📚 Iterative Methods for Approximating Roots of Equations | 迭代法求方程近似解
Many equations, especially those involving transcendental functions such as eˣ + x = 0 or x³ – 2x – 5 = 0, cannot be solved exactly using elementary algebraic techniques. Iterative methods provide a powerful numerical alternative, allowing us to approximate roots to any desired degree of accuracy. In this article, we explore the principles, convergence criteria, and practical implementation of fixed-point iteration — one of the most fundamental iterative techniques in A-Level mathematics and beyond.
许多方程，尤其是涉及超越函数的方程，如 eˣ + x = 0 或 x³ – 2x – 5 = 0，无法通过初等代数技巧精确求解。迭代法提供了一种强大的数值替代方案，使我们能够以任意所需的精度近似求根。本文将探讨不动点迭代的原理、收敛判据及实际应用 — 这是 A-Level 数学及其后续学习中最基本的迭代技术之一。
1. Why Do We Need Iterative Methods? | 为什么需要迭代法？
Polynomial equations of degree five or higher generally have no closed-form solution expressible in radicals, as established by the Abel–Ruffini theorem. Even simpler-looking equations such as x = cos x must be handled numerically. Furthermore, in engineering and the sciences, we frequently encounter equations whose roots cannot be found by factoring or algebraic manipulation.
五次及以上的多项式方程通常没有可用根式表示的闭式解，这一结论由阿贝尔–鲁菲尼定理所确立。即使是看似简单的方程如 x = cos x，也必须借助数值方法处理。此外，在工程和科学领域，我们经常遇到无法通过因式分解或代数变形求出根的方程。
For such equations, iterative methods work by generating a sequence of approximations that, under suitable conditions, converge to a root. Each step uses the previous approximation to produce a better one, and the process is repeated until the change between successive approximations falls below a chosen tolerance.
对于此类方程，迭代法通过生成一个近似值序列来工作，该序列在适当条件下收敛到根。每一步都使用前一个近似值来产生更优的近似值，这一过程不断重复，直到相邻两次近似值之差低于所选容差。
2. The Fixed-Point Iteration Method | 不动点迭代法
The fixed-point iteration method is built upon rewriting the original equation f(x) = 0 in the equivalent form x = g(x). A solution of x = g(x) is called a fixed point of g, since applying g to the solution leaves it unchanged. For example, the equation x³ – 2x – 5 = 0 can be rearranged as x = ∛(2x + 5).
不动点迭代法建立在将原方程 f(x) = 0 改写成等价形式 x = g(x) 的基础之上。方程 x = g(x) 的解称为 g 的不动点，因为对解应用 g 后保持不变。例如，方程 x³ – 2x – 5 = 0 可以变形为 x = ∛(2x + 5)。
The iteration then proceeds by selecting an initial approximation x₀ and applying the recurrence relation xₙ₊₁ = g(xₙ) repeatedly. If the sequence converges, its limit α satisfies α = g(α), meaning α is a root of the original equation f(x) = 0.
迭代过程从选取初始近似值 x₀ 开始，反复应用递推关系 xₙ₊₁ = g(xₙ)。如果该序列收敛，其极限 α 满足 α = g(α)，即 α 是原方程 f(x) = 0 的根。
xₙ₊₁ = g(xₙ), n = 0, 1, 2, …
3. Convergence Condition | 收敛条件
A crucial question arises: does the iteration always converge? The answer is no. Convergence depends on the choice of g and the initial value x₀. According to the contraction mapping theorem, if g is continuously differentiable on an interval containing the fixed point α and satisfies |g′(x)| ≤ L < 1 for all x in that interval, then the iteration converges to α from any starting point within the interval.
一个关键问题随之而来：迭代是否总是收敛？答案是否定的。收敛性取决于 g 的选择和初始值 x₀。根据压缩映射定理，如果 g 在包含不动点 α 的区间上连续可微，且该区间内所有 x 都满足 |g′(x)| ≤ L < 1，那么从该区间内的任意起点出发，迭代都收敛到 α。
In practice, a simpler test is often used: check whether |g′(x)| < 1 near the expected root. If this condition holds, the iteration moves toward the root; if |g′(x)| > 1, the iteration moves away from it, and the sequence diverges.
