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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|Second-Order Gear Method

C. William Gear [5] published a book in 1971 that has since become one of the classics. He constructs a set of difference equations with varying truncation error that has some really nice properties. He shows how one can systematically formulate a difference equation of higher order and in the early days of SPICE one can incorporate several of them in the numerical solver. In recent decades, the second-order version has proven to be the most widely used, and in modern simulators, the Gear2 option is a standard integration method.

In the second-order Gear implementation of a derivative, we have
$$\frac{d f}{d t}(t+\Delta t) \approx \frac{1}{\Delta t}\left(\frac{3}{2} f(t+\Delta t)-2 f(t)+\frac{1}{2} f(t-\Delta t)\right) .$$
It looks quite a bit different compared to the earlier implementations. This formulation is somewhat less precise than the trap method (we will see why later in Chap. 4) but does not suffer from its “ringing” weakness. Numerically, this is also straightforward to implement; we just need an initial derivative in addition to initial value and knowledge of the solution two timesteps back. We have for the full formulation using the previous example
$$\frac{i(t+\Delta t)}{C}=\frac{3 u(t+\Delta t) / 2-2 u(t)+u(t-\Delta t) / 2}{\Delta t}, \quad u(0)=1,$$
or after reformulation
$$u(t+\Delta t)=u(t) \frac{4}{3}-\frac{u(t-\Delta t)}{3}+\frac{2 \Delta t}{3 C} i(t+\Delta t), \quad u(0)=1,$$
As before, ignoring the current update routine, a pseudocode will look like subroutine SolveDiffGear2
The second-order Gear method is an example of a multistep difference method where we need to know the solution two timesteps back.

## 电气工程代写|模拟电路代写analog circuit代考|Accuracy

The accuracy of the solution will depend on the circuit to be solved, the timestep, and which integration method was used. The accuracy of numerical approximations to differential equations cun be estimated by their truncution error. One way to construct difference approximations is to use a Taylor series of a function around some point:
$$f(t+\Delta t)=f(t)+f^{\prime}(t) \Delta t+\frac{1}{2} f^{\prime \prime}(t) \Delta t^2+\ldots+\frac{1}{n !} f^n(t) \Delta t^n$$
A first-order accurate approximation will now be found by assuming all higherorder derivatives are negligible, and we are left with
$$f(t+\Delta t)=f(t)+f^{\prime}(t) \Delta t+\frac{1}{2} f^{\prime \prime}(t) \Delta t^2+\ldots$$
which after some rewrite gives
$$f^{\prime}(t)=\frac{f(t+\Delta t)-f(t)}{\Delta t}+o(\Delta t)$$
where the $o$ symbol indicates behavior for small arguments.
This is the Euler forward approximation which is accurate to first order. It means the truncation error is $\sim \Delta t$. One can similarly show that the trapezoidal approximation and Gear2 approximation are of second-order accuracy where the error scales as $\Delta t^2$. What this means practically is that if the solution is a straight line, the first derivative is not changing with time, so the second-order derivative is zero, and all the three methods we have been discussing will have no truncation error in the evaluation of the derivative. If the solution is a second-order polynomial, the Euler method will start to have truncation errors, and one must reduce the timestep to reduce their impact. The trap and Gear2 method can trace out a second-order polynomial with no truncation error but will have errors when higher-order solutions are encountered. Generally the higher the order, the better when it comes to accuracy. The penalty is lengthy evaluations in time, so most simulators use up to third-order approximations in the derivative calculation.

# 模拟电路代考

## 电气工程代写|模拟电路代写analog circuit代考|Second-Order Gear Method

C. William Gear [5] 于 1971 年出版了一本书，自此成为经典之一。他构造了一组具有不同截断误差的差分方程，这些方程具有一些非常好的特性。 他展示了如何系统地制定一个高阶差分方程，并且在 SPICE 的早期，人们可以将其中的几个合并到数值求解器中。近几十年来，二阶版本已被证明 是使用最广泛的版本，在现代模拟器中，Gear2 选项是一种标准的集成方法。

$$\frac{d f}{d t}(t+\Delta t) \approx \frac{1}{\Delta t}\left(\frac{3}{2} f(t+\Delta t)-2 f(t)+\frac{1}{2} f(t-\Delta t)\right)$$

$$\frac{i(t+\Delta t)}{C}=\frac{3 u(t+\Delta t) / 2-2 u(t)+u(t-\Delta t) / 2}{\Delta t}, \quad u(0)=1,$$

$$u(t+\Delta t)=u(t) \frac{4}{3}-\frac{u(t-\Delta t)}{3}+\frac{2 \Delta t}{3 C} i(t+\Delta t), \quad u(0)=1,$$

## 电气工程代写|模拟电路代写analog circuit代考|Accuracy

$$f(t+\Delta t)=f(t)+f^{\prime}(t) \Delta t+\frac{1}{2} f^{\prime \prime}(t) \Delta t^2+\ldots+\frac{1}{n !} f^n(t) \Delta t^n$$

$$f(t+\Delta t)=f(t)+f^{\prime}(t) \Delta t+\frac{1}{2} f^{\prime \prime}(t) \Delta t^2+\ldots$$

$$f^{\prime}(t)=\frac{f(t+\Delta t)-f(t)}{\Delta t}+o(\Delta t)$$

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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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