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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

物理代写|计算物理代写Computational Physics代考|Linear Interpolation

The principles behind interpolation and extrapolation are something every scientist should understand. Most measurements of a system, whether that is a physical experiment or theoretical calculation, will consist of pairs of discrete values; an independent variable $\mathrm{x}$, which will vary, and a dependent variable $\mathrm{y}$, which is measure. To extract information from these pairs of values one would, ideally, find an analytical function that would give $y$ for any arbitrary $x$. Often an analytical solution does not exist or is too tedious or complicated to solve. In this case, how to find a value for $y$ that sits between measured values in $x$ ? We can either try to fit the data to some function (typically a polynomial) or interpolate the data. The data should be extrapolated to find a y beyond measured range in $\mathrm{x}$. The difference between the two methods is that interpolation is constrained so that the function used to approximate the data must pass through the measured data points, whereas data fitting only requires that some error function is minimized.

As the data points can be approximated by any number of functions, we must have some guidelines that outline a reasonable approximation. As a rule, these guidelines usually rely on the consistency of the gradients or derivatives of the approximation and as a result may not be suitable for functions that have rapid variations, such as those with oscillatory behavior. Sometimes, an important detail about the behavior of a function may be missed should the measurements be too sparsely spread. As a crude example of this, think about measuring the displacement of a mass on a spring as a function of time. If the sample frequency (how often you take a measurement) matches the period of oscillation then the interpolated result would show that the mass does not move at all, which is clearly an error.

物理代写|计算物理代写Computational Physics代考|Polynomial Interpolation

Equation (3.3) is called a first-order polynomial. By adding higher powers of $x$, one can modify this to higher-order polynomials. For instance, if the highest power of $\mathrm{x}$ were two then it would be a second-order polynomial (also called a quadratic) and so on. Higher-order polynomials will be better at approximating rapidly changing functions but there is a practical limit to this, which will be discussed in the subsequent sections.

First, we can extend Equation (3.3) so that it forms an n ordered polynomial
$$g(x)=a_0+a_1 x+a_2 x^2+\cdots+a_n x^n$$
Using our interpolation constraint that the approximation must pass through the measured values gives
$$f\left(x_j\right)=f_j=g\left(x_j\right)=a_0+a_1 x_j+a_2 x_j^2+\cdots+a_n x_j^n$$
This is a system of $\mathrm{n}+1$ linear equations (you may know them as simultaneous equations) that we would use to solve for the coefficients. Notice that to perform an $\mathrm{n}$ ordered interpolation you need $\mathrm{n}+1$ data points. For instance, the first-order (linear) interpolation requires two points; a second-order interpolation requires three points, and so forth. How a linear system of equations can be solved explicitly using a LAPACK routine is discussed later in this chapter. For the moment, one could formulate the coefficients using an alternate method.

计算物理代考

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物理代写|计算物理代写Computational Physics代考|多项式插值

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$$g(x)=a_0+a_1 x+a_2 x^2+\cdots+a_n x^n$$

$$f\left(x_j\right)=f_j=g\left(x_j\right)=a_0+a_1 x_j+a_2 x_j^2+\cdots+a_n x_j^n$$

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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