（四）误差分析

本文主要内容如下：

• 1. 矩阵的条件数
• 2. 条件数的性质
• 3. 误差与残差

1. 矩阵的条件数

线性方程组
A x ⃗ = b ⃗ \bold A\vec{x}=\vec{b} Ax =b
其中，$\mathbf{A}$ 为可逆矩阵， x ⃗ \vec{x} x 为该线性方程的精确解。

定义 若系数矩阵 $\mathbf{A}$ 与常数项 b ⃗ \vec{b} b 的微小变化就引起线性方程组的解发生相对较大的改变，则称该线性方程组为病态方程组，相应的系数矩阵称作病态矩阵，否则称作良态方程组、良态矩阵。

(1) 设常数项有微小改变 δ b ⃗ \delta\vec{b} δb ，使得方程组的解发生改变 δ x ⃗ \delta\vec{x} δx ，即：
A ( x ⃗ + δ x ⃗ ) = b ⃗ + δ b ⃗ ⟹ δ x ⃗ = A − 1 ( δ b ⃗ ) ⟹ ∣ ∣ δ x ⃗ ∣ ∣ ≤ ∣ ∣ A − 1 ∣ ∣ ⋅ ∣ ∣ δ b ⃗ ∣ ∣ \bold A(\vec{x}+\delta\vec{x})=\vec{b}+\delta\vec{b}\Longrightarrow\delta\vec{x}=\bold A^{-1}(\delta\vec{b})\Longrightarrow||\delta\vec{x}||\le||\bold A^{-1}||\cdot||\delta\vec{b}|| A(x +δx )=b +δb ⟹δx =A−1(δb )⟹∣∣δx ∣∣≤∣∣A−1∣∣⋅∣∣δb ∣∣
又由于
∣ ∣ b ⃗ ∣ ∣ = ∣ ∣ A x ⃗ ∣ ∣ ≤ ∣ ∣ A ∣ ∣ ⋅ ∣ ∣ x ∣ ∣ ⃗ ⟹ 1 ∣ ∣ x ⃗ ∣ ∣ ≤ ∣ ∣ A ∣ ∣ ∣ ∣ b ⃗ ∣ ∣ ||\vec{b}||=||\bold A\vec{x}||\le||\bold A||\cdot||\vec{x||}\Longrightarrow\frac{1}{||\vec{x}||}\le\frac{||\bold A||}{||\vec{b}||} ∣∣b ∣∣=∣∣Ax ∣∣≤∣∣A∣∣⋅∣∣x∣∣ ​⟹∣∣x ∣∣1​≤∣∣b ∣∣∣∣A∣∣​
故
∣ ∣ δ x ⃗ ∣ ∣ ∣ ∣ x ⃗ ∣ ∣ ≤ ∣ ∣ A − 1 ∣ ∣ ⋅ ∣ ∣ A ∣ ∣ ∣ ∣ δ b ⃗ ∣ ∣ ∣ ∣ b ⃗ ∣ ∣ ( a − 1 ) \frac{||\delta\vec{x}||}{||\vec{x}||}\le||\bold A^{-1}||\cdot||\bold A||\frac{||\delta\vec{b}||}{||\vec{b}||}\quad(a-1) ∣∣x ∣∣∣∣δx ∣∣​≤∣∣A−1∣∣⋅∣∣A∣∣∣∣b ∣∣∣∣δb ∣∣​(a−1)
另外
∣ ∣ δ b ⃗ ∣ ∣ = ∣ ∣ A ( δ x ⃗ ) ∣ ∣ ≤ ∣ ∣ A ∣ ∣ ⋅ ∣ ∣ δ x ⃗ ∣ ∣ ∣ ∣ x ⃗ ∣ ∣ = ∣ ∣ A − 1 b ⃗ ∣ ∣ ≤ ∣ ∣ A − 1 ∣ ∣ ⋅ ∣ ∣ b ⃗ ∣ ∣ ||\delta\vec{b}||=||\bold A(\delta\vec{x})||\le||\bold A||\cdot||\delta\vec{x}||\\\ \\ ||\vec{x}||=||\bold{A}^{-1}\vec{b}||\le||\bold A^{-1}||\cdot||\vec{b}|| ∣∣δb ∣∣=∣∣A(δx )∣∣≤∣∣A∣∣⋅∣∣δx ∣∣ ∣∣x ∣∣=∣∣A−1b ∣∣≤∣∣A−1∣∣⋅∣∣b ∣∣
故
∣ ∣ δ x ⃗ ∣ ∣ ∣ ∣ x ⃗ ∣ ∣ ≥ ∣ ∣ δ b ⃗ ∣ ∣ ∣ ∣ A ∣ ∣ 1 ∣ ∣ x ⃗ ∣ ∣ ≥ 1 ∣ ∣ A − 1 ∣ ∣ ⋅ ∣ ∣ A ∣ ∣ ∣ ∣ δ b ⃗ ∣ ∣ ∣ ∣ b ⃗ ∣ ∣ ( a − 2 ) \frac{||\delta\vec{x}||}{||\vec{x}||}\ge\frac{||\delta\vec{b}||}{||\bold A||}\frac{1}{||\vec{x}||}\ge\frac{1}{||\bold A^{-1}||\cdot||\bold A||}\frac{||\delta\vec{b}||}{||\vec{b}||}\quad(a-2) ∣∣x ∣∣∣∣δx ∣∣​≥∣∣A∣∣∣∣δb ∣∣​∣∣x ∣∣1​≥∣∣A−1∣∣⋅∣∣A∣∣1​∣∣b ∣∣∣∣δb ∣∣​(a−2)
上面的讨论给出了，常数项波动时，解的相对误差的上、下界，且有
∣ ∣ A − 1 ∣ ∣ ⋅ ∣ ∣ A ∣ ∣ ≥ 1 ||\bold A^{-1}||\cdot||\bold A||\ge1 ∣∣A−1∣∣⋅∣∣A∣∣≥1

