📚 AS Maths: Numerical Methods Revision | AS 数学：数值方法 考点精讲
Numerical methods are a collection of techniques used to find approximate solutions to mathematical problems that cannot easily be solved analytically. In AS Mathematics, you will encounter methods for solving equations and estimating integrals, as well as understanding the iterative processes behind them. This guide covers all the essential topics for the numerical methods section, from locating roots using sign changes to the Newton-Raphson method and the trapezium rule, with clear explanations and exam-focused tips.
数值方法是一系列用于求解难以解析求解的数学问题的近似解的技巧。在 AS 数学中，你将学习求解方程和估计积分的方法，并理解其背后的迭代过程。本指南涵盖了数值方法部分的所有重要考点，从利用符号变化定位根、牛顿-拉夫森法到梯形法则，并配有清晰的解释和针对考试的技巧。
1. What Are Numerical Methods? | 什么是数值方法？
Numerical methods are algorithms that produce successively better approximations to an exact solution. They are particularly useful when equations cannot be solved using algebraic manipulation alone, for example, finding x such that eˣ = 3x + 1. In AS level, we focus on solving f(x) = 0 and numerical integration. These methods rely on iteration, convergence, and an understanding of errors, and are often implemented on calculators or computers.
数值方法是能够逐步产生越来越接近精确解的近似值的算法。当方程无法通过代数操作直接求解时，它们特别有用，例如求解 eˣ = 3x + 1。在 AS 阶段，我们主要关注 f(x) = 0 的求解和数值积分。这些方法依赖于迭代、收敛以及对误差的理解，通常通过计算器或计算机来实现。
2. Locating Roots: Change of Sign | 利用符号变化定位根
If 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 the equation f(x) = 0 in the interval (a, b). This is a direct consequence of the Intermediate Value Theorem. Always check that the function is continuous – a change of sign guarantees a root only if there is no break in the curve. To show that a root lies in a given interval, evaluate f at the endpoints and verify that one is positive and the other negative.
如果 f(x) 在区间 [a, b] 上连续，且 f(a) 与 f(b) 符号相反，则方程 f(x) = 0 在区间 (a, b) 内至少有一个根。这是介值定理的直接推论。务必检查函数是否连续——只有在曲线没有间断的情况下，符号改变才能保证根的存在。要证明某个区间内存在根，只需计算端点的函数值，并确认一个为正、另一个为负。
For example, to show that the equation x³ – 5x + 3 = 0 has a root between 1 and 2: f(1) = 1 – 5 + 3 = –1 (negative), f(2) = 8 – 10 + 3 = +1 (positive). Since f is continuous and the sign changes, a root exists in (1, 2).
例如，要证明方程 x³ – 5x + 3 = 0 在 1 和 2 之间有一个根：f(1) = 1 – 5 + 3 = –1（负），f(2) = 8 – 10 + 3 = +1（正）。由于 f 连续且符号改变，因此在 (1, 2) 内存在一个根。
3. Simple Iteration and Staircase/Cobweb Diagrams | 简单迭代与阶梯图/蛛网图
An iterative formula of the form xₙ₊₁ = g(xₙ) can be used to find an approximate solution to the equation f(x) = 0 by rearranging it into the form x = g(x). Starting with an initial value x₀, we repeatedly apply the formula to generate a sequence x₁, x₂, x₃, … If this sequence converges to a limit L, then L is a solution of the original equation. The geometry of iteration can be illustrated with staircase and cobweb diagrams, obtained by drawing y = x and y = g(x) and tracing vertical and horizontal lines from the starting point.
形如 xₙ₊₁ = g(xₙ) 的迭代公式可用于求方程 f(x) = 0 的近似解，方法是将方程重组为 x = g(x) 的形式。从一个初始值 x₀ 开始，反复使用该公式生成序列 x₁、x₂、x₃……如果该序列收敛到某个极限 L，则 L 就是原方程的解。迭代的几何意义可用阶梯图和蛛网图来说明，即先画出 y = x 和 y = g(x)，然后从起点开始作竖直和水平线。
The convergence of the iteration depends on the gradient of g(x) near the root. If |g'(x)| < 1 in the neighbourhood of the root, the iteration will converge; if |g'(x)| > 1, it diverges. Always rearrange f(x) = 0 so that the derivative condition holds for the intended starting value.
