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

In this section, we introduce an interpolation that operates in a multicriteria manner. In this interpolator, we assume that at the interpolation points, the value of the function $f$ is equal to the value of the interpolation function, as well as the value of the finite derivatives of $f$ is equal to the value of the finite derivatives of the interpolation function, and this is the reason for smoothness and accuracy of the interpolation function.
Suppose that $x_{i}$ s are real numbers for $i=0, \ldots, n$, so that:
$$x_{0} \leq x_{i} \leq \cdots \leq x_{n}$$
Therefore, for $i=0, \ldots, m$, there are points like $\xi_{i}$ with a new arrangement so that:
$$\xi_{0}<\xi_{1}<\cdots<\xi_{m}$$
So, we consider the interpolation points as follows:
$$\left(\xi_{i}, y_{i}^{(k)}\right), \quad i=0, \ldots, m, \quad k=0, \ldots, n_{i}-1, \quad n_{i} \in \mathbb{N}$$
The polynomial of at most degree $n$ of
$$p(x)=\sum_{i=0}^{m} \sum_{k=0}^{m_{i}-1} y_{i}^{(k)} L_{i k}(x)$$
is called Hermit interpolation polynomial if it satisfies the following interpolation conditions:
$$p^{(k)}\left(\xi_{i}\right)=y_{i}^{(k)}, \quad i=0, \ldots, m, \quad k=0, \ldots, n_{i}-1$$

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

In this section, we will investigate on how to interpolate functions if they have poles. Obviously, it is not possible to interpolate these functions with the methods mentioned in the previous sections, so if it is a fractional function, its best interpolators are asymptotical interpolating functions.

Now we want to introduce the general form of an asymptotical interpolator. Suppose that $\left(x_{i}, f_{i}\right) \mathrm{s}$ are interpolation points for $i=0,1, \ldots, \mu+v$. In this case, we define the fractional function $\phi^{\mu, v}$ as follows:

$$\varphi^{\mu, v}(x)=\frac{p^{\mu}(x)}{q^{v}(x)}=\frac{a_{0}+a_{1} x+\cdots+a_{\mu} x^{\mu}}{b_{0}+b_{1} x+\cdots+b_{v} x^{\nabla}}$$
According to the above relation, it is clear that the unknown parameters of the fractional function of (4.4) are $a_{0}, \ldots, a_{\mu}, b_{0}, \ldots, b_{v}$, so that their number is $\mu+v+2$. Therefore, it is sufficient to determine the unknown parameters in such a way that (4.4) become the interpolating function. In order for the fractional function (4.4) to interpolate the above-mentioned points, the following interpolation problem must be established:
$$\phi^{\mu, v}\left(x_{i}\right)=f_{i,} \quad i=0, \ldots, \mu+v$$
If $\phi^{\mu, v}\left(x_{i}\right) \neq 0$, we can write:
$$\phi^{\mu, v}\left(x_{i}\right)=\frac{p^{\mu}\left(x_{i}\right)}{q^{v}\left(x_{i}\right)}=f_{i}, \quad i=0, \ldots \mu+v$$
In this case,
$$p^{\mu}\left(x_{i}\right)-f_{i} q^{v}\left(x_{i}\right)=0, \quad i=0, \ldots, \mu+v$$
where the relation (4.6) is a homogenized system of (4.5). It is clear that the system (4.6) has $\mu+v+2$ unknown and $\mu+v+1$ equations, so to solve (4.6), we need a known parameter. For this purpose, we assume one of the denominator parameters as a known parameter and we obtain the fractional function (4.4) by solving the system (4.6).

数值分析代考

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

$$x_{0} \leq x_{i} \leq \cdots \leq x_{n}$$

$$\xi_{0}<\xi_{1}<\cdots<\xi_{m}$$

$$\left(\xi_{i}, y_{i}^{(k)}\right), \quad i=0, \ldots, m, \quad k=0, \ldots, n_{i}-1, \quad n_{i} \in \mathbb{N}$$

$$p(x)=\sum_{i=0}^{m} \sum_{k=0}^{m_{i}-1} y_{i}^{(k)} L_{i k}(x)$$

$$p^{(k)}\left(\xi_{i}\right)=y_{i}^{(k)}, \quad i=0, \ldots, m, \quad k=0, \ldots, n_{i}-1$$

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

$$\varphi^{\mu, v}(x)=\frac{p^{\mu}(x)}{q^{v}(x)}=\frac{a_{0}+a_{1} x+\cdots+a_{\mu} x^{\mu}}{b_{0}+b_{1} x+\cdots+b_{v} x^{\nabla}}$$

$$\phi^{\mu, v}\left(x_{i}\right)=f_{i}, \quad i=0, \ldots, \mu+v$$

$$\phi^{\mu, v}\left(x_{i}\right)=\frac{p^{\mu}\left(x_{i}\right)}{q^{v}\left(x_{i}\right)}=f_{i}, \quad i=0, \ldots \mu+v$$

$$p^{\mu}\left(x_{i}\right)-f_{i} q^{v}\left(x_{i}\right)=0, \quad i=0, \ldots, \mu+v$$

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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