实践中，通常使用更简单的检验方法：检查预期根附近是否满足 |g′(x)| < 1。如果该条件成立，迭代趋向于根；如果 |g′(x)| > 1，迭代则远离根，序列发散。
4. Order of Convergence | 收敛阶
The speed at which an iterative sequence approaches the root is measured by its order of convergence. If the error eₙ = |xₙ – α| satisfies eₙ₊₁ ≈ C·eₙ, the method has linear convergence (order 1). If eₙ₊₁ ≈ C·eₙ², the method has quadratic convergence (order 2), meaning the number of correct decimal digits roughly doubles with each step.
迭代序列接近根的速度用其收敛阶来衡量。如果误差 eₙ = |xₙ – α| 满足 eₙ₊₁ ≈ C·eₙ，则该方法具有线性收敛（一阶）；如果 eₙ₊₁ ≈ C·eₙ²，则具有二次收敛（二阶），即每一步正确小数位数大约翻倍。
For fixed-point iteration, if g′(α) ≠ 0, we have linear convergence. However, if g′(α) = 0 and g″(α) ≠ 0, the convergence becomes quadratic. This insight leads to improved methods such as Newton–Raphson, which deliberately constructs g so that g′(α) = 0.
对于不动点迭代，若 g′(α) ≠ 0，则为线性收敛。然而，若 g′(α) = 0 且 g″(α) ≠ 0，则收敛变为二次。这一洞察催生了改进方法，如牛顿–拉弗森法，该方法刻意构造 g 使得 g′(α) = 0。
5. Choosing a Suitable Rearrangement | 选择合适的变形形式
The same equation f(x) = 0 can often be rearranged into several different forms x = g(x). Some of these forms converge; others do not. For instance, the equation x² = 2 can be written as x = 2/x or x = ½(x + 2/x). The first form has |g′(x)| = 2/x², which equals 1 at x = √2, making convergence marginal or non-existent. The second form, known as the Babylonian method, has g′(x) = ½(1 – 2/x²), which equals 0 at x = √2, giving rapid quadratic convergence.
同一个方程 f(x) = 0 往往可以变形为多种形式 x = g(x)。其中一些形式收敛，另一些则发散。例如，方程 x² = 2 可以写成 x = 2/x 或 x = ½(x + 2/x)。第一种形式 |g′(x)| = 2/x²，在 x = √2 处等于 1，收敛性微弱或不收敛。第二种形式即巴比伦方法，g′(x) = ½(1 – 2/x²)，在 x = √2 处等于 0，具有快速的二次收敛。
Therefore, when applying fixed-point iteration, we should test the derivative of each candidate g and select the one satisfying |g′(x)| < 1 in a neighbourhood of the root. Graphical methods, such as plotting y = x and y = g(x), help visualise intersections that correspond to fixed points.
因此，在应用不动点迭代时，需要对每个候选 g 检验其导数，选择在根附近满足 |g′(x)| < 1 的形式。图形方法，如绘制 y = x 和 y = g(x)，有助于直观观察与不动点对应的交点。
6. Illustrative Example | 示例讲解
Consider the equation x³ – 3x + 1 = 0. Between 0 and 1, there is a root. We rearrange the equation as x = (x³ + 1)/3, so g(x) = (x³ + 1)/3. Then g′(x) = x², which is less than 1 in absolute value on (0, 1). Starting with x₀ = 0.5, we compute:
考虑方程 x³ – 3x + 1 = 0。在 0 和 1 之间存在一个根。我们将方程变形为 x = (x³ + 1)/3，因此 g(x) = (x³ + 1)/3。则 g′(x) = x²，其在 (0, 1) 上绝对值小于 1。取初始值 x₀ = 0.5，计算得到：
| n | xₙ |
| 0 | 0.500000 |
| 1 | 0.375000 |
| 2 | 0.350912 |
| 3 | 0.347729 |
| 4 | 0.347400 |
| 5 | 0.347363 |
The sequence converges to approximately 0.347296, the exact value of 2cos(100°)–related trigonometric expression. The successive approximations stabilise to six decimal places after about 8–10 iterations, demonstrating linear convergence in action.