(2) 设系数矩阵有微小的扰动 $\delta \mathbf{A}$，使得方程组的解发生改变 δ x ⃗ \delta\vec{x} δx ，即：
( A + δ A ) ( x ⃗ + δ x ⃗ ) = b ⃗ ⟹ ( A + δ A ) ( δ x ⃗ ) = A [ E + ( δ A ) A − 1 ] ( δ x ⃗ ) = − ( δ A ) x ⃗ (\bold A+\delta\bold A)(\vec{x}+\delta\vec{x})=\vec{b}\Longrightarrow(\bold A+\delta\bold A)(\delta\vec{x})=\bold A[\bold E+(\delta\bold A)\bold A^{-1}](\delta\vec{x})=-(\delta\bold{A})\vec{x} (A+δA)(x +δx )=b ⟹(A+δA)(δx )=A[E+(δA)A−1](δx )=−(δA)x
若 ∣ ∣ ( δ A ) A − 1 ∣ ∣ v < 1 ||(\delta\bold A)\bold A^{-1}||_v<1 ∣∣(δA)A−1∣∣v​<1 ，则 E + ( δ A ) A − 1 \bold E+(\delta\bold A)\bold A^{-1} E+(δA)A−1 可逆，且
∣ ∣ ( E + ( δ A ) A − 1 ) − 1 ∣ ∣ v ≤ 1 1 − ∣ ∣ ( δ A ) A − 1 ∣ ∣ v ||(\bold E+(\delta\bold A)\bold A^{-1})^{-1}||_v\le\frac{1}{1-||(\delta\bold A)\bold A^{-1}||_v} ∣∣(E+(δA)A−1)−1∣∣v​≤1−∣∣(δA)A−1∣∣v​1​
那么
∣ ∣ δ x ⃗ ∣ ∣ = ∣ ∣ [ E + ( δ A ) A − 1 ] − 1 ∣ ∣ v ⋅ ∣ ∣ A − 1 ∣ ∣ v ⋅ ∣ ∣ δ A ∣ ∣ v ⋅ ∣ ∣ x ⃗ ∣ ∣ ≤ ∣ ∣ A − 1 ∣ ∣ v ⋅ ∣ ∣ δ A ∣ ∣ v ⋅ ∣ ∣ x ⃗ ∣ ∣ 1 − ∣ ∣ ( δ A ) A − 1 ∣ ∣ v ||\delta\vec{x}||=||[\bold E+(\delta\bold A)\bold A^{-1}]^{-1}||_v\cdot||\bold A^{-1}||_v\cdot||\delta\bold{A}||_v\cdot||\vec{x}||\le\frac{||\bold A^{-1}||_v\cdot||\delta\bold{A}||_v\cdot||\vec{x}||}{1-||(\delta\bold A)\bold A^{-1}||_v} ∣∣δx ∣∣=∣∣[E+(δA)A−1]−1∣∣v​⋅∣∣A−1∣∣v​⋅∣∣δA∣∣v​⋅∣∣x ∣∣≤1−∣∣(δA)A−1∣∣v​∣∣A−1∣∣v​⋅∣∣δA∣∣v​⋅∣∣x ∣∣​
若更严格的要求 ∣ ∣ ( δ A ) A − 1 ∣ ∣ v < ∣ ∣ δ A ∣ ∣ v ⋅ ∣ ∣ A − 1 ∣ ∣ v < 1 ||(\delta\bold A)\bold A^{-1}||_v<||\delta\bold A||_v\cdot||\bold A^{-1}||_v<1 ∣∣(δA)A−1∣∣v​<∣∣δA∣∣v​⋅∣∣A−1∣∣v​<1，则有：
∣ ∣ δ x ⃗ ∣ ∣ ∣ ∣ x ⃗ ∣ ∣ ≤ ∣ ∣ A − 1 ∣ ∣ v ⋅ ∣ ∣ A ∣ ∣ v ⋅ ∣ ∣ δ A ∣ ∣ v ∣ ∣ A ∣ ∣ 1 − ∣ ∣ A − 1 ∣ ∣ v ⋅ ∣ ∣ A ∣ ∣ v ⋅ ∣ ∣ δ A ∣ ∣ v ∣ ∣ A ∣ ∣ ( b ) \frac{||\delta\vec{x}||}{||\vec{x}||}\le\frac{||\bold A^{-1}||_v\cdot||\bold A||_v\cdot\dfrac{||\delta\bold{A}||_v}{||\bold A||}}{1-||\bold A^{-1}||_v\cdot||\bold{A}||_v\cdot\dfrac{||\delta\bold{A}||_v}{||\bold A||}}\quad(b) ∣∣x ∣∣∣∣δx ∣∣​≤1−∣∣A−1∣∣v​⋅∣∣A∣∣v​⋅∣∣A∣∣∣∣δA∣∣v​​∣∣A−1∣∣v​⋅∣∣A∣∣v​⋅∣∣A∣∣∣∣δA∣∣v​​​(b)
或者

