📚 The Amendment Process in Numerical Methods | 数值方法中的修正过程
In Edexcel A Level Mathematics, many polynomial, trigonometric and exponential equations cannot be solved algebraically. A numerical method therefore starts with a rough approximation and applies a rule to amend that approximation, producing a sequence that hopefully approaches the true root.
在 Edexcel A Level 数学中，许多多项式、三角和指数方程无法用代数方法求解。因此数值方法从一个粗略的近似值开始，并应用规则来修正该近似值，产生一个有望逼近真实根的序列。
1. What “Amendment” Means in Numerical Work | 数值工作中“修正”的含义
An amendment process is any systematic rule that takes a current estimate xₙ and returns an improved estimate xₙ₊₁. We repeat the rule until the change between successive estimates is small enough for the accuracy required by the question.
修正过程是任何系统化规则：将当前估计值 xₙ 代入，返回一个改进后的估计值 xₙ₊₁。我们重复该规则，直到相邻两次估计值之间的变化足够小，满足题目所要求的精度。
The sequence x₀, x₁, x₂, … is called an iterative sequence. Each new value is an amendment of the previous value, not a completely new calculation. This is why calculators are often used with the ANS key to carry the previous answer into the next iteration.
序列 x₀, x₁, x₂, … 称为迭代序列。每个新值都是对前一个值的修正，而不是一次全新的计算。这就是为什么考试中常常使用计算器的 ANS 键，将前一个答案带入下一次迭代。
In Edexcel questions, the amendment rule may be given directly, or you may be asked to construct it from an equation. In both cases, the principle is the same: substitute, amend, substitute again.
在 Edexcel 的题目中，修正规则可能直接给出，也可能要求你从方程中构造出来。两种情况下原理相同：代入、修正、再代入。
2. Locating a Root by Sign Change | 通过符号变化定位根
Before amending values, we need evidence that a root exists in an interval. If f is continuous on [a, b] and f(a) × f(b) < 0, then there is at least one root in the interval (a, b).
在修正数值之前，我们需要先证明某区间内存在根。如果 f 在 [a, b] 上连续，且 f(a) × f(b) < 0，则在区间 (a, b) 内至少存在一个根。
This sign-change condition is the starting point for the amendment process. It gives a bracketing interval, and bisection or decimal search can then shrink that interval by repeatedly testing the midpoint or endpoints.
这个符号变化条件是修正过程的起点。它给出一个包含根的区间，然后二分法或十进制搜索可以通过反复检验中点或端点来缩小区间。
For example, for f(x) = x³ − 3x + 1, we find f(0) = 1 and f(1) = −1, so the sign changes and a root lies between 0 and 1. This interval can now be narrowed step by step.
例如，对于 f(x) = x³ − 3x + 1，我们有 f(0) = 1 而 f(1) = −1，因此符号发生变化，根位于 0 和 1 之间。这个区间现在可以逐步缩小。
3. The Iteration Formula xₙ₊₁ = g(xₙ) | 迭代公式 xₙ₊₁ = g(xₙ)
To amend an estimate automatically, we rearrange the equation f(x) = 0 into the form x = g(x). The rearrangement is not unique; different choices of g can lead to different convergence behaviour.
为了自动修正估计值，我们将方程 f(x) = 0 改写为 x = g(x) 的形式。这种改写并不唯一；不同的 g 选择会导致不同的收敛行为。
The iteration formula is then:
然后迭代公式为：
xₙ₊₁ = g(xₙ)
Starting from x₀, the calculator or spreadsheet repeatedly applies g. Each application is an amendment that should move the estimate closer to the fixed point α, where α = g(α) and f(α) = 0.
从 x₀ 开始，计算器或电子表格反复应用 g。每一次应用都是一次修正，应当使估计值更接近不动点 α，其中 α = g(α) 且 f(α) = 0。
For the equation x³ − 3x + 1 = 0, one possible rearrangement is x = (3x − 1)^(1/3). Another is x = (x³ + 1)/3. They are algebraically equivalent as fixed-point equations, but their gradients differ greatly near the root.
对于方程 x³ − 3x + 1 = 0，一种可能的改写是 x = (3x − 1)^(1/3)。另一种是 x = (x³ + 1)/3。它们作为不动点方程在代数上等价，但在根附近的梯度差异很大。
4. Convergence and the Gradient Condition | 收敛性与梯度条件
The amendment process does not always work. For the iteration xₙ₊₁ = g(xₙ), the sequence converges to the root α if the derivative g′(x) satisfies |g′(x)| < 1 in an interval around α.
修正过程并不总是有效。对于迭代 xₙ₊₁ = g(xₙ)，如果导数 g′(x) 在 α 附近的区间内满足 |g′(x)| < 1，则序列收敛到根 α。
If |g′(x)| > 1 near the root, the amendments become larger rather than smaller, and the sequence diverges or oscillates away from the root. If |g′(x)| = 1, convergence is uncertain.
如果 |g′(x)| > 1 在根附近成立，则修正量会越来越大，序列会发散或振荡远离根。如果 |g′(x)| = 1，收敛性不确定。
Cobweb and staircase diagrams illustrate this behaviour. A staircase occurs when g′(x) is positive, while a cobweb occurs when g′(x) is negative; both can converge if the gradient condition holds.
蛛网图和阶梯图可以说明这种行为。当 g′(x) 为正时出现阶梯图，当 g′(x) 为负时出现蛛网图；只要梯度条件成立，两者都可以收敛。
In an exam, you may be asked to show that a given iteration converges because |g′(α)| < 1. You can often estimate α first by testing signs or using a graph.
在考试中，题目可能要求你证明给定迭代收敛，因为 |g′(α)| < 1。你通常可以先通过检验符号或使用图像来估计 α。
5. Newton-Raphson as a Special Amendment | 牛顿-拉夫森法作为一种特殊修正
The Newton-Raphson method is a more powerful amendment rule. It uses the tangent to the curve at the current estimate to find where that tangent crosses the x-axis, giving the next estimate.
牛顿-拉夫森法是一种更有效的修正规则。它利用当前估计值处曲线的切线，求出该切线与 x 轴的交点，从而得到下一个估计值。
The formula is:
公式为：
xₙ₊₁ = xₙ − f(xₙ) / f′(xₙ)
This is derived from the tangent equation y − f(xₙ) = f′(xₙ)(x − xₙ). Setting y = 0 and solving for x gives the amendment term −f(xₙ)/f′(xₙ).
该公式由切线方程 y − f(xₙ) = f′(xₙ)(x − xₙ) 推导而来。令 y = 0 并解出 x，就得到修正项 −f(xₙ)/f′(xₙ)。
Newton-Raphson converges very rapidly when it works, typically doubling the number of correct decimal places at each step. However, it requires f′(xₙ) ≠ 0, and a poor starting value can cause failure.
牛顿-拉夫森法在有效时收敛非常快，通常每一步能使正确的小数位数翻倍。但它要求 f′(xₙ) ≠ 0，且较差的初始值可能导致失败。
6. Worked Example: Newton-Raphson | 例题：牛顿-拉夫森法
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