该序列收敛到约 0.347296，这个值与 2cos(100°) 相关的三角表达式精确对应。逐次近似在大约 8–10 次迭代后稳定到六位小数，体现了线性收敛的实际过程。
7. Graphical Interpretation | 图解意义
Visually, fixed-point iteration corresponds to a staircase or cobweb diagram. Starting at (x₀, 0), we move vertically to the curve y = g(x), obtaining the point (x₀, g(x₀)) = (x₀, x₁). Then we move horizontally to the line y = x, arriving at (x₁, x₁), and repeat the process.
从图形上看，不动点迭代对应于阶梯图或蛛网图。从 (x₀, 0) 出发，垂直移动到曲线 y = g(x)，得到点 (x₀, g(x₀)) = (x₀, x₁)。然后水平移动到直线 y = x，到达 (x₁, x₁)，并重复此过程。
If |g′(x)| < 1 near the root, the cobweb spirals inward toward the fixed point. If |g′(x)| > 1, the cobweb spirals outward, clearly showing divergence. This diagram offers an intuitive and quick way to assess whether a given iteration will converge.
如果在根附近 |g′(x)| < 1，蛛网图向不动点盘旋收敛。如果 |g′(x)| > 1，蛛网图向外盘旋，清晰地显示发散。该图提供了一种直观而快速的评估方式，判断给定迭代是否收敛。
8. Choosing the Initial Value x₀ | 选择初始值 x₀
The initial guess x₀ plays a critical role in determining which root the iteration finds, especially when the equation has multiple roots. A common strategy is to first locate the root approximately using the intermediate value theorem: if f(a) and f(b) have opposite signs, then a root exists between a and b. A midpoint or a value near the expected root is then taken as x₀.
初始猜测 x₀ 在决定迭代找到哪个根方面起着关键作用，特别是当方程有多个根时。常用策略是先用中值定理大致定位根：如果 f(a) 和 f(b) 异号，则根存在于 a 和 b 之间。然后取中点或接近预期根的值作为 x₀。
It is also advisable to ensure that x₀ lies within the interval where |g′(x)| < 1. A poor initial guess may cause the iteration to diverge or converge to a different root than intended. Thus, initial sketching of the graph is highly recommended before numerical computation begins.
同时建议确保 x₀ 位于 |g′(x)| < 1 的区间内。不佳的初始猜测可能导致迭代发散，或者收敛到非预期根。因此，在数值计算之前，强烈推荐先绘制函数草图。
9. Stopping Criteria and Accuracy | 终止判据与精度控制
In practice, we cannot iterate infinitely; we stop when the successive approximations are close enough. The most common stopping criteria are |xₙ₊₁ – xₙ| < ε, where ε is a preselected tolerance such as 10⁻⁶, and |f(xₙ)| < ε, which verifies how close xₙ is to being an actual root.
在实践中，我们无法无限迭代；当相邻近似值足够接近时停止。最常用的终止判据是 |xₙ₊₁ – xₙ| < ε（其中 ε 是预设容差，如 10⁻⁶）和 |f(xₙ)| < ε，后者用于验证 xₙ 接近真实根的程度。
However, a small difference |xₙ₊₁ – xₙ| does not always guarantee that xₙ is close to the root, especially when the function is very flat near the root. Therefore, checking both criteria where feasible provides greater confidence in the result.
然而，小的差值 |xₙ₊₁ – xₙ| 并不总是保证 xₙ 接近根，尤其是当函数在根附近非常平缓时。因此，在可行的情况下同时检查两个判据，可以为结果提供更大的置信度。
10. Applications in A-Level Problems | A-Level 试题中的应用
In A-Level mathematics, iterative methods commonly appear in numerical methods questions. Students are often given a rearrangement x = g(x) and asked to perform several iterations starting from a given x₀, then state the root to a specified degree of accuracy. Typical values of tolerance include 0.001 or 0.0001.