迭代的收敛性取决于 g(x) 在根附近的导数。如果在根附近 |g'(x)| < 1，迭代就会收敛；如果 |g'(x)| > 1，则会发散。务必对 f(x) = 0 进行重组，使得在预期的起始值处满足这一导数条件。
4. The Newton-Raphson Method | 牛顿-拉夫森法
The Newton-Raphson method is a powerful iteration for finding roots of f(x) = 0. It uses the formula:
xₙ₊₁ = xₙ – f(xₙ) / f'(xₙ)
牛顿-拉夫森法是一种求解 f(x) = 0 的根的强大迭代方法。其公式为：
xₙ₊₁ = xₙ – f(xₙ) / f'(xₙ)
Starting from an initial guess x₀, each iteration produces a better approximation by following the tangent to the curve at xₙ down to the x-axis. The method converges extremely quickly when the initial guess is reasonably close to the true root and f'(x) is not zero. It usually achieves quadratic convergence, meaning the number of correct digits roughly doubles with each iteration.
从初始猜测值 x₀ 开始，每次迭代通过追踪曲线在 xₙ 处的切线到 x 轴，产生一个更好的近似值。当初始猜测值足够接近真实根且 f'(x) 不为零时，该方法收敛速度极快。它通常能实现二次收敛，即每次迭代正确的数字位数大约翻倍。
Example: to find √2, we solve f(x) = x² – 2 = 0. Then f'(x) = 2x, so the iteration becomes xₙ₊₁ = xₙ – (xₙ² – 2)/(2xₙ). With x₀ = 1, we get x₁ = 1.5, x₂ = 1.41666…, converging rapidly to 1.41421356…
示例：求 √2，则求解 f(x) = x² – 2 = 0。f'(x) = 2x，因此迭代公式为 xₙ₊₁ = xₙ – (xₙ² – 2)/(2xₙ)。取 x₀ = 1，可得 x₁ = 1.5，x₂ = 1.41666…，迅速收敛到 1.41421356…
5. When Newton-Raphson Fails | 牛顿-拉夫森法的失效情形
The Newton-Raphson method can fail in several situations: if the initial guess is near a stationary point where f'(x) ≈ 0, the tangent is nearly horizontal and the next approximation shoots far away; if the root is a multiple root, convergence becomes slow; or if the iteration enters a cycle without approaching the root. Awareness of these pitfalls is key in exam questions that ask you to explain why the method may not be suitable.
牛顿-拉夫森法在以下几种情况可能失效：初始猜测值位于驻点附近，此时 f'(x) ≈ 0，切线几乎水平，下一个近似值会跑得很远；如果根是多重根，收敛会变慢；或者迭代进入循环而无法趋近于根。了解这些陷阱对于解答要求解释该方法为何可能不合适的考题至关重要。
A typical exam problem gives an example where x₀ is chosen such that f'(x₀) is very small, causing x₁ to be undefined or far from the desired root. You should be able to identify this from the function values and explain the geometric reasoning.
常见的考题会给出这样的例子：所选 x₀ 使得 f'(x₀) 非常小，导致 x₁ 无定义或远离所需的根。你应当能够根据函数值识别这一点，并从几何角度加以解释。
6. Numerical Integration: The Trapezium Rule | 数值积分：梯形法则
When an integral is difficult to evaluate analytically, we can approximate its value using the trapezium rule. For a function y = f(x) on the interval [a, b] divided into n strips of equal width h = (b – a)/n, the approximate area is given by:
A ≈ (h/2)[y₀ + yₙ + 2(y₁ + y₂ + … + yₙ₋₁)]
其中 y₀ = f(x₀), y₁ = f(x₁), …, yₙ = f(xₙ) 且 x₀ = a, xₙ = b。
当积分难以解析计算时，我们可以用梯形法则近似其值。对于区间 [a, b] 上的函数 y = f(x)，将其等分为 n 个宽度为 h = (b – a)/n 的小条，近似面积由下式给出：
A ≈ (h/2)[y₀ + yₙ + 2(y₁ + y₂ + … + yₙ₋₁)]
where y₀ = f(x₀), y₁ = f(x₁), …, yₙ = f(xₙ) and x₀ = a, xₙ = b.
The rule works by approximating the area under the curve with a series of trapeziums. The greater the number of strips, the more accurate the approximation, assuming the function is smooth. In exams, you will often be asked to apply the rule with a given number of ordinates (usually 3, 4, or 5) and to use the result to estimate the integral.
该法则通过用一系列梯形近似曲线下的面积来工作。假设函数光滑，条数越多，近似值越精确。在考试中，你通常会被要求使用给定数量的纵坐标（通常 3、4 或 5 个）应用此法则，并用结果估计积分值。
7. Over- and Under-estimates of the Trapezium Rule | 梯形法则的高估与低估
Whether the trapezium rule gives an overestimate or underestimate depends on the curvature of the graph. If the function is convex (curving upwards) on the interval, i.e. f”(x) > 0, the trapeziums will lie above the curve, producing an overestimate. If the function is concave (curving downwards), f”(x) < 0, the trapeziums lie below, giving an underestimate. For a straight line, the rule gives the exact area.
