Locating Roots | 定位方程的根

📚 Locating Roots | 定位方程的根

In A-Level Mathematics, solving equations exactly is not always possible. Many real-world problems lead to equations that cannot be rearranged to give a neat algebraic solution. When we cannot find an exact answer, numerical methods allow us to home in on approximate solutions. The very first step in almost all numerical equation-solving is locating a root — determining roughly where a solution lies so that a more refined method can take over. This article explores the change-of-sign approach, interval bisection, and the underlying theory you need for Edexcel.

在A-Level数学中，很多方程无法精确求解。实际生活中的许多问题会导出无法整理出简洁代数解的方程。当无法得到精确解时，数值方法可以帮助我们逐步逼近近似解。差不多所有数值解方程的第一步都是“定位根”——先大致判定解落在哪个区间，然后才能用更精细的方法逼近。本文围绕Edexcel考纲，探讨变号法、区间二分法以及背后的理论基础。

1. What Does “Locating a Root” Mean? | “定位根”意味着什么？

Consider an equation f(x) = 0. A root is a value x = α such that f(α) = 0. Locating a root means finding an interval [a, b] within which we are confident a root lies. We do not need the exact value at this stage — only a narrow enough bracket to start a systematic search or an iterative method.

考虑方程 f(x)=0。根就是使得 f(α)=0 的 x=α。定位根就是找到一个区间 [a, b]，可以确信其中含有一个根。此时我们还不需要精确值，只需要一个足够窄的“括号”，以便开始系统搜索或迭代方法。

2. The Change-of-Sign Principle | 变号原理

If a continuous function f(x) changes sign between x = a and x = b, i.e. f(a) and f(b) have opposite signs, then there must be at least one root in the interval (a, b). This is the intuitive foundation of root‑locating.

如果一个连续函数 f(x) 在 x=a 和 x=b 处异号，即 f(a) 与 f(b) 符号相反，那么在区间 (a,b) 内至少存在一个根。这就是定位根最直观的依据。

We can express this sign change as f(a) × f(b) < 0. For example, if f(1) = −2 and f(2) = 3, the product is negative, so a root lies between 1 and 2.

我们可以用 f(a) × f(b) < 0 来表达这个变号。例如 f(1)=−2，f(2)=3，乘积为负，表明 1 和 2 之间存在一个根。

3. The Intermediate Value Theorem | 介值定理

The formal justification is the Intermediate Value Theorem (IVT): if f is continuous on the closed interval [a, b] and N is any number between f(a) and f(b), then there exists at least one c in (a, b) such that f(c) = N. For root locating we set N = 0, so if f(a) and f(b) straddle zero, we are guaranteed a root.

这一说法的严格依据是介值定理 (IVT)：若 f 在闭区间 [a,b] 上连续，且 N 是介于 f(a) 和 f(b) 之间的任意数，则至少存在一个 c∈(a,b) 使得 f(c)=N。定位根时取 N=0，所以只要 f(a) 与 f(b) 分居零的两侧，就一定存在根。

Continuity is essential — the IVT does not apply if the function has a gap, jump or asymptote in the interval. Always check that the function is well‑behaved over the range you are considering.

连续性是关键——若函数在区间内有间断、跳跃或渐近线，IVT 便不适用。务必确认所考察区间内函数是“良态”的。

4. Step-by-Step: Sign Change Method | 逐步：变号法

To locate a root of f(x) = 0 using a sign change, you can follow these steps:

用变号法定位 f(x)=0 的根，可以遵循以下步骤：

Choose two starting x‑values, say a and b, and evaluate f(a) and f(b).

选取两个初始 x 值 a 和 b，计算 f(a)、f(b)。
Check the sign of f(a) × f(b). If negative, a root lies between a and b. If positive, try a different pair or adjust the interval.

检查 f(a)×f(b) 的符号。若为负，则区间内含根；若为正，更换 x 值对或调整区间。
Once a sign change is confirmed, you can either proceed to bisection or feed the interval into an iterative formula. For Edexcel exams, you may be asked to show that a root lies between two given integers.

