牛顿-拉夫森方法

Newton-Raphson Method | 牛顿-拉夫森方法

Definition | 定义

The Newton-Raphson method is an iterative numerical technique used to find the roots of a real-valued function. It is based on the first-order Taylor series expansion of the function.

牛顿-拉夫森方法是一种迭代数值技术，用于求解实值函数的根。它基于函数的一阶泰勒级数展开。

Formula | 公式

The iterative formula for the Newton-Raphson method is:

牛顿-拉夫森方法的迭代公式为：

$$x_{n+1}=x_n-\frac{f(x_n)}{f^{\prime }(x_n)}$$

Where:

其中：

• $x_n$ is the current approximation $x_n$ 是当前的近似值
• $f(x_n)$ is the function value at $x_n$ $f(x_n)$ 是函数在 $x_n$ 处的值
• $f^{\prime }(x_n)$ is the derivative of the function at $x_n$ $f^{\prime }(x_n)$ 是函数在 $x_n$ 处的导数

Steps | 步骤

1. Initialization: Choose an initial guess $x_0$. 初始化：选择一个初始猜测值 $x_0$。

2. Iteration: Compute the next approximation $x_{n+1}$ using the formula until the difference between $x_{n+1}$ and $x_n$ is less than a predetermined tolerance. 迭代：使用公式计算下一个近似值 $x_{n+1}$，直到 $x_{n+1}$ 和 $x_n$ 之间的差值小于预定的容差。

3. Convergence Check: The process is repeated until convergence is achieved, meaning the value of the function at the current approximation is close to zero. 收敛检查：重复该过程直到收敛，即函数在当前近似值处的值接近于零。

Common Uses | 常见用途

The Newton-Raphson method is widely used in various scientific and engineering applications to solve equations numerically, such as finding the square roots, solving nonlinear equations, and optimizing functions.

牛顿-拉夫森方法广泛应用于各种科学和工程领域，用于数值求解方程，例如求平方根、求解非线性方程和优化函数。

Advantages and Disadvantages | 优点和缺点

Advantages:

• Fast convergence when close to the root 收敛速度快，尤其是接近根时
• Requires only the first derivative 只需计算一阶导数

Disadvantages:

• Requires a good initial guess 需要较好的初始猜测值
• May not converge if the derivative is zero or the function is not differentiable at the guess 如果导数为零或函数在猜测点不可导，可能不收敛

Pseudocode | 伪代码

def newton_raphson(f, df, x0, tol):
    x = x0
    while abs(f(x)) > tol:
        x = x - f(x) / df(x)
    return x

This pseudocode demonstrates a simple implementation of the Newton-Raphson method, where f is the function, df is its derivative, x0 is the initial guess, and tol is the tolerance level.

这个伪代码演示了牛顿-拉夫森方法的简单实现，其中 f 是函数，df 是其导数，x0 是初始猜测值，tol 是容差水平。