i_{0}, \ldots, i_{k}, i_{k}}(x)-\left(x-x_{i_{i}}\right) P_{i_{0}, \ldots, i_{i}, i_{i}}(x)}{x_{i_{4}}-x_{i_{i}}}$$
The Aitken method works with the last two indices of the data and moves in the data table as a right triangle.

By comparing these two methods in terms of error propagation, it can be claimed that the Aitken method has a more relative delete error because if $x_{i}$ are approximate; at the denominator of the Aitken method, we will have the difference of approximate numbers with the same sign, and this increases the error propagation.

Because the use of computer programming is better for recursive relationships, it will be easier to work with Neville and Aitken methods. Also, in these two methods, in addition to the interpolation polynomial, the value of the function at the desired point is also obtained using the data table, and unlike the Lagrange method, the degree of the interpolation polynomial can be determined with simpler calculations.

数学代写|数值分析代写numerical analysis代考|Interpolation by Newton’s Divided Differences

This type of interpolation is type (1) interpolation based on the Neville recursive method. The polynomial
$$p(x)=f_{0}+\left(x-x_{0}\right) f\left[x_{0}, x_{1}\right]+\cdots+\left(x-x_{0}\right) \ldots\left(x-x_{n-1}\right) f\left[x_{0}, \ldots, x_{11}\right]$$
is the Newton’s interpolation polynomial where
$$f\left[x_{0}, \ldots, x_{n}\right]=\frac{f\left[x_{1}, \ldots, x_{n}\right]-f\left[x_{0, \ldots,} x_{n-1}\right]}{x_{n}-x_{0}}$$
are $n$th order divided differences between points $x_{0}, \ldots, x_{n}$.

It should be noted that the Newton’s divided differences are invariant under permutations of the indices, and the previous calculations can still be used by adding a point. So, the coefficients of this interpolation polynomial are stable.

数值分析代考

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

. 需要注意的是，出现两个索引的原因是点的顺序并不重 要。多项式 $P_{i 0, \ldots, i_{n}}(x)$ 最多程度 $n$ 被称为上述点的内维尔揷值器，其中
$$P_{i_{0, \ldots, i n}}(x)=\frac{\left(x-x_{i_{0}}\right) P_{i_{1}, \ldots, i_{n}}(x)-\left(x-x_{i_{n}}\right) P_{i_{1}, \ldots, i_{\underline{H}{1}-1}}(x)}{x{i_{q}}-x_{i 0}}$$

$$P_{i_{6, \ldots, i}, i_{i} i_{i} i}(x)=\frac{\left(x-x_{i j}\right) P_{i_{0, \ldots, i k}, i_{k}}(x)-\left(x-x_{i j}\right) P_{i_{0, \ldots}, i_{i}, i_{i}}(x)}{x_{i_{4}}-x_{i_{i}}}$$
Aitken 方法使用数据的最后两个索引，并在数据表中以直角三角形移动。

数学代写|数值分析代写numerical analysis代考|Interpolation by Newton’s Divided Differences

$$p(x)=f_{0}+\left(x-x_{0}\right) f\left[x_{0}, x_{1}\right]+\cdots+\left(x-x_{0}\right) \ldots\left(x-x_{n-1}\right) f\left[x_{0}, \ldots, x_{11}\right]$$

$$f\left[x_{0}, \ldots, x_{n}\right]=\frac{f\left[x_{1}, \ldots, x_{n}\right]-f\left[x_{0, \ldots,} x_{n-1}\right]}{x_{n}-x_{0}}$$

有限元方法代写

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MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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