# 条件数

## 2. 定义

$\begin{array}{c} E(\tilde{f}(x)) = |f(x) - \tilde{f}(x)| \end{array}$

$\begin{array}{c} RE(\tilde{f}(x)) = \frac{E(\tilde{f}(x))}{|f(x)|} = \frac{|f(x)-\tilde{f}(x)|}{|f(x)|} \end{array}$

$\begin{array}{c} \lim_{\varepsilon \rightarrow 0} \sup_{E(\tilde{x}) \leq \varepsilon } \frac{E(\tilde{f}(x))}{E(\tilde{x})} \end{array}$

$\begin{array}{c} \lim_{\varepsilon \rightarrow 0} \sup_{E(\tilde{x}) \leq \varepsilon } \frac{RE(\tilde{f}(x))}{RE(\tilde{x})} \approx \frac{|f^{'}(x)|}{|f(x)|} \cdot |x| \end{array}$

【注】相对条件数更能准确地描述函数输出对输入微小变化的敏感程度，因此一般都采取相对条件数（简称条件数）刻画条件数。

## 3. 常见函数条件数

$x + a$ $\|{x \over x+a}\|$
$ax$ $1$
$\frac{1}{x}$ $1$
$x^n$ $\|n\|$
$e^x$ $\|x\|$
$\ln(x)$ $\|\frac{1}{\ln(x)}\|$
$\sin(x)$ $\|x \cot(x)\|$
$\cos(x)$ $\|x \tan(x)\|$
$\tan(x)$ $\|x(\tan(x)+\cot(x))\|$
$\arcsin(x)$ $\frac{x}{\sqrt{1-x^2}\arcsin(x)}$
$\arccos(x)$ $\frac{\|x\|}{\sqrt{1-x^2}\arccos(x)}$
$\arctan(x)$ $\frac{x}{(1+x^2)\arctan(x)}$

### 3.1 矩阵范数

$\begin{array}{c} \kappa(A) = |A^{-1}| \cdot |A| \end{array}$

• $| \cdot |$$L_2$ 诱导出的矩阵范数，则

$\begin{array}{c} \kappa(A) = \frac{\sigma_{\max}(A)}{\sigma_{\min}(A)} \end{array}$

1. $A$ 是正规矩阵，则

$\begin{array}{c} \kappa(A) = |\frac{\lambda_{\max}(A)}{\lambda_{\min}(A)} | \end{array}$

1. $A$ 是酋矩阵，则

$\begin{array}{c} \kappa(A) = 1 \end{array}$