📚 Numerical Solution of Equations: Newton's Method | 方程数值解：牛顿法
is one of the most fundamental activities in mathematics and science. Although exact algebraic solutions exist for simple functions, many equations arising in real-world applications cannot be solved in closed form. Newton' phson method, is a powerful numerical technique that generates successively better approximations to the roots of a real-valued function. This article explores the derivation, geometric interpretation, algorithm, convergence properties, worked examples and common pitfalls of Newton' thod, providing a comprehensive revision resource for IB Mathematics and A Level students.
在数学和科学的许多领域中，求解形如 f(x) = 0 的方程至关重要。虽然简单函数存在解析解，但现实应用中的许多方程无法精确求解。牛顿法（也称牛顿-拉弗森方法）是一种强大的数值技术，可逐次逼近实值函数的根。本文探讨牛顿法的推导、几何解释、算法、收敛性质、典型例题及常见陷阱，为 IB 数学与 A Level 学生提供全面的复习资料。
1. Introduction to Numerical Methods | 数值方法简介
tten in the form f(x) = 0, these methods generate a sequence of approximations x₀, x₁, x₂, … ations performed, so we can in principle obtain as many correct decimal places as we wish.
数值方法为无法解析求解的数学问题提供近似解。对于方程 f(x) = 0，这些方法生成一个逼近根的序列 x₀, x₁, x₂, …。答案的精度由迭代次数控制，因此理论上我们可以获得任意多位正确的小数。
e typical situations: the equation has no closed-form solution (for example x = cos x or eˣ n practice; or the function is only known through experimental data rather than a formula.
在三种典型情形下必须使用数值方法：方程没有解析解（例如 x = cos x 或 eˣ = 3x）；精确解过于复杂，在实际中不实用；或者函数仅由实验数据给出，而没有具体表达式。
Newton's method is one of the most efficient and widely used iterative techniques. Unlike the bisection method, which only uses the sign of the function, Newton's method exploits both the function value and its derivative, and this extra information produces extremely rapid convergence near the root.
牛顿法是最有效且应用最广的迭代技术之一。与只利用函数符号的二分法不同，牛顿法同时利用函数值及其导数，这些额外信息使得它在根附近收敛极快。
2. The Newton-Raphson Formula | 牛顿-拉弗森公式
The core iteration formula of Newton's method is:
牛顿法的核心迭代公式为：
xₙ₊₁ = xₙ − f(xₙ) / f′(xₙ)
Here xₙ is the current approximation, f(xₙ) is the value of the function at that point, and f′(xₙ) is the value of the derivative there. The method requires f′(xₙ) ≠ 0 at every step; if the derivative vanishes, the formula is undefined and the iteration stops.
其中 xₙ 是当前近似值，f(xₙ) 是该点的函数值，f′(xₙ) 是该点的导数值。该方法要求每一步都有 f′(xₙ) ≠ 0；若导数等于零，公式无意义，迭代无法继续。
Two conditions must be satisfied for the method to work well. First, the function must be differentiable near the root. Second, the initial guess x₀ must be reasonably close to the true root r; if it is not, the sequence may diverge or behave chaotically.
要使该方法表现良好，必须满足两个条件。第一，函数在根附近必须可导。第二，初始猜测值 x₀ 必须与真实根 r 足够接近；否则序列可能发散或出现混沌行为。
In examinations, the formula is usually quoted directly, but you must still define f(x) and f′(x) clearly before substituting any numbers. Always write down the value of xₙ₊₁ to at least one more decimal place than the final answer requires.
在考试中，公式通常可以直接引用，但在代入数值之前，你仍必须清楚地写出 f(x) 与 f′(x)。每一步算出的 xₙ₊₁ 至少要比最终答案多保留一位小数。
3. Deriving the Formula | 公式的推导
Newton' Consider the tangent line to the curve y = f(x) at the point (xₙ, f(xₙ)). Its equation is:
牛顿法可以从线性化的思想推导出来。考虑曲线 y = f(x) 在点 (xₙ, f(xₙ)) 处的切线，其方程为：
y = f(xₙ) +
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