## Problem Statement > Given some functions $f(x)$, determine the value(s) of x where $f(x) = 0$ Especially when $f(x)$ is not analytically. ## Root finding algorithms Two main types: - Bracketing Methods - Open Methods Following steps are essential for any root find algorithms: 1. Guess: one more more initial values for $x$ 2. Iterate: using the scheme to improve the guess 3. Check the convergence ### Bracketing method  Put it simply, it a kind of method like binary search. We can test if $f(a)f(b) < 0$ to determine whether we need to discard half of the interval. The approximation relative error is $$ \epsilon_r = \frac{x_{i+1} - x_i}{x_{i+1}} $$ When $|\epsilon_r|$ becomes less than a specified threshold $\epsilon_s$ the iterations are stopped. [](https://pic1.imgdb.cn/item/67d84f2e88c538a9b5c005c8.png) For this algorithms, we are able to set the iteration number $n$. The absolute value of error $|E| = |V_t - V_a|$. Let $L = b- a$ be the length of the interval, so we know the interval length after $n$ iteration his $\frac{L}{2^n}$ Therefore, $$ |E| \leq \frac{L}{2^n} $$ So we can determine the iteration number $n$ as: $$ n = \frac{\ln \frac{L}{E_{max}}}{\ln 2} $$ ## Improvement To improve the root position estimation, we can construct a straight line between point $a, b$, and then use the intersection point with x axis as the initial point.  As shown in the figure, $$ x_i = b - f(b) \frac{b-a}{f(b)-f(a)} $$ Loading... ## Problem Statement > Given some functions $f(x)$, determine the value(s) of x where $f(x) = 0$ Especially when $f(x)$ is not analytically. ## Root finding algorithms Two main types: - Bracketing Methods - Open Methods Following steps are essential for any root find algorithms: 1. Guess: one more more initial values for $x$ 2. Iterate: using the scheme to improve the guess 3. Check the convergence ### Bracketing method  Put it simply, it a kind of method like binary search. We can test if $f(a)f(b) < 0$ to determine whether we need to discard half of the interval. The approximation relative error is $$ \epsilon_r = \frac{x_{i+1} - x_i}{x_{i+1}} $$ When $|\epsilon_r|$ becomes less than a specified threshold $\epsilon_s$ the iterations are stopped. [](https://pic1.imgdb.cn/item/67d84f2e88c538a9b5c005c8.png) For this algorithms, we are able to set the iteration number $n$. The absolute value of error $|E| = |V_t - V_a|$. Let $L = b- a$ be the length of the interval, so we know the interval length after $n$ iteration his $\frac{L}{2^n}$ Therefore, $$ |E| \leq \frac{L}{2^n} $$ So we can determine the iteration number $n$ as: $$ n = \frac{\ln \frac{L}{E_{max}}}{\ln 2} $$ ## Improvement To improve the root position estimation, we can construct a straight line between point $a, b$, and then use the intersection point with x axis as the initial point.  As shown in the figure, $$ x_i = b - f(b) \frac{b-a}{f(b)-f(a)} $$ 最后修改：2025 年 03 月 20 日 © 允许规范转载 打赏 赞赏作者 支付宝微信 赞 如果觉得我的文章对你有用，请随意赞赏
