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# 数学代写|数值分析代写Numerical analysis代考|STAT434 Newton’s method

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## 数学代写数值分析代写Numerical analysis代考|Newton’s method

The next method we study is Newton’s method. While the bisection method has the attractive property that it will converge if the method is continuous and has a sign change, it has a disadvantage in that it is not smart enough to take advantage of simple functions. For example, if we know a function is linear, then finding the root should take just one step. But we also know that if you “zoom in” on a smooth function, it looks linear. Newton figured out how to take advantage of this idea and create a faster root finding algorithm.

Newton’s idea is simple: To find the root of a function, find the tangent line of the function at your current guess for the root, then your next guess will be where the tangent line hits the x-axis. Note that if the function is linear, then the root will be found in just one step, and if the function is close to linear, then this will give a very good approximation of the root. To start the algorithm, we need an initial guess, and the better it is, the better the algorithm will work. The idea is illustrated in Figure 5.2.

If we know the previous guess $x_k$, Newton’s idea allows us to explicitly find $x_{k+1}$. The tangent line to $f$ at $x_k$ is given by
$$y-f\left(x_k\right)=f^{\prime}\left(x_k\right)\left(x-x_k\right) .$$
The next Newton iteration is defined to be where this line crosses the $x$-axis, and we call this point $\left(x_{k+1}, 0\right)$. Plugging this point into the tangent line gives
$$0-f\left(x_k\right)=f^{\prime}\left(x_k\right)\left(x_{k+1}-x_k\right) .$$
Now solve for $x_{k+1}$ :
$$x_{k+1}=x_k-\frac{f\left(x_k\right)}{f^{\prime}\left(x_k\right)} .$$

## 数学代写|数值分析代写Numerical analysis代考|Secant method

Although Newton’s method is much faster than bisection, it has a disadvantage in that it explicitly uses the derivative of the function. In many processes where rootfinding is needed (e. g., optimization processes), we may not know an analytical expression for the function, and thus may not be able to find an expression for the derivative. That is, a function evaluation may be the result of a process or experiment. While we may theoretically expect the function to be differentiable, we cannot find the derivative explicitly. For situations like this, the secant method has been developed, and one can think of the secant method as being the same as Newton’s method, except that instead of using the tangent line at $x_k$, you use the secant line at $x_k$ and $x_{k-1}$. This defines the following algorithm:
Algorithm 46 (Secant method).
Given: $f$, tol, $x_0, x_1$
while $\left(\left|x_{k+1}-x_k\right|>\right.$ tol $)$ :
$$x_{k+1}=x_k-\frac{f\left(x_k\right)}{\frac{f\left(x_k\right)-f\left(x_{k-1}\right)}{x_k-x_{k-1}}} .$$

## 数学代写数值分析代写Numerical analysis代考|Newton’s method

$$y-f\left(x_k\right)=f^{\prime}\left(x_k\right)\left(x-x_k\right) .$$

$$0-f\left(x_k\right)=f^{\prime}\left(x_k\right)\left(x_{k+1}-x_k\right) .$$

$$x_{k+1}=x_k-\frac{f\left(x_k\right)}{f^{\prime}\left(x_k\right)} .$$

## 数学代写|数值分析代写Numerical analysis代考|Secant method

(例如，优化过程)，我们可能不知道函数的解析表达式，因此可能无法找到导数的表达式。即，功能评估可 以是过程或实验的结果。虽然理论上我们可能期望函数是可微的，但我们无法明确地找到导数。对于这种情 况，割线法已经发展起来，人们可以认为割线法与牛顿法相同，只是不使用切线在 $x_k$ ，你使用割线在 $x_k$ 和 $x_{k-1}$. 这定义了以下算法:

$$x_{k+1}=x_k-\frac{f\left(x_k\right)}{\frac{f\left(x_k\right)-f\left(x_{k-1}\right)}{x_k-x_{k-1}}} .$$

## 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。