Iteration: Numerical Methods for Solving Equations | 迭代法：解方程的数值方法

📚 Iteration: Numerical Methods for Solving Equations | 迭代法：解方程的数值方法

In A Level Mathematics, iteration is a numerical technique for locating approximate solutions of equations. It is especially useful when an equation cannot be rearranged into a closed algebraic form, or when an exact root is irrational and a decimal approximation is sufficient. Edexcel questions often ask you to use a given iterative formula, carry out repeated substitutions, and judge whether the process converges to a root.

在 A Level 数学中，迭代法是一种求方程近似解的数值技术。当方程难以写成闭合的代数形式，或者精确根是无理数且只需要小数近似值时，迭代法特别有用。Edexcel 考试题经常要求考生使用给定的迭代公式进行反复代入，并判断该过程是否收敛到某个根。

1. What Is Iteration? | 什么是迭代法

Iteration means repeating a calculation according to a fixed rule. Starting from an initial estimate x₀, we generate a sequence x₁, x₂, x₃, … by applying the same formula xₙ₊₁ = g(xₙ) at each step. If the sequence approaches a fixed number α, we say the iteration converges to α.

迭代的意思是按照固定规则重复计算。从初始估计值 x₀ 开始，我们通过每一步应用同一个公式 xₙ₊₁ = g(xₙ) 来生成数列 x₁、x₂、x₃……如果这个数列逐渐接近某个固定数 α，我们就说迭代收敛到 α。

This method is part of numerical methods in Edexcel Pure Mathematics. It allows us to approximate roots of equations such as f(x) = 0 when algebraic methods are difficult, slow, or impossible at this level.

这种方法属于 Edexcel 纯数学中的数值方法。当代数方法困难、耗时或者在当前阶段无法使用时，它可以用来近似求解 f(x) = 0 这类方程的根。

2. Rearranging f(x) = 0 into x = g(x) | 将 f(x) = 0 改写为 x = g(x)

To use iteration, we first rearrange the equation f(x) = 0 into the form x = g(x). This is not unique; the same equation can often be rearranged in several different ways. The iterative formula is then written as:

要使用迭代法，我们首先要把方程 f(x) = 0 改写成 x = g(x) 的形式。这种改写并不是唯一的，同一个方程通常可以有多种不同的改写方式。迭代公式随后写成：

xₙ₊₁ = g(xₙ)

For example, the quadratic equation x² − 5x + 3 = 0 can be rearranged by writing 5x = x² + 3, which gives:

例如，二次方程 x² − 5x + 3 = 0 可以通过写成 5x = x² + 3 来改写，得到：

x = (x² + 3)/5

This is now in the required form with g(x) = (x² + 3)/5. Other rearrangements are possible, but some will converge and others will not.

这样就得到了所需的形式，其中 g(x) = (x² + 3)/5。也可以有其他改写方式，但有些会收敛，有些则不会。

3. Existence of Roots: The Sign Change Rule | 根的存在性：符号变化法则

Before applying iteration, we usually need to show that a root exists in a given interval. If f(x) is continuous on [a, b] and f(a) × f(b) < 0, then there is at least one root in the interval (a, b). This is the sign change rule.

在应用迭代法之前，我们通常需要证明在给定区间内存在一个根。如果 f(x) 在 [a, b] 上连续，并且 f(a) × f(b) < 0，那么在区间 (a, b) 内至少存在一个根。这就是符号变化法则。

For f(x) = x² − 5x + 3, we have f(0) = 3 and f(1) = −1. Since the sign changes between x = 0 and x = 1, there must be a root in the interval [0, 1]. The smaller root is approximately 0.6972.

对于 f(x) = x² − 5x + 3，我们有 f(0) = 3 和 f(1) = −1。由于符号在 x = 0 与 x = 1 之间发生变化，因此区间 [0, 1] 内必定存在一个根。较小的根约为 0.6972。

The sign change rule is sufficient for existence but is not necessary in every case. A repeated root where the curve touches the x-axis may not produce a sign change, so care is needed when interpreting results.

符号变化法则对于根的存在性是充分条件，但并非在所有情况下都是必要条件。如果曲线刚好与 x 轴相切而产生重根，则可能不会出现符号变化，因此解释结果时需要小心。

4. Fixed Points and Roots | 不动点与根

If a number α satisfies α = g(α), then α is called a fixed point of the function g. Because the equation f(x) = 0 was rearranged into x = g(x), a fixed point of g corresponds to a root of the original equation f(x) = 0.

如果某个数 α 满足 α = g(α)，那么 α 就称为函数 g 的不动点。由于方程 f(x) = 0 已被改写为 x = g(x)，因此 g 的不动

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