( A + δ A ) ( δ x ⃗ ) + ( δ A ) x ⃗ = A ( δ x ⃗ ) + ( δ A ) ( δ x ⃗ + x ⃗ ) = 0 ⟹ δ x ⃗ = − A − 1 ( δ A ) ( δ x ⃗ + x ⃗ ) (\bold A+\delta\bold A)(\delta\vec{x})+(\delta\bold A)\vec{x}=\bold A(\delta\vec{x})+(\delta\bold{A})(\delta\vec{x}+\vec{x})=0\Longrightarrow\delta\vec{x}=-\bold A^{-1}(\delta\bold{A})(\delta\vec{x}+\vec{x}) (A+δA)(δx )+(δA)x =A(δx )+(δA)(δx +x )=0⟹δx =−A−1(δA)(δx +x )
故
∣ ∣ δ x ⃗ ∣ ∣ ∣ ∣ x ⃗ + δ x ⃗ ∣ ∣ ≤ ∣ ∣ A − 1 ∣ ∣ ⋅ ∣ ∣ A ∣ ∣ ⋅ ∣ ∣ δ A ∣ ∣ ∣ ∣ A ∣ ∣ ( c ) \frac{||\delta\vec{x}||}{||\vec{x}+\delta\vec{x}||}\le||\bold{A}^{-1}||\cdot||\bold{A}||\cdot\frac{||\delta\bold{A}||}{||\bold{A}||}\quad(c) ∣∣x +δx ∣∣∣∣δx ∣∣​≤∣∣A−1∣∣⋅∣∣A∣∣⋅∣∣A∣∣∣∣δA∣∣​(c)
上式给出了系数矩阵发生改变时，方程解的扰动。

结合 $(a)(c)$ 式可以发现方程解相对误差的上界均与系数 ∣ ∣ A − 1 ∣ ∣ ⋅ ∣ ∣ A ∣ ∣ ||\bold A^{-1}||\cdot||\bold{A}|| ∣∣A−1∣∣⋅∣∣A∣∣ 相关。该系数越大，可能引起的相对误差就越大，它反映了方程组的病态程度。

定义 对于非奇异矩阵 A ∈ R n × n A\in\mathbb{R}^{n\times n} A∈Rn×n 而言，将 ∣ ∣ A − 1 ∣ ∣ ⋅ ∣ ∣ A ∣ ∣ ||\bold A^{-1}||\cdot||\bold{A}|| ∣∣A−1∣∣⋅∣∣A∣∣ 称作它的条件数，记作 $cond(\mathbf{A})$ ，即
c o n d ( A ) = ∣ ∣ A − 1 ∣ ∣ ⋅ ∣ ∣ A ∣ ∣ cond(\bold A)=||\bold A^{-1}||\cdot||\bold{A}|| cond(A)=∣∣A−1∣∣⋅∣∣A∣∣

