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数学代写|数值分析代写Numerical analysis代考|MATH408 Bisection

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数学代写数值分析代写Numerical analysis代考|Bisection

Given a bracket $[a, b]$ for which $f$ takes opposite sign at $a$ and $b$, the simplest technique for finding $x_*$ is the bisection algorithm:
For $k=0,1,2, \ldots$

Compute $f\left(c_k\right)$ for $c_k=\frac{1}{2}\left(a_k+b_k\right)$.

If $f\left(c_k\right)=0$, exit; otherwise, repeat with
$$\left[a_{k+1}, b_{k+1}\right]:= \begin{cases}{\left[a_k, c_k\right],} & \text { if } f\left(a_k\right) f\left(c_k\right)<0 ; \ {\left[c_k, b_k\right],} & \text { if } f\left(c_k\right) f\left(b_k\right)<0 .\end{cases}$$

Stop when the interval $b_{k+1}-a_{k+1}$ is sufficiently small, or if $f\left(c_k\right)=0$.

How does this method converge? Not bad for such a simple idea. At the $k$ th stage, there must be a root in the interval $\left[a_k, b_k\right]$. Take $c_k=$ $\frac{1}{2}\left(a_k+b_k\right)$ as the next estimate to $x_$, giving the error $e_k=c_k-x_$. The worst possible error would occur if $x_$ was close to $a_k$ or $b_k$, half the bracket’s width away from $c_k$, and hence $\left|e_k\right|=\left|c_k-x_\right| \leq$ $\frac{1}{2}\left(b_k-a_k\right)=2^{-k-1}\left(b_0-a_0\right)$.

Theorem 4.1. Given a bracket $[a, b] \subset \mathbb{R}$ for which $f(a) f(b)<0$, there exists $x_* \in[a, b]$ such that $f\left(x_\right)=0$ and the bisection point $c_k$ satisfies $$\left|c_k-x_\right| \leq \frac{b-a}{2^{k+1}} .$$
We say this iteration converges linearly (the log of the error is bounded by a straight line when plotted against iteration count – see the next example) with rate $\rho=1 / 2$. Practically, this means that the error is cut in half at each iteration, independent of the behavior of $f$. Reduction of the initial bracket width by ten orders of magnitude would require roughly $\log _2 10^{10} \approx 33$ iterations. If $f$ is fast to evaluate, this convergence will be pretty quick; moreover, since the algorithm only relies on our ability to compute the sign of $f(x)$ accurately, the algorithm is robust to strange behavior in $f$ (such as local minima).

数学代写|数值分析代写Numerical analysis代考|Regula Falsi

A simple adjustment to bisection can often yield much quicker convergence. The name of the resulting algorithm, regula falsi (literally ‘false rule’) hints at the technique. As with bisection, begin with an interval $\left[a_0, b_0\right] \subset \mathbb{R}$ such that $f\left(a_0\right) f\left(b_0\right)<0$. The goal is to be more sophisticated about the choice of the root estimate $c_k \in\left(a_k, b_k\right)$. Instead of simply choosing the middle point of the bracket as in bisection, we approximate $f$ with the line $p_k \in \mathcal{P}_1$ that interpolates $\left(a_k, f\left(a_k\right)\right)$ and $\left(b_k, f\left(b_k\right)\right)$, so that $p_k\left(a_k\right)=f\left(a_k\right)$ and $p\left(b_k\right)=f\left(b_k\right)$. This unique polynomial is given (in the Newton form) by
$$p_k(x)=f\left(a_k\right)+\frac{f\left(b_k\right)-f\left(a_k\right)}{b_k-a_k}\left(x-a_k\right) .$$
Now approximate the zero of $f$ in $\left[a_k, b_k\right]$ by the zero of the linear model $p_k$ :
$$c_k=\frac{a_k f\left(b_k\right)-b_k f\left(a_k\right)}{f\left(b_k\right)-f\left(a_k\right)} .$$
The algorithm then takes the following form:
For $k=0,1,2, \ldots$

1. Compute $f\left(c_k\right)$ for $c_k=\frac{a_k f\left(b_k\right)-b_k f\left(a_k\right)}{f\left(b_k\right)-f\left(a_k\right)}$.
2. If $f\left(c_k\right)=0$, exit; otherwise, repeat with
$$\left[a_{k+1}, b_{k+1}\right]:= \begin{cases}{\left[a_k, c_k\right],} & \text { if } f\left(a_k\right) f\left(c_k\right)<0 ; \ {\left[c_k, b_k\right],} & \text { if } f\left(c_k\right) f\left(b_k\right)<0 .\end{cases}$$
3. Stop when $f\left(c_k\right)$ is sufficiently small, or the maximum number of iterations is exceeded.

数学代写数值分析代写Numerical analysis代考|Bisection

\left 或额外的 \right }

数学代写|数值分析代写Numerical analysis代考|Regula Falsi

$$p_k(x)=f\left(a_k\right)+\frac{f\left(b_k\right)-f\left(a_k\right)}{b_k-a_k}\left(x-a_k\right) .$$

$$c_k=\frac{a_k f\left(b_k\right)-b_k f\left(a_k\right)}{f\left(b_k\right)-f\left(a_k\right)} .$$

1. 计算 $f\left(c_k\right)$ 为了 $c_k=\frac{a_k f\left(b_k\right)-b_k f\left(a_k\right)}{f\left(b_k\right)-f\left(a_k\right)}$.
2. 如果 $f\left(c_k\right)=0$ ， 出口; 否则，重复
$$\left[a_{k+1}, b_{k+1}\right]:=\left{\left[a_k, c_k\right], \quad \text { if } f\left(a_k\right) f\left(c_k\right)<0 ;\left[c_k, b_k\right], \quad \text { if } f\left(c_k\right) f\left(b_k\right)<0 .\right.$$
3. 停止时 $f\left(c_k\right)$ 足够小，或者超过最大迭代次数。

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。