在 A-Level 数学中，迭代法常见于数值方法考题。学生通常会被给定一个变形形式 x = g(x)，要求从给定 x₀ 开始进行若干次迭代，然后以指定精度给出根。常见容差为 0.001 或 0.0001。
An example from past papers: show that x = ln(2x + 5) can be used to find a root of eˣ = 2x + 5. Starting with x₀ = 2.5, candidates perform iterations to obtain x₁ = ln(10) ≈ 2.302585, then continue to x₂, x₃, … until the value stabilises to three decimal places. Examiners look for consistent iteration values, correct rounding, and a clear statement of the final root.
一个来自历年真题的例子：证明 x = ln(2x + 5) 可用于求 eˣ = 2x + 5 的根。从 x₀ = 2.5 出发，考生迭代得 x₁ = ln(10) ≈ 2.302585，然后继续计算 x₂、x₃… 直到该值稳定到三位小数。阅卷者注重一致的迭代值、正确的四舍五入，以及对最终根的清晰陈述。
11. Common Mistakes and How to Avoid Them | 常见错误与防范
A frequent error is using a rearrangement with |g′(x)| > 1 near the root, causing the values to explode. Another common mistake is rounding intermediate results too early, which propagates errors through subsequent iterations. Students also sometimes stop the iteration one step too soon, obtaining a value that has not yet converged to the required accuracy.
一个常见错误是使用在根附近 |g′(x)| > 1 的变形，导致数值爆炸。另一个常见错误是过早对中间结果进行四舍五入，使误差在后续迭代中传播。学生有时也会提前一步停止迭代，获得的数值尚未收敛到所需精度。
To avoid these pitfalls: always check g′(x) before starting; retain at least six significant figures during intermediate calculations; and continue iterating until at least two consecutive values agree to the required accuracy. Drawing a cobweb diagram can also reveal convergence issues quickly.
为避免这些陷阱：开始前务必检验 g′(x)；中间计算至少保留六位有效数字；继续迭代直到至少两个连续值在所需精度下一致。绘制蛛网图也能快速揭示收敛问题。
12. Comparison with Newton–Raphson Method | 与牛顿–拉弗森法对比
The Newton–Raphson method is another iterative technique, given by the formula xₙ₊₁ = xₙ – f(xₙ)/f′(xₙ). It converges quadratically, making it significantly faster than standard fixed-point iteration. However, it requires the derivative f′(x), and it may fail when f′(x) ≈ 0 or when the initial guess is far from the root.
牛顿–拉弗森法是另一种迭代技术，其公式为 xₙ₊₁ = xₙ – f(xₙ)/f′(xₙ)。它二次收敛，因此比标准不动点迭代快得多。然而，它需要导函数 f′(x)，并且当 f′(x) ≈ 0 或初始猜测远离根时可能失效。
Fixed-point iteration, on the other hand, does not require differentiation, which makes it easier to apply to functions that are hard to differentiate. Its linear convergence is slower, but it is more robust in certain contexts and simpler to derive. In essence, Newton–Raphson can be seen as a specially chosen fixed-point iteration with g(x) = x – f(x)/f′(x), deliberately constructed to achieve rapid convergence.
另一方面，不动点迭代不需要求导，这使得它更易于应用于难以微分的函数。其线性收敛速度较慢，但在某些情境下更加稳健，且推导更简单。本质上，牛顿–拉弗森法可视为一种特殊选择的不动点迭代，其中 g(x) = x – f(x)/f′(x)，旨在实现快速收敛。
Iterative methods transform the problem of solving equations from an algebraic puzzle into a computational process. By understanding the convergence condition |g′(x)| < 1, choosing a good rearrangement, selecting a sensible initial value, and applying appropriate stopping criteria, we can reliably approximate roots of equations that otherwise resist exact solution. Mastery of these techniques is an essential component of A-Level numerical methods and provides a foundation for more advanced computational mathematics.
迭代法将求解方程的问题从代数谜题转化为计算过程。通过理解收敛条件 |g′(x)| < 1、选择合适的变形、选取合理的初始值以及应用恰当的终止判据，我们能可靠地逼近那些否则无法精确求解的方程的根。掌握这些技术是 A-Level 数值方法的重要组成部分，也为更高级的计算数学奠定了基础。
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