📚 Iteration | 迭代法
Iteration is a powerful numerical technique used to find approximate solutions to equations that cannot be solved algebraically. By repeatedly applying a formula, we can generate a sequence of values that converges to a root, providing accurate results step by step.
迭代法是一种强大的数值方法，用于求解无法通过代数方式解出的方程。通过反复应用一个公式，我们可以生成一个逐渐逼近根的数值序列，从而逐步得到精确的结果。
1. Introduction to Iteration | 迭代法简介
Many equations, especially those involving exponentials, logarithms, or trigonometric functions, do not have neat algebraic solutions. Iteration offers an alternative: start with an initial guess and use a recurrence relation to improve the estimate until the root is pinned down with desired accuracy.
许多方程，特别是包含指数、对数或三角函数的方程，往往没有整洁的代数解。迭代法提供了一种替代方案：从一个初始猜测值开始，利用递推关系逐步改进估计值，直到将根锁定到所需精度。
In the context of Edexcel A‑Level Mathematics, iteration mainly appears as fixed-point iteration and the Newton–Raphson method, both of which are core tools for solving f(x)=0.
在Edexcel A‑Level数学中，迭代主要以不动点迭代和牛顿-拉弗森方法出现，两者都是求解 f(x)=0 的核心工具。
2. Rearranging Equations into the Form x = g(x) | 将方程改写为 x = g(x) 的形式
e form x = g(x). For example, to solve x³ − 3x − 5 = 0, we can write x = (x³ − 5)/3 or x = ∛ many possible rearrangements, and the choice affects whether the iteration will converge.
要用不动点迭代法，必须先将方程 f(x)=0 改写为 x = g(x) 的形式。例如，要求解 x³ − 3x − 5 = 0，可以写成 x = (x³ − 5)/3 或 x = ∛(3x + 5)。通常存在多种可能的改写方式，而不同的选择会影响迭代是否会收敛。
Finding a suitable rearrangement is a skill. A good starting point is to isolate the simplest occurrence of x and express everything else in terms of x.
寻找合适的重新排列是一项技能。一个好的出发点是分离出最简单的 x 项，并将其余部分用 x 表示。
3. The Iterative Process | 迭代过程
dly entering the expression. For instance, setting x₀ = 2, we compute x₁ = g(2), then x₂ = g(x₁), continuing until the values stabilise to a required number of decimal places.
一旦得到 x = g(x)，我们选择一个初始近似值 x₀，然后使用 xₙ₊₁ = g(xₙ) 生成序列。借助计算器可以通过反复输入表达式快速完成。例如，设 x₀ = 2，计算 x₁ = g(2)，然后 x₂ = g(x₁)，持续进行直到数值稳定在所需的小数位数。
xₙ₊₁ = g(xₙ)
xₙ₊₁ = g(xₙ)
Convergence can be seen when the difference between successive terms becomes very small, say |xₙ₊₁ − xₙ| < 10⁻⁵. The final value is taken as an approximate root.
当相邻两项的差值变得非常小，例如 |xₙ₊₁ − xₙ| < 10⁻⁵ 时，就可以认为收敛了。最终值即作为近似根。
4. Conditions for Convergence | 收敛条件
Not every rearrangement leads to a convergent sequence. A key condition for convergence near a root α is that |g'(α)| < 1. If |g'(x)| is small around the root, the iteration converges rapidly; if |g'(x)| is close to 1, convergence is slow; if |g'(x)| > 1, the process diverges.
并非每一种重排都能导致收敛序列。在根 α 附近收敛的一个关键条件是 |g'(α)| < 1。如果 |g'(x)| 在根附近很小，迭代就会快速收敛；如果 |g'(x)| 接近 1，收敛缓慢；若 |g'(x)| > 1，过程就会发散。
This condition can be checked by differentiating g(x) and evaluating the derivative near the suspected root or over the interval of interest.
