Numerical Methods and Their Applications | 数值方法及其应用

📚 Numerical Methods and Their Applications | 数值方法及其应用

In A-Level Mathematics, numerical methods provide powerful techniques for solving equations and evaluating integrals that cannot be handled by analytical methods alone. This article explores the core numerical methods in the syllabus — root location, interval bisection, fixed-point iteration, the Newton-Raphson method, and numerical integration — together with their applications and common pitfalls.

在A-Level数学中，数值方法为解决无法用解析方法处理的方程和积分提供了强有力的技术。本文深入探讨考纲中的核心数值方法——根的定位、二分法、不动点迭代、牛顿-拉弗森法以及数值积分——并讨论它们的应用与常见误区。

1. Why Numerical Methods? | 为什么要学习数值方法？

Many mathematical problems have no closed-form solution. For example, the equation x³ − 2x − 5 = 0 has no simple factorisation, and equations such as eˣ = 3x cannot be solved by elementary algebra. In such cases, we resort to numerical methods that produce approximate answers to any desired degree of accuracy.

许多数学问题没有解析解。例如，方程 x³ − 2x − 5 = 0 无法简单因式分解，而 eˣ = 3x 这类方程也无法通过初等代数求解。在这种情况下，我们求助于数值方法，它可以按所需精度给出近似答案。

Numerical methods are iterative: they start from an initial guess and refine it step by step. They are essential in engineering, physics and economics, where real-world models are rarely solvable exactly. Even when an analytic solution exists, numerical methods are often faster and more convenient for practical computation.

数值方法是迭代式的：从一个初始猜测出发，逐步改进。它们在工程、物理和经济学中不可或缺，因为现实世界的模型很少能被精确求解。即使存在解析解，数值方法在实践中往往更快、更方便。

2. Locating Roots: The Change of Sign | 根的定位：变号法

For a continuous function f(x), if 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 the intermediate value theorem in action.

对于连续函数 f(x)，若 f(a) 与 f(b) 异号，则方程 f(x) = 0 在区间 (a, b) 内至少存在一个根。这就是介值定理的体现。

Example: Let f(x) = x³ − 2x − 5. Evaluating:

例：设 f(x) = x³ − 2x − 5。计算得：

• f(2) = 8 − 4 − 5 = −1 < 0
• f(3) = 27 − 6 − 5 = 16 > 0

f(2) × f(3) < 0 ⟹ a root lies between 2 and 3

f(2) × f(3) < 0 ⟹ 根位于 2 与 3 之间

However, the change-of-sign test has limitations. If f has two roots close together, or a repeated root where the curve touches the axis without crossing it, the sign may not change. Always sketch the graph or evaluate several points before concluding that no root exists.

然而，变号法有局限性。若函数有两个很接近的根，或存在重根（曲线与x轴相切而不穿越），符号可能不改变。在得出结论之前，务必画图或取多点评测。

3. Interval Bisection | 二分法

Once a root is located in (a, b), the bisection method repeatedly halves the interval. Let m = (a + b)/2, then:

一旦确定根在 (a, b) 内，二分法不断将区间减半。令 m = (a + b)/2，则：

• If f(m) = 0, then m is the exact root.
• If f(a) × f(m) < 0, the root lies in (a, m); set b = m.
• Otherwise, the root lies in (m, b); set a = m.

• 若 f(m) = 0，则 m 即为精确根。
• 若 f(a) × f(m) < 0，根在 (a, m) 内；令 b = m。
• 否则，根在 (m, b) 内；令 a = m。

Each iteration halves the width of the interval containing the root. After n iterations, the uncertainty is at most (b − a)/2ⁿ. For the example above, starting with a = 2, b = 3:

每迭代一次，包含根的区间宽度减半。经过 n 次迭代后，误差不超过 (b − a)/2ⁿ。对上面的例子，从 a = 2, b = 3 开始：

• m₁ = 2.5, f(2.5) = 15.625 − 5 − 5 = 5.625 > 0 ⟹ new interval (2, 2.5)
• m₂ = 2.25, f(2.25) = 11.39 − 4.5 − 5 = 1.89 > 0 ⟹ new interval (2, 2.25)
• m₃ = 2.125, f(2.125) = 9.596 − 4.25 − 5 = 0.346 > 0 ⟹ new interval (2, 2.125)
• m₄ = 2.0625, f(2.0625) ≈ −0.351 < 0 ⟹ new interval (2.0625, 2.125)

Bisection is slow but guaranteed to converge for any continuous function with a sign change. In an exam, you may be asked to perform three or four bisection steps and give the resulting interval.

二分法收敛速度较慢，但只要有变号且函数连续，它就必定收敛。考试中常要求你做三四步二分，并给出所得区间。

4. Fixed-Point Iteration: x = g(x) | 不动点迭代法

Rearrange f(x) = 0 into the form x = g(x). Then

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）