Edexcel Maths: Numerical Methods Revision | Edexcel 数学：数值方法考点精讲

📚 Edexcel Maths: Numerical Methods Revision | Edexcel 数学：数值方法考点精讲

Numerical methods are essential tools in A-level Mathematics for approximating solutions to equations and definite integrals that cannot be solved analytically. In Edexcel exams, questions often require you to apply iteration formulas, trapezoidal rule, or Simpson’s rule, and to understand convergence and error bounds.

在 A-level 数学中，数值方法是近似求解解析法无法处理的方程和定积分的必备工具。在 Edexcel 考试中，常需应用迭代公式、梯形法则或辛普森法则，并理解收敛性和误差界。

1. Iterative Methods for Solving Equations | 解方程的迭代法

An iterative method generates a sequence of approximations x₀, x₁, x₂, … that approach a root of f(x)=0. We start from an initial guess x₀ and repeatedly apply an iteration formula.

迭代法生成一列近似值 x₀, x₁, x₂, …，逐步逼近 f(x)=0 的根。我们从初始猜测值 x₀ 开始，反复使用迭代公式。

The general fixed-point iteration is xₙ₊₁ = g(xₙ), where g(x) is obtained by rearranging f(x)=0 into the form x = g(x).

一般的不动点迭代形式为 xₙ₊₁ = g(xₙ)，其中 g(x) 由 f(x)=0 变形为 x = g(x) 得到。

For convergence, we need |g'(x)| < 1 near the root. If |g'(x)| ≥ 1, the sequence may diverge.

收敛条件要求根附近 |g'(x)| < 1。若 |g'(x)| ≥ 1，序列可能发散。

xₙ₊₁ = g(xₙ)

Example: To solve x³ − 5x + 1 = 0, one possible rearrangement is x = (x³ + 1)/5, so g(x) = (x³ + 1)/5.

例如：求解 x³ − 5x + 1 = 0，一种变形为 x = (x³ + 1)/5，于是 g(x) = (x³ + 1)/5。

2. Newton-Raphson Method | 牛顿-拉弗森法

The Newton-Raphson method uses the tangent line to improve an estimate. The formula is derived from the linear approximation of f(x).

牛顿-拉弗森法利用切线来改进估计值，其公式源自 f(x) 的线性近似。

xₙ₊₁ = xₙ − f(xₙ)/f'(xₙ)

This method converges quadratically (very fast) if the initial guess is sufficiently close to the root and f'(x) ≠ 0.

若初始猜测足够接近根且 f'(x) ≠ 0，该法二次收敛（速度很快）。

In Edexcel exams, you may be given an iteration formula and asked to apply it, or to show how it is derived from a given equation. Always use the exact derivative, not a numerical approximation.

Edexcel 考试中，你可能需要应用给定的迭代公式，或展示如何从方程导出该公式。务必使用精确导数，而非数值近似。

Be careful when f'(xₙ) is very small — the method may overshoot or fail.

当 f'(xₙ) 非常小时要小心，方法可能跳离或失效。

3. Bisection Method | 二分法

The bisection method works by repeatedly halving an interval [a, b] that contains a sign change of f(x), i.e., f(a) × f(b) < 0.

二分法通过反复对分含 f(x) 符号变化的区间 [a, b] 来工作，即 f(a) × f(b) < 0。

At each step, compute the midpoint c = (a+b)/2. If f(c) ≈ 0 (within tolerance), stop. Otherwise, replace either a or b with c, maintaining the sign change.

每一步计算中点 c = (a+b)/2。若 f(c) ≈ 0（在容差内），停止。否则将 a 或 b 替换为 c，保持符号变化。

The interval width halves each iteration, so after n steps the error is at most (b−a)/2ⁿ. This guarantees linear convergence but is slower than Newton-Raphson.

区间宽度每次迭代减半，因此 n 步后误差至多为 (b−a)/2ⁿ。这保证了线性收敛，但比牛顿法慢。

Bisection is robust — it always converges provided f is continuous — and is often used as a backup method.

二分法很稳健——只要 f 连续就一定收敛——常作为备份方法使用。

4. Trapezoidal Rule | 梯形法则

The trapezoidal rule approximates a definite integral ∫ₐᵇ f(x) dx by dividing the area into n equal-width strips of width h = (b−a)/n and using trapezia.

梯形法则将定积分 ∫ₐᵇ f(x) dx 近似为 n 个等宽条带面积之和，带宽 h = (b−a)/n，并使用梯形。

∫ₐᵇ f(x) dx ≈ (h/2)[y₀ + 2(y₁ + y₂ + … + yₙ₋₁) + yₙ]

where yᵢ = f(xᵢ) and xᵢ = a + i·h.

其中 yᵢ = f(xᵢ)，xᵢ = a + i·h。

In Edexcel questions, you must show the formula substitution, often with a table of y-values. Always use the exact h and list the y-values correctly.

Edexcel 题目中，你需要展示公式代入，常需列出 y 值表格。务必使用精确 h，并正确列出 y 值。

Tip: For a quick check, the trapezoidal rule usually overestimates when the curve is concave up, and underestimates when concave down.

