📚 Numerical Methods | 数值方法
Numerical methods are essential tools for finding approximate solutions to equations and integrals when exact analytical methods are difficult or impossible. In the Edexcel A-Level Mathematics specification, this topic focuses on locating roots, using iterative formulas, applying the Newton-Raphson method, and estimating areas with the trapezium rule.
数值方法是在精确解析方法难以或无法使用时，寻找方程和积分近似解的重要工具。在 Edexcel A-Level 数学考试大纲中，本专题重点关注根的定位、迭代公式的使用、牛顿-拉弗森方法的应用，以及用梯形法则估算面积。
1. What Are Numerical Methods? | 什么是数值方法？
Numerical methods produce approximate answers by repeated calculation. They are used when an equation cannot be solved algebraically, or when a function has no elementary antiderivative. The methods studied in Edexcel Pure Mathematics are iterative and can be implemented efficiently on a calculator or computer.
数值方法通过重复计算得到近似答案。当方程无法用代数方法求解，或函数没有初等原函数时，就需要使用数值方法。Edexcel 纯数学中学习的数值方法都是迭代型的，可以在计算器或计算机上高效执行。
In an exam, you are expected to know both how to use these methods and why they work. You must also understand their limitations, especially when a method fails to converge or gives a misleading result.
在考试中，你既要会使用这些方法，也要理解它们为什么有效。你还必须了解它们的局限性，尤其是当方法无法收敛或给出误导性结果时。
2. Locating Roots by Change of Sign | 用符号变化定位根
If a function f(x) is continuous on the interval [a, b] and f(a) and f(b) have opposite signs, then there is at least one root of f(x) = 0 in the interval (a, b). This is known as the sign-change rule.
如果函数 f(x) 在区间 [a, b] 上连续，且 f(a) 与 f(b) 符号相反，那么 f(x) = 0 在区间 (a, b) 内至少有一个根。这就是符号变化法则。
f(a) × f(b) < 0 ⇒ ∃ α ∈ (a, b) such that f(α) = 0
To prove that a root lies between two values, evaluate the function at both endpoints and state that the sign change and continuity guarantee the existence of a root. For example, for f(x) = x³ − 2x − 5, we have:
要证明两个值之间存在根，需要计算函数在这两个端点的值，并说明符号变化和连续性保证了根的存在。例如，对于 f(x) = x³ − 2x − 5，我们有：
f(2) = 2³ − 2×2 − 5 = −1 < 0 and f(3) = 3³ − 2×3 − 5 = 16 > 0
Therefore, f(x) = 0 has a root in the interval (2, 3). The actual root is approximately 2.0946.
因此，f(x) = 0 在区间 (2, 3) 内有一个根。实际根约为 2.0946。
3. Limitations of Sign-Change Methods | 符号变化法的局限
A sign change is sufficient to prove the existence of at least one root, but it is not necessary. If f(a) and f(b) have the same sign, the interval may still contain 0, 2, 4, or another even number of roots.
符号变化是证明至少存在一个根的充分条件，但不是必要条件。如果 f(a) 和 f(b) 同号，该区间仍可能包含 0、2、4 或另一个偶数个根。
Another important limitation is that a discontinuity can produce a sign change without a root. For example, the function f(x) = 1/x changes sign between x = −1 and x = 1, but there is no root because the function is not continuous at x = 0.
另一个重要局限是，不连续可能导致符号变化但没有根。例如，函数 f(x) = 1/x 在 x = −1 与 x = 1 之间符号发生改变，但由于函数在 x = 0 处不连续，因此没有根。
Edexcel questions often ask you to explain why a sign change in a table of values may not guarantee a unique root, or why further investigation is needed when the graph has a vertical asymptote.
