误差的传播¶
有一个计算公式：
\[y = f(x_1, x_2, \cdots, x_n)\]
其中 \(x_1, x_2, \cdots, x_n\) 是输入，\(y\) 是输出。
Motivating Questions
- 输入有误差 \(x_1^*, x_2^*, \cdots, x_n^*\)，和输出的误差 \(y^*\) 关系？
- \(e(x_1^*), e(x_2^*), \cdots, e(x_n^*)\) 和 \(e(y^*)\) 关系？
- \(|e_{\mathrm{r}}(x_i^*)| \leq \varepsilon, \, i = 1, 2, \cdots, n\)，\(|e(y^*)| \leq ?\)
\(f\) 需要的性质：在 \((x_1^*, x_2^*, \cdots, x_n^*)\) 附近可微。
\[ \begin{aligned} e(y^*) &= y - y^* = f(x_1, x_2, \cdots, x_n) - f(x_1^*, x_2^*, \cdots, x_n^*) \\ &\approx \vec{k} \cdot (x_1 - x_1^*, x_2 - x_2^*, \cdots, x_n - x_n^*) \\ &= k_1 (x_1 - x_1^*) + k_2 (x_2 - x_2^*) + \cdots + k_n (x_n - x_n^*) \\ \end{aligned} \]
- 一元
\[y = y^* + k (x - x^*)\]
\[\Delta y = k \Delta x\]
- 二元
\[y = y^* + k_1 (x_1 - x_1^*) + k_2 (x_2 - x_2^*)\]
\[\Delta y = k_1 \Delta x_1 + k_2 \Delta x_2\]
\[\Rightarrow k_1 = \frac{\Delta y}{\Delta x_1} - \frac{k_2 \Delta x_2}{\Delta x_1} \to k_1 = \frac{\partial y}{\partial x_1} + 0 = \frac{\partial y}{\partial x_1}\]
- 多元
\[k_i = \frac{\partial y}{\partial x_i}, \quad i = 1, 2, \cdots, n\]
\[e(y^*) = \sum_{i=1}^{n} \left.\frac{\partial y}{\partial x_i}\right|_{(x_1^*, x_2^*, \cdots, x_n^*)} (x_i - x_i^*) + o(||x - x^*||)\]
例子
+：输入 \(x_1, x_2\)，输出 \(f(x_1, x_2) = x_1 + x_2\)。
\[ \begin{aligned} e(x_1 + x_2) &= \frac{\partial f}{\partial x_1} e(x_1) + \frac{\partial f}{\partial x_2} e(x_2) \\ &= 1 \cdot e(x_1) + 1 \cdot e(x_2) \\ &= e(x_1) + e(x_2) \end{aligned} \]
此即书上 (1.7) 式。