一旦确认变号，就可进入二分法或将区间代入迭代公式。Edexcel 考试常要求学生证明根位于某两个整数之间。

5. Example: f(x) = x³ − x − 1 | 例题：f(x) = x³ − x − 1

Take f(x) = x³ − x − 1. We try integer values: f(1) = 1 − 1 − 1 = −1; f(2) = 8 − 2 − 1 = 5. The sign changes from negative to positive, so there is a root between x = 1 and x = 2.

以 f(x)=x³ − x − 1 为例。尝试整数值：f(1)=1−1−1=−1；f(2)=8−2−1=5。符号由负变正，因此在 x=1 与 x=2 之间存在一个根。

To narrow the interval, we might test x = 1.5: f(1.5) = 3.375 − 1.5 − 1 = 0.875 (positive). Now the sign changes between x = 1 (negative) and x = 1.5 (positive), so the root is in [1, 1.5].

要缩小区间，可试 x=1.5：f(1.5)=3.375−1.5−1=0.875（正）。此时 x=1（负）与 x=1.5（正）之间变号，所以根位于 [1,1.5] 内。

6. Interval Bisection (Bisection Method) | 区间二分法

Bisection is the simplest systematic method for narrowing down a root. Once you have an interval [a, b] with f(a) × f(b) < 0, you repeatedly halve the interval. At each step, compute the midpoint m = ½(a + b) and evaluate f(m). Then replace either a or b with m, keeping the sub‑interval where the sign change persists.

二分法是一种最简单的系统性缩根方法。有了一个 f(a)×f(b)<0 的区间 [a,b] 后，重复将区间对半分。每一步计算中点 m=½(a+b) 及 f(m)，然后用 m 取代 a 或 b，保留仍然发生变号的那个子区间。

This guarantees the root stays trapped in an interval whose width halves each time. After n bisections, the interval width is (b − a) / 2ⁿ.

这确保了根始终被套在一个区间内，且区间宽度每步减半。经过 n 次二分后，区间宽度为 (b−a)/2ⁿ。

7. Performing Bisection Efficiently | 高效执行二分法

In practice, use a table to keep track of a, b, m, f(a), f(b), f(m) and the sign of f(m). A typical layout for the bisection of x³ − x − 1 starting with [1, 2] would look like this:

实际操作中可用表格跟踪 a、b、m、f(a)、f(b)、f(m) 以及 f(m) 的符号。以 [1,2] 开始二分 x³−x−1 的典型表格如下：

a	b	m	f(m)	New interval
1	2	1.5	0.875 (+)	[1, 1.5]
1	1.5	1.25	−0.296875 (−)	[1.25, 1.5]
1.25	1.5	1.375	0.2246… (+)	[1.25, 1.375]

Always check which sub‑interval retains the sign change — do not guess from the size of f(m).

始终要根据哪一侧保持变号来决定区间，不要凭 f(m) 的绝对值大小猜测。

8. Accuracy and Stopping Criteria | 精度与停止准则

A root located by bisection is typically given as the midpoint of the final interval, with an error bound of ± half the interval width. For example, after 10 bisections starting from [1,2], width = 1/2¹⁰ ≈ 0.00098, so the error is about ±0.0005. You can state the root correct to a certain number of decimal places when the interval is sufficiently narrow.

用二分法定位的根通常取最后区间的中点，误差界为 ± 半区间宽度。例如从 [1,2] 开始经过 10 次二分后，宽度约为 0.00098，误差约为 ±0.0005。当区间足够窄时，你可声明根精确到若干小数位。

Edexcel questions may ask you to carry out bisection to a specified accuracy, e.g. to show the root is 1.32 correct to 2 decimal places. This means the interval containing the root must be entirely within [1.315, 1.325).

Edexcel 考题可能要求通过二分法证明根精确到某一位，例如证明根为 1.32（精确至两位小数）。这意味着含根区间必须完全落在 [1.315,1.325) 之内。

9. Graphical Insight | 图形洞察

Plotting the graph of y = f(x) gives immediate visual clues about where roots might lie. A sign change between two x‑coordinates appears as the curve crossing the x‑axis. You can sometimes spot nearby integer bounds by looking at the graph.

画出 y=f(x) 的图像能迅速给出根所在区间的可视化线索。两 x 坐标间的变号即表现为曲线穿过 x 轴。有时通过看图就能直接找到邻近的整数界限。

Many calculators and software tools allow you to zoom in. In an exam, a sketch is sufficient to justify your initial choice of a and b, but the formal justification must come from evaluating f(a) and f(b).