Edexcel 试题经常要求你解释为什么数值表中的符号变化不能保证唯一根，或者当图像存在竖直渐近线时为什么需要进一步研究。
4. Fixed Point Iteration | 不动点迭代
To solve an equation of the form f(x) = 0 using fixed point iteration, first rearrange it into the form x = g(x). Starting from an initial value x₀, the iteration formula is applied repeatedly:
要使用不动点迭代法求解 f(x) = 0 形式的方程，首先将它改写为 x = g(x)。从初始值 x₀ 开始，反复应用迭代公式：
xₙ₊₁ = g(xₙ)
Each new value is substituted back into g(x) to produce the next value. If the process settles on a fixed value α, then α satisfies α = g(α), which means f(α) = 0.
每个新值都会被代回 g(x) 以产生下一个值。如果该过程稳定在某个固定值 α 上，那么 α 满足 α = g(α)，这意味着 f(α) = 0。
There are usually several possible rearrangements of the same equation. One rearrangement may converge quickly, another may converge slowly, and another may diverge entirely.
同一个方程通常有多种可能的改写形式。一种改写可能收敛很快，另一种可能收敛很慢，还有一种可能完全发散。
5. Convergence of Fixed Point Iteration | 不动点迭代的收敛性
The behaviour of fixed point iteration near a root α depends on the derivative of g(x). If |g′(x)| < 1 in a neighbourhood of α, the iteration converges to α. If |g′(x)| > 1, the iteration moves away from α and diverges.
不动点迭代在根 α 附近的行为取决于 g(x) 的导数。如果在 α 的邻域内 |g′(x)| < 1，迭代会收敛到 α；如果 |g′(x)| > 1，迭代会远离 α 并发散。
|g′(α)| < 1 ⇒ iteration converges
When 0 < g′(α) < 1, the iteration approaches the root from one side, producing a staircase diagram. When −1 < g′(α) < 0, the values alternate on either side of the root, producing a cobweb diagram.
当 0 < g′(α) < 1 时，迭代从一侧接近根，形成阶梯图。当 −1 < g′(α) < 0 时，数值在根的两侧交替，形成蛛网图。
In an exam, you may be asked to choose a suitable rearrangement by checking the derivative near the root. A rearrangement with a derivative close to zero usually converges more rapidly.
在考试中，你可能会被要求通过检查根附近的导数来选择合适的改写形式。导数接近零的改写通常收敛更快。
6. Newton-Raphson Method | 牛顿-拉弗森方法
The Newton-Raphson method uses tangents to approximate the root of f(x) = 0. Starting from x₀, the next approximation is the point where the tangent to y = f(x) at x₀ crosses the x-axis.
牛顿-拉弗森方法利用切线来逼近 f(x) = 0 的根。从 x₀ 开始，下一个近似值就是 y = f(x) 在 x₀ 处的切线与 x 轴的交点。
xₙ₊₁ = xₙ − f(xₙ) / f′(xₙ)
For the equation f(x) = x³ − 2x − 5, the derivative is f′(x) = 3x² − 2, so the Newton-Raphson formula becomes:
对于方程 f(x) = x³ − 2x − 5，其导数为 f′(x) = 3x² − 2，因此牛顿-拉弗森公式变为：
xₙ₊₁ = xₙ − (xₙ³ − 2xₙ − 5) / (3xₙ² − 2)
Starting with x₀ = 2, the method produces x₁ ≈ 2.1, x₂ ≈ 2.0946, which is already correct to four decimal places. This illustrates the quadratic convergence often seen when the starting point is close to the root.
从 x₀ = 2 开始，该方法得到 x₁ ≈ 2.1，x₂ ≈ 2.0946，已经精确到小数点后四位。这说明当起点接近根时，牛顿-拉弗森方法通常具有二次收敛速度。
7. When Newton-Raphson Fails | 牛顿-拉弗森方法的失效情形
Newton-Raphson is powerful, but it does not always work. One failure occurs when f′(xₙ) = 0 at the current approximation, because the tangent is horizontal and never meets the x-axis.
牛顿-拉弗森方法很强大，但并不总是有效。一种失效情况发生在当前近似值处 f′(xₙ) = 0 时，因为此时切线是水平的，永远不会与 x 轴相交。
Another failure occurs if the starting value is far from the root or if the function has a local maximum or minimum near the root. The iteration may converge to a different root or oscillate without settling.