可以通过对 g(x) 求导，并在怀疑的根附近或所关注的区间内评估导数值来检验这一条件。
5. Testing for Convergence Using Derivatives | 利用导数检验收敛性
e we have an equation x = g(x) and an initial guess x₀ = 1.5. If g(x) = (x³ − 5)/3, then g'(x) = x². Near x = 2, g' so this rearrangement will probably diverge from that starting point. In contrast, g(x) = ∛(3x + 5) gives g'(x) = 1 / (∛(3x + 5)²), which is less than 1 near the root, promising convergence.
假设我们有一个方程 x = g(x)，初始猜测 x₀ = 1.5。若 g(x) = (x³ − 5)/3，则 g'(x) = x²。在 x = 2 附近，g'(2) = 4，大于 1，因此该重排很可能从这个起点开始发散。相反，g(x) = ∛(3x + 5) 给出 g'(x) = 1 / (∛(3x + 5)²)，在根附近小于 1，预示着会收敛。
The smaller |g'(α)| is, the faster the iteration converges. This is why selecting an appropriate rearrangement is crucial to the success of the method.
|g'(α)| 越小，迭代收敛得越快。这就是为什么选择适当的重排对方法的成功至关重要。
6. Graphical Interpretation of Iteration | 迭代的图形解释
tion can be visualised by plotting the lines y = x and y = g(x). Starting from x₀ on the x‑ ly to the curve y = g(x), then horizontally to y = x, giving the next value x₁. This stair‑se or spiral pattern illustrates convergence toward the intersection point where x = g(x).
不动点迭代可以通过绘制直线 y = x 和曲线 y = g(x) 来直观展示。从 x 轴上的 x₀ 出发，垂直移动到曲线 y = g(x)，再水平移动到直线 y = x，即得到下一个值 x₁。这种阶梯状或螺旋状的图形清楚地展示了向交点 x = g(x) 的收敛过程。
ess than 1 in magnitude, a spiral staircase forms, converging in an oscillatory manner. If |g'(x)| ≥ 1, the steps move away from the intersection – divergence.
如果 g(x) 在根附近的斜率为负且绝对值小于 1，则会形成螺旋状阶梯，以振荡方式收敛。若 |g'(x)| ≥ 1，各步便会远离交点——这便是发散。
7. The Newton–Raphson Method | 牛顿–拉弗森方法
The Newton– e. It is designed to solve f(x) = 0 directly and typically requires fewer steps than fixed‑point iteration.
牛顿–拉弗森法是一种特殊形式的迭代法，它利用 f(x) 的导数来实现快速收敛。该方法直接求解 f(x) = 0，通常比不动点迭代所需步骤更少。
xₙ₊₁ = xₙ − f(xₙ) / f'(xₙ)
xₙ₊₁ = xₙ − f(xₙ) / f'(xₙ)
This formula assumes f'(xₙ) ≠ 0. Starting from a sensible initial guess, each successive approximation is computed by subtracting the ratio of the function value to its derivative from the current estimate.
该公式假设 f'(xₙ) ≠ 0。从一个合理的初始猜测出发，每次将当前估计值减去函数值与导数之比，从而得到新的近似值。
8. Derivation of the Newton–Raphson Formula | 牛顿–拉弗森公式的推导
The method comes from the first‑order Taylor expansion of f(x) about xₙ: f(x) ≈ f(xₙ) + f'(xₙ)(x − xₙ). Setting f(x) = 0 and solving for x gives x ≈ xₙ − f(xₙ)/f'xₙ). The next approximation is taken as this improved value, giving the iterative formula.
该方法源自 f(x) 在 xₙ 附近的一阶泰勒展开：f(x) ≈ f(xₙ) + f'(xₙ)(x − xₙ)。令 f(x) = 0 并求解 x，得到 x ≈ xₙ − f(xₙ)/f'(xₙ)。将这一更好的值作为下一个近似值，即得到迭代公式。
Geometrically, Newton–Raphson works by following the tangent line at xₙ to where it crosses the x‑axis. The x‑intercept of the tangent becomes the new estimate. This explains why the method is so efficient when the starting point is close to the actual root.