很多计算器和软件工具都可以放大图形。在考试中，画出草图足以支撑你对 a 和 b 的初始选择，但正式的论证必须来自计算 f(a) 与 f(b)。

10. Common Pitfalls and Special Cases | 常见陷阱及特殊情况

Sign change is a sufficient but not necessary condition for a root. You can have a root without a sign change: a repeated root (where the graph touches but does not cross the x‑axis) gives f(a) and f(b) the same sign, yet a root exists. For example, f(x) = (x − 2)² has a root at x = 2, but f(1) = 1 and f(3) = 1 — no sign change.

变号是根存在的充分非必要条件。没有变号也可能有根：重根（图像与 x 轴相切而不穿过）会导致 f(a) 与 f(b) 同号，但根确实存在。例如 f(x)=(x−2)² 在 x=2 处有重根，但 f(1)=1，f(3)=1，没有变号。

Discontinuities can fake a sign change. If f(x) has a vertical asymptote between a and b, the function may change sign without having a root. Always check that f is continuous on [a, b] before applying the IVT.

间断点会伪装成变号。如果 f(x) 在 a、b 间有垂直渐近线，函数可能变号但并无根。应用 IVT 前一定要确认 f 在 [a,b] 上连续。

More than one root in the interval: a sign change only guarantees an odd number of roots if the function is continuous. You might have 1, 3, 5, … roots. Bisection will converge to one of them, but you may need to subdivide if multiple roots are suspected.

区间内有多个根：连续函数变号只保证有奇数个根（可能是1,3,5…个）。二分法会收敛到其中一个，但若怀疑有多根，可能需要进一步细分。

11. Linking to Iterative Methods (Edexcel Context) | 与迭代方法的联系（Edexcel背景）

In Edexcel Pure Mathematics, locating a root is usually the first part of a question on numerical methods. Once an interval is established, you may be asked to use an iterative formula such as xₙ₊₁ = g(xₙ) or the Newton‑Raphson method. A good initial approximation from bisection improves the chance of rapid convergence.

在Edexcel纯数学中，定位根通常是数值方法题目的第一部分。确立区间以后，考题可能让你使用迭代公式，如 xₙ₊₁=g(xₙ) 或牛顿‑拉夫森法。通过二分法获得一个好的初始近似，可以提高快速收敛的机会。

When writing an iterative formula, you are often rearranging f(x)=0 into the form x = g(x). The root you located ensures that the iteration starts near the intended solution, avoiding divergence to another root.

在构造迭代公式时，通常要把 f(x)=0 改写为 x=g(x) 的形式。你定位到的根可以确保迭代起点落在目标解附近，避免发散到别的根。

12. Exam Tips and Summary | 考试提示与总结

Key points to remember for Edexcel exams:

Edexcel 考试需牢记的要点：

Always state clearly that a sign change implies a root, and mention continuity. Write: “Since f(a) < 0 and f(b) > 0 and f is continuous, there is a root in [a, b].”

务必明确说明变号意味着根的存在，并提及连续性。写下：“因为 f(a)<0 且 f(b)>0，且 f 连续，故 [a,b] 内存在一个根。”
For bisection, keep values to enough decimal places to avoid rounding errors, but round the final answer sensibly.

进行二分法计算时，保留足够的小数位以避免舍入误差累积，但最终答案应合理舍入。
If asked to find a root correct to, say, 2 decimal places, continue bisection until the entire interval is within ±0.005 of the candidate decimal.

若要求根精确到两位小数，应继续二分直到整个区间落在候选小数的 ±0.005 范围内。
Use a table to show your working — it demonstrates method and makes checking easier.

用表格展示计算过程——既能体现方法又便于检查。
Watch out for discontinuities and repeated roots. The IVT requires continuity; a missing sign change does not necessarily mean no root.

提防间断点和重根。IVT 以连续性为前提；没有变号不一定没有根。

Locating roots is a fundamental numerical skill. Mastering the sign‑change test and bisection lays the groundwork for all the iterative techniques that follow.

定位根是一项基本的数值计算技能。掌握变号检验和二分法，将为后续所有迭代技术打下坚实基础。

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Locating Roots | 定位方程的根