梯形法则是高估还是低估取决于曲线的凹凸性。如果函数在区间上是凸的（向上弯曲），即 f”(x) > 0，梯形会位于曲线上方，导致高估。如果函数是凹的（向下弯曲），f”(x) < 0，梯形位于下方，导致低估。对于直线，该法则给出精确面积。
You can be asked to state whether a particular trapezium rule value is an overestimate or an underestimate by examining the shape of the graph or using a second derivative sign. A sketch is often helpful to support your reasoning.
题目可能会要求你通过观察图形形状或利用二阶导数的符号，判断某个梯形法则值是高估还是低估。画一个草图通常有助于支撑你的推理。
8. Improving Accuracy and Error Bounds | 提高精度与误差界
To improve the accuracy of a numerical method, you can increase the number of iterations (for equation solving) or increase the number of strips (for integration). In the Newton-Raphson method, providing a better initial guess also improves convergence. For the trapezium rule, doubling the number of strips roughly quarters the error, provided the function is sufficiently smooth.
要提高数值方法的精度，可以增加迭代次数（对于方程求解）或增加条数（对于积分）。在牛顿-拉夫森法中，提供更好的初始猜测值也能改善收敛性。对于梯形法则，只要函数足够光滑，将条数翻倍大约可使误差减小到原来的四分之一。
The error in the trapezium rule can be bounded by the formula |E| ≤ (b–a)³/(12n²) M, where M is the maximum value of |f”(x)| on [a, b]. Although derivation is not required at AS, you may be asked to find or use an error bound.
梯形法则的误差可用公式 |E| ≤ (b–a)³/(12n²) M 界定，其中 M 是 |f”(x)| 在 [a, b] 上的最大值。虽然 AS 阶段不要求推导，但你可能会被要求找出或使用误差界。
9. Practical Tips for Exam Questions | 考试题目实用技巧
When working through numerical methods questions, always write down the formula you are using and show a clear, step-by-step substitution. For iteration, keep values to a high degree of accuracy (at least 6 decimal places) to avoid rounding errors swamping the convergence. Label your iterations clearly: x₁, x₂, x₃, …
在做数值方法题目时，一定要写出所使用的公式，并展示清晰、逐步的代入过程。对于迭代，保留较高的精度（至少 6 位小数），以避免舍入误差破坏收敛性。清楚标出你的迭代值：x₁、x₂、x₃……
- For change of sign: state that f is continuous, give the values, and note the sign change.
- 符号变化法：声明 f 连续，给出函数值，并指出符号变化。
- Newton-Raphson: find f'(x) first, then use the formula; be careful with differentiation.
- 牛顿-拉夫森法：先求 f'(x)，然后使用公式；注意微分。
- Trapezium rule: organize your y-values in a table; remember to multiply the middle ordinates by 2.
- 梯形法则：将 y 值整理成表格；记住中间的纵坐标要乘以 2。
- Staircase/cobweb: clearly indicate whether the diagram shows convergence or divergence.
- 阶梯图/蛛网图：清楚表明该图是显示收敛还是发散。
Exam questions often combine a small part of algebraic rearrangement with several numerical computations, so practice switching smoothly between exact and approximate forms.
考题常常将一小部分代数重组与若干数值计算结合在一起，因此要练习在精确形式和近似形式之间流畅切换。
10. Summary of Key Formulas and Concepts | 关键公式与概念总结
| Concept / 概念 | Formula / Key Point / 公式/要点 |
| Change of sign / 符号变化 | f(a)f(b) < 0 ⇒ root in (a,b) for continuous f |
| Simple iteration / 简单迭代 | xₙ₊₁ = g(xₙ); converges if |g'(x)| < 1 near root |
| Newton-Raphson / 牛顿-拉夫森 | xₙ₊₁ = xₙ – f(xₙ)/f'(xₙ) |
| Trapezium rule / 梯形法则 | ∫ₐᵇ f(x)dx ≈ (h/2)[y₀ + yₙ + 2∑yᵢ] |
| Over/underestimate / 高估/低估 | f” > 0 ⇒ overestimate; f” < 0 ⇒ underestimate |
| Error bound / 误差界 | |E| ≤ (b–a)³/(12n²) max|f”(x)| |
Mastery of numerical methods requires both algebraic accuracy and a solid grasp of the geometric interpretation. Practise past paper questions to become fluent in recognising which method to apply and how to comment on its validity.
掌握数值方法既需要代数运算的准确性，又需要对几何解释的扎实理解。练习往年真题，以熟练识别该使用何种方法以及如何评价其有效性。
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