另一种失效情况发生在起始值离根太远，或者函数在根附近有局部极大值或极小值时。迭代可能收敛到另一个根，或者振荡而无法稳定。
Some functions can also cause the iteration to enter a cycle. For example, certain choices of x₀ may lead to x₁ = x₃ and x₂ = x₄, so the sequence repeats instead of converging.
某些函数还会使迭代进入循环。例如，某些 x₀ 的选择可能导致 x₁ = x₃ 且 x₂ = x₄，于是序列重复而不是收敛。
In exam questions, you may be asked to explain why the Newton-Raphson formula cannot be used at a given value, usually by showing that f′(x₀) = 0.
在试题中，你可能会被要求解释为什么在给定值处不能使用牛顿-拉弗森公式，通常是通过证明 f′(x₀) = 0。
8. Numerical Integration: The Trapezium Rule | 数值积分：梯形法则
The trapezium rule estimates the area under a curve by dividing it into n strips of equal width and approximating each strip by a trapezium. The width of each strip is:
梯形法则通过将曲线下方面积分割为 n 个等宽条带，并用梯形近似每个条带来估算面积。每个条带的宽度为：
h = (b − a) / n
If the function values at the ordinates are y₀, y₁, y₂, …, yₙ, the trapezium rule gives:
如果各纵坐标处的函数值为 y₀, y₁, y₂, …, yₙ，梯形法则给出：
∫ₐᵇ f(x) dx ≈ h/2 [y₀ + 2(y₁ + y₂ + … + yₙ₋₁) + yₙ]
The first and last ordinates are used once, while every internal ordinate is used twice. This is why y₀ and yₙ are multiplied by 1, but all intermediate values are multiplied by 2.
第一个和最后一个纵坐标使用一次，而每个内部纵坐标使用两次。这就是为什么 y₀ 和 yₙ 乘以 1，而所有中间值都乘以 2。
In Edexcel exams, the trapezium rule is often applied to functions that cannot be integrated exactly, or to data given in a table. Always show the full substitution into the formula to gain method marks.
在 Edexcel 考试中，梯形法则常用于无法精确积分的函数，或表格中给出的数据。始终要展示完整代入公式的过程，以获得方法分。
9. Accuracy and Error in the Trapezium Rule | 梯形法则的精度与误差
Increasing the number of strips reduces the width h and usually improves the accuracy of the trapezium rule. However, more strips require more calculations and may still not give an exact answer.
增加条带数量会减小宽度 h，通常会提高梯形法则的精度。然而，条带越多，计算量越大，而且结果可能仍不精确。
The direction of the error depends on the curvature of the graph. If f(x) is convex, meaning f″(x) > 0, the trapezium rule gives an overestimate. If f(x) is concave, meaning f″(x) < 0, it gives an underestimate.
误差的方向取决于曲线的弯曲方向。如果 f(x) 是凸函数，即 f″(x) > 0，梯形法则会给出高估值。如果 f(x) 是凹函数，即 f″(x) < 0，它会给出低估值。
You may be asked in the exam whether the trapezium rule gives an overestimate or underestimate, and to justify your answer by referring to the shape of the graph or the sign of f″(x).
考试中你可能会被问到梯形法则给出的是高估还是低估，并要求你通过图像的形状或 f″(x) 的符号来证明答案。
10. Choosing and Combining Methods | 选择与综合运用数值方法
The three main numerical techniques are often used together in a structured problem. A typical Edexcel question might ask you to show that a root exists in a given interval using sign change, then use an iteration or Newton-Raphson to find the root to a required accuracy, and finally use the trapezium rule to estimate a related integral.
三种主要的数值技巧通常在一个结构化问题中综合使用。一道典型的 Edexcel 题目可能会要求你用符号变化证明某区间内存在根，然后用迭代法或牛顿-拉弗森方法求出达到所需精度的根，最后用梯形法则估算相关的积分。
| Method 方法 | Main Use 主要用途 | Key Condition 关键条件 |
|---|---|---|
| Sign change 符号变化 | Locate an interval containing a root 定位包含根的区间 | f(x) continuous 连续 |
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