📚 Numerical Root-Finding Methods in Modelling | 数值求根法在建模中的应用
Numerical root-finding methods are used when a mathematical model produces an equation in the form f(x) = 0 that cannot be solved exactly by factorising or rearrangement. These methods generate successive approximations that can be made as accurate as needed.
数值求根法用于当数学模型导出形如 f(x) = 0 的方程，而该方程无法通过因式分解或代数变形精确求解时。这类方法会逐步逼近根，并可根据需要达到任意精度。
1. Why Root Finding Appears in Models | 为什么建模中需要求根
Many real-world problems are set up as equations: finding a break-even point, balancing supply and demand, finding a chemical equilibrium, or calculating the time at which a moving object returns to ground level.
许多实际问题都会被转化为方程：求盈亏平衡点、平衡供需关系、寻找化学平衡，或计算运动物体何时回到地面。
In every case, the model is arranged as a function f(x), and the important answer is the value of x for which f(x) = 0. This value is called a root of the equation.
在每种情况中，模型都被整理成函数 f(x)，而关键答案就是使 f(x) = 0 的 x 值。这个值被称为方程的根。
Sometimes f(x) is simple, but in realistic models it often includes powers, exponentials, logarithms or trigonometric terms. Then an exact method is either impossible or very time-consuming, so numerical methods are preferred.
有时 f(x) 很简单，但在实际模型中它常常涉及幂函数、指数函数、对数函数或三角函数。此时精确解法要么不存在，要么非常耗时，因此数值方法更受青睐。
2. Setting Up a Model as a Root Problem | 将模型转化为求根问题
The first step in numerical modelling is to define a continuous variable and write down an equation that describes the condition of interest. The condition is usually something like “revenue equals cost” or “net force equals zero”.
数值建模的第一步是定义连续变量，并写出描述目标条件的方程。该条件通常类似“收入等于成本”或“合外力为零”。
For a break-even model, if revenue is R(x) and total cost is C(x), the profit is P(x) = R(x) − C(x). The break-even point is found when profit is zero.
对于盈亏平衡模型，若收入为 R(x)，总成本为 C(x)，则利润为 P(x) = R(x) − C(x)。盈亏平衡点出现在利润为零时。
P(x) = R(x) − C(x) = 0
The equation is already in root-finding form. The model may also include constraints such as x ≥ 0, and the final numerical answer must lie inside the physical domain of the model.
该方程已经具有求根形式。模型中通常还会包含 x ≥ 0 等约束，最终数值答案必须落在该模型的实际定义域内。
3. The Sign-Change Principle | 变号原理
If a function f is continuous on the closed 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 open interval (a, b).
若函数 f 在闭区间 [a, b] 上连续，且 f(a) 与 f(b) 异号，那么方程 f(x) = 0 在开区间 (a, b) 内至少存在一个根。
f(a) × f(b) < 0
This is the basis of interval bisection and decimal search. It does not find the root exactly, but it guarantees that a root is trapped between two known values.
这是二分法和十等分搜索法的基础。变号检验不能直接求出根，但可以保证根被夹在两个已知值之间。
However, the sign-change rule only works if f is continuous. If the function has a vertical asymptote, the sign can change without the function ever taking the value zero. Always check continuity and domain.
但变号规则只在 f 连续时才成立。如果函数有垂直渐近线，符号可能改变，但函数永远不会取零值。因此必须检查连续性和定义域。
4. Bisection Method | 二分法
The bisection method is the most reliable numerical root-finding method. It begins with an interval [a, b] that contains a sign change and repeatedly halves that interval.
二分法是最可靠的求根方法。它从一个包含变号的区间 [a, b] 开始，然后不断将该区间减半。
Step 1: Calculate the midpoint m = (a + b) / 2.
第一步：计算中点 m = (a + b) / 2。
Step 2: Evaluate f(m). If f(m) has the same sign as f(a), replace a by m. Otherwise replace b by m.
第二步：计算 f(m)。若 f(m) 与 f(a) 同号，则将 a 替换为 m；否则将 b 替换为 m。
Step 3: Repeat until the interval width is small enough.
第三步：重复以上过程，直到区间宽度足够小。
After n iterations, the width of the interval is (b − a) / 2ⁿ, so the maximum error in the root is halved at every step.
经过 n 次迭代后，区间宽度为 (b − a) / 2ⁿ，因此最大误差每一步都会缩小一半。
For example, take f(x) = x³ + x − 3. Since f(1) = −1 and f(2) = 7, there is a root between 1 and 2.
例如，取 f(x) = x³ + x − 3。由于 f(1) = −1，f(2) = 7，因此该函数在 1 和 2 之间有一个根。
| a | b | m | Sign of f(m) | New interval |
|---|---|---|---|---|
| 1 | 2 | 1.5 | Positive | (1, 1.5) |
| 1 | 1.5 | 1.25 | Positive | (1, 1.25) |
| 1 | 1.25 | 1.125 | Negative | (1.125, 1.25) |
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