技巧：快速检验时，当曲线凹向上时梯形法则通常高估，凹向下时低估。

5. Simpson’s Rule | 辛普森法则

Simpson’s rule gives a more accurate approximation by fitting quadratic arcs. It requires an even number of strips n (i.e., odd number of ordinates).

辛普森法则通过拟合二次弧段给出更精确的近似，要求 n 为偶数（即纵标数目为奇数）。

∫ₐᵇ f(x) dx ≈ (h/3)[y₀ + 4y₁ + 2y₂ + 4y₃ + 2y₄ + … + yₙ]

The pattern of coefficients is 1, 4, 2, 4, 2, …, 4, 1. Again, h = (b−a)/n.

系数模式为 1, 4, 2, 4, 2, …, 4, 1。同样，h = (b−a)/n。

When the function is a polynomial of degree ≤ 3, Simpson’s rule gives the exact integral (within rounding). For other functions, the error is O(h⁴).

当函数为次数 ≤3 的多项式时，辛普森法则给出精确积分（忽略舍入误差）。对于其他函数，误差为 O(h⁴)。

Edexcel often asks you to apply Simpson’s rule with a given number of intervals and to comment on accuracy. Ensure you correctly alternate coefficients.

Edexcel 常要求用给定区间数应用辛普森法则并评论精度。务必正确交替系数。

6. Error Estimation | 误差估计

For the trapezoidal rule, the theoretical error bound is

|Eₜ| ≤ (b−a)³/(12n²) · max|f”(x)| for x∈[a,b]

梯形法则的理论误差界为

|Eₜ| ≤ (b−a)³/(12n²) · max|f”(x)|，x∈[a,b]

For Simpson’s rule, the error bound involves the fourth derivative:

|Eₛ| ≤ (b−a)⁵/(180n⁴) · max|f⁽⁴⁾(x)|

辛普森法则误差界涉及四阶导数：

|Eₛ| ≤ (b−a)⁵/(180n⁴) · max|f⁽⁴⁾(x)|

In iterative methods, you can estimate error by observing |xₙ₊₁ − xₙ|. When this difference is less than the required tolerance, the approximation is likely accurate enough.

在迭代法中，可通过观察 |xₙ₊₁ − xₙ| 估计误差。当该差值小于要求的容差时，近似值可能已足够精确。

Edexcel questions may ask you to justify the number of iterations or intervals needed to achieve a given accuracy. Use the error formulas to find n.

Edexcel 题目可能要求论证达到给定精度所需的迭代次数或区间数。使用误差公式求 n。

7. Convergence and Order | 收敛性与收敛阶

A sequence converges if xₙ → α as n → ∞. The order of convergence describes the speed: linear if |xₙ₊₁ − α| ≤ c|xₙ − α| for c<1; quadratic if |xₙ₊₁ − α| ≤ M|xₙ − α|².

序列收敛是指当 n → ∞ 时 xₙ → α。收敛阶描述速度：线性：|xₙ₊₁ − α| ≤ c|xₙ − α|，c<1；二次：|xₙ₊₁ − α| ≤ M|xₙ − α|²。

Newton-Raphson exhibits quadratic convergence (doubling correct digits each step) under suitable conditions, while fixed-point iteration often shows linear convergence.

在适当条件下，牛顿-拉弗森法呈二次收敛（每步正确位数翻倍），而不动点迭代通常为线性收敛。

Bisection always has linear convergence with constant factor 1/2 per step — very predictable but slow.

二分法始终具有线性收敛，每步缩放因子 1/2——非常可预测但缓慢。

For fixed-point iteration, the magnitude of g'(α) determines the convergence rate: if |g'(α)| < 1, convergence is linear with factor ≈ |g'(α)|.

对不动点迭代，|g'(α)| 的大小决定收敛速率：若 |g'(α)| < 1，收敛是线性的，因子 ≈ |g'(α)|。

8. Application Tips and Common Mistakes | 应用技巧与常见错误

Always check that your iteration formula rearranges correctly from the given equation, and that the root lies within the interval you intend to use.

始终检查迭代公式是否从给定方程正确变形，以及根是否在你打算使用的区间内。

For Newton-Raphson, double-check the derivative — an algebraic slip can break the method. Use exact values or high precision during calculations; do not round intermediate steps too early.

对于牛顿法，仔细核对导数——代数错误可使方法失效。计算时使用精确值或高精度；不要过早舍入中间步骤。

When applying trapezoidal or Simpson’s rules, set out a table of x and y values clearly. For Simpson, be certain n is even — if not, the formula is invalid.

应用梯形或辛普森法则时，清晰列出 x 和 y 值表格。对辛普森，务必确保 n 为偶数——否则公式无效。

If an exam question asks ‘show that the iteration formula is …’ or ‘use this to find an approximation’, show all substitutions and at least 4–5 decimal places.

若考题要求“证明迭代公式为…”或“用此求近似值”，展示所有代入，至少保留 4–5 位小数。

Do not confuse the error bound formulas for trapezoidal and Simpson — they involve different powers of n and derivatives. Know which is which.

不要混淆梯形和辛普森的误差界公式——它们涉及不同次方的 n 和导数。区分清楚。

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Edexcel Maths: Numerical Methods Revision | Edexcel 数学：数值方法 考点精讲