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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

## 电气工程代写|模拟电路代写analog circuit代考|General Stability Theory

What one means by a system being stable can be different from application to application. Here, as in most circuit theory books, we will adopt the bounded definition of stability, meaning the solution is considered stable if the signals will remain within a certain limited bound, $B$, at all times. We will follow the common thread in the literature and use simple linear systems to investigate the numerical stability.
To analyze stability, let us then consider a homogeneous linear differential system with constant coefficients
$$\frac{d x}{d t}=\boldsymbol{A x}$$
where we assume $|x(t)|{\max } \rightarrow \leq B$ as $t \rightarrow \infty$. This equation is referred to as a test equation. A set of linear equations like the test equation has the exact general solution $$x(t)=\sum{i=1}^n c_i e^{\lambda t} v_i$$
where $\lambda_i, v_i$ are the eigenvalues/vectors to the system obtained by solving
$$\boldsymbol{A} v_i=\lambda_i v_i$$
$\lambda_i$ are in general complex numbers. If we assume the system is stable, it implies that $\operatorname{Re}\left(\lambda_i\right) \leq 0 \forall i$. All eigenvalues are in the left half plane or on the imaginary axes. Let us illustrate in Fig. 2.2. It shows how the solution with positive real eigenvalues explodes beyond bounds due to an exponent that increases linearly with time on the right-hand plane, while eigenvalues with negative real values decreases with time on the left-hand side (Fig. 2.2).

## 电气工程代写|模拟电路代写analog circuit代考|Newton-Raphson

Let us look at the following one-dimensional equation
$$f(x)=0$$
where we need to solve for $x$. Let us do a Taylor expansion around a point $x_0$ which does not solve the equation but is close. We find
$$f\left(x=x_0+\Delta x\right)=f\left(x_0\right)+\frac{d f}{d x}\left(x_0\right) \Delta x=0$$
which can be written as
$$\Delta x=-\frac{f\left(x_0\right)}{d f / d x\left(x_0\right)}$$
Clearly, if the higher-order derivatives are zero, we have a linear equation, and we have arrived at the solution with this calculation of $\Delta x$. In practice, the higherorder terms still contribute, and we need to iterate a few times to arrive at the correct solution. The method is generally easy to implement, and it is a standard work horse in almost all numerical implementations of nonlinear solvers. We will also use the multidimensional version later on in Chap. 5. Newton’s method is guaranteed to converge if the starting point is close enough to the solution and the solution function is smooth (no discontinuities of the function itself or its derivatives) since then the higher-order derivatives are small. Clearly for functions that are not continuous or the derivative is not continuous, the method will easily get confused. This was historically one of the most notorious reasons transistor models had convergence issues. The continuity of the derivatives has been guaranteed in the latest transistor model implementations (Fig. 2.7).

# 模拟电路代考

## 电气工程代写|模拟电路代写analog circuit代考|General Stability Theory

$$\frac{d x}{d t}=\boldsymbol{A} \boldsymbol{x}$$

$$x(t)=\sum i=1^n c_i e^{\lambda t} v_i$$

$$\boldsymbol{A} v_i=\lambda_i v_i$$
$\lambda_i$ 一般都是复数。如果我们假设系统是稳定的，这意味着 $\operatorname{Re}\left(\lambda_i\right) \leq 0 \forall i$. 所有特征值都在左半平面或虚轴上。让我们在图 2.2 中进行说明。它显示了 具有正实特征值的解如何由于指数在右侧平面上随时间线性增加而爆炸超出界限，而在左侧平面上具有负实值的特征值随时间减小 (图 2.2)。

## 电气工程代写|模拟电路代写analog circuit代考|Newton-Raphson

$$f(x)=0$$

$$f\left(x=x_0+\Delta x\right)=f\left(x_0\right)+\frac{d f}{d x}\left(x_0\right) \Delta x=0$$

$$\Delta x=-\frac{f\left(x_0\right)}{d f / d x\left(x_0\right)}$$

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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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