从几何上看，牛顿–拉弗森法是沿着 xₙ 处的切线移动，直到它与 x 轴相交。切线与 x 轴的交点就成为新的估计值。这也解释了为什么当初始点接近实际根时，该方法效率特别高。
9. Applying the Newton–Raphson Method | 应用牛顿–拉弗森方法
To solve eˣ − 4x = 0, we set f(x) = eˣ − 4x and f'(x) = eˣ − 4). With a calculator, repeated application quickly homes in on the root near x ≈ 0.357.
为求解 eˣ − 4x = 0，设 f(x) = eˣ − 4x，f'(x) = eˣ − 4。选择 x₀ = 0.5，计算 x₁ = 0.5 − (e⁰.⁵ − 2)/(e⁰.⁵ − 4)。借助计算器重复应用，很快就能逼近 x ≈ 0.357 附近的根。
It is wise to check that the function changes sign around the initial guess, indicating a root is indeed present. This reduces the risk of the method wandering off.
明智的做法是检查函数在初始猜测附近是否变号，以确认确实存在根。这样可以降低方法跑偏的风险。
10. When Iteration Fails | 迭代失败的情形
Both fixed‑point iteration and Newton–Raphson can fail. Fixed‑point iteration diverges if |g'(x)| > 1 near the root or if the initial guess is too far away. Newton–Raphson can fail when f'(xₙ) is near zero, leading to huge jumps, or when the sequence oscillates without settling.
不动点迭代和牛顿–拉弗森法都可能失败。如果根附近 |g'(x)| > 1，或初始猜测距离太远，不动点迭代就会发散。牛顿–拉弗森法在 f'(xₙ) 接近零时可能失效，导致大幅跳跃，或者序列振荡不定。
Additionally, the Newton–Raphson method may converge to a different root if the starting value is not sufficiently close to the intended one. This is why drawing a rough graph or using sign‑change checks is recommended.
此外，如果起始值不够接近目标根，牛顿–拉弗森法可能会收敛到另一个根。因此建议先画草图或使用符号变化检查来指引。
11. Choosing a Suitable Rearrangement | 选择合适的重新排列
For fixed‑point iteration, always test |g'(x)| at the suspected root. A rearrangement that gives a derivative near 0 will converge extremely quickly. If you have a choice, pick the one with the smallest magnitude of g'(x) over the interval containing the root.
对于不动点迭代，一定要在猜测的根处检验 |g'(x)|。导数接近 0 的重新排列将收敛得非常快。如果有选择余地，就在包含根的区间内选取 |g'(x)| 最小的那个重排。
It is also beneficial to write the iteration in a form that makes calculator entry easy: the entire expression should be directly typeable, and brackets should be used clearly to avoid misinterpretation.
把迭代式写成便于在计算器中输入的形式也很有帮助：整个表达式应能直接输入，并且要清晰使用括号，避免误解。
12. Summary and Exam Tips | 总结与考试技巧
Iteration is a core numerical method in Edexcel A‑Level Maths. You need to be able to rearrange equations, perform fixed‑point iteration, justify convergence using derivatives, apply the Newton–Raphson method, and explain why a method may fail. Practice with a variety of functions is essential.
迭代法是 Edexcel A‑Level 数学中的核心数值方法。你需要能够重新排列方程、执行不动点迭代、利用导数证明收敛性、应用牛顿–拉弗森法，并解释方法为何可能失败。对各种函数进行练习是必不可少的。
- Memorise the Newton–Raphson formula and its derivation.
- 记住牛顿–拉弗森公式及其推导。
- Always show a check of sign change to confirm a root exists.
- 始终展示符号变化的检验，以确认根的存在。
- State the convergence condition |g'(x)| < 1 explicitly in written answers.
- 在书面答案中明确陈述收敛条件 |g'(x)| < 1。
- Use exact values or a high number of decimal places until the final rounding step.
- 在最终舍入步之前使用精确值或较多的小数位数。
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