常用的矩阵条件数——谱条件数：
c o n d ( A ) v = ∣ ∣ A ∣ ∣ 2 ∣ ∣ A − 1 ∣ ∣ 2 = λ m a x ( A T A ) λ m i n ( A A T ) = λ m a x ( A T A ) λ m i n ( A T A ) cond(\bold A)_v=||\bold A||_2||\bold A^{-1}||_2=\sqrt{\frac{\lambda_{max}(\bold A^T\bold A)}{\lambda_{min}(\bold A\bold A^T)}}=\sqrt{\frac{\lambda_{max}(\bold A^T\bold A)}{\lambda_{min}(\bold A^T\bold A)}} cond(A)v​=∣∣A∣∣2​∣∣A−1∣∣2​=λmin​(AAT)λmax​(ATA)​ ​=λmin​(ATA)λmax​(ATA)​ ​

证明：由于$\mathbf{A}$ 可逆，故 A A T \bold{AA^T} AAT 为正定矩阵，其所有特征值均为正，又
∣ ∣ A − 1 ∣ ∣ 2 = λ m a x ( A − T A − 1 ) = λ m a x [ ( A A T ) − 1 ] = 1 λ m i n ( A A T ) ||\bold A^{-1}||_2=\sqrt{\lambda_{max}(\bold A^{-T}\bold A^{-1})}=\sqrt{\lambda_{max}[(\bold A\bold A^{T})^{-1}]}=\sqrt{\frac{1}{\lambda_{min}(\bold A\bold A^T)}}\quad ∣∣A−1∣∣2​=λmax​(A−TA−1) ​=λmax​[(AAT)−1] ​=λmin​(AAT)1​ ​
而 A A T \bold{AA^T} AAT 与 A T A \bold{A^TA} ATA 具有相同的特征值，因为：
( A B ) u ⃗ = λ u ⃗ ⟹ ( B A ) B u ⃗ = λ ( B u ⃗ ) (\bold{AB})\vec{u}=\lambda\vec{u}\Longrightarrow\bold{(BA)B}\vec{u}=\lambda(\bold{B}\vec{u}) (AB)u =λu ⟹(BA)Bu =λ(Bu )
令 B = A T \bold B=\bold A^T B=AT 即可。(证毕)

2. 条件数的性质

( 1 ) c o n d ( A ) ≥ 1 ; ( 2 ) ∀ c ∈ R ， c ≠ 0 ， c o n d ( c A ) = ∣ ∣ c A ∣ ∣ ⋅ ∣ ∣ 1 c A − 1 ∣ ∣ = c o n d ( A ) ; ( 3 ) 正交矩阵的谱条件数为 1 ，即 c o n d ( Q ) 2 = λ m a x ( Q T Q ) λ m i n ( Q Q T ) = 1 ( 4 ) 若 A 为非奇异矩阵， Q 为正交矩阵，则  c o n d ( A ) 2 = c o n d ( Q A ) 2 = c o n d ( A Q ) 2 c o n d ( Q A ) 2 = λ m a x ( A T Q T Q A ) λ m i n ( Q A A T Q T ) = λ m a x ( A T A ) λ m i n ( A A T ) = c o n d ( A ) 2 c o n d ( A Q ) 2 = λ m a x ( Q T A T A Q ) λ m i n ( A Q Q T A T ) = λ m a x ( A T A ) λ m i n ( A A T ) = c o n d ( A ) 2 注：正交变换不改变矩阵的特征值。 \begin{aligned} & (1)\ cond(\bold A)\ge1;\\\\ & (2)\ \forall c\in\mathbb{R}，c\ne0，cond(c\bold A)=||c\bold A||\cdot||\frac{1}{c}\bold{A}^{-1}||=cond(\bold A);\\\\ & (3)\ 正交矩阵的谱条件数为1，即\\\\ & \qquad\qquad cond(\bold Q)_2=\sqrt{\frac{\lambda_{max}(\bold Q^T\bold Q)}{\lambda_{min}(\bold Q\bold Q^T)}}=1\\\\ & (4)\ 若\bold A为非奇异矩阵，\bold Q为正交矩阵，则\ cond(\bold A)_2=cond(\bold{QA})_2=cond(\bold{AQ})_2\\\\ &\qquad\qquad cond(\bold QA)_2=\sqrt{\frac{\lambda_{max}(\bold {A^TQ^T}\bold {QA})}{\lambda_{min}(\bold{QA}\bold {A^TQ^T})}}=\sqrt{\frac{\lambda_{max}(\bold {A^T}\bold {A})}{\lambda_{min}(\bold{A}\bold {A^T})}}=cond(\bold A)_2\\\\ &\qquad\qquad cond(\bold {AQ})_2=\sqrt{\frac{\lambda_{max}(\bold {Q^TA^T}\bold {AQ})}{\lambda_{min}(\bold {AQ}\bold {Q^TA^T})}}=\sqrt{\frac{\lambda_{max}(\bold {A^T}\bold {A})}{\lambda_{min}(\bold {A}\bold {A^T})}}=cond(\bold A)_2\\\\ &注：正交变换不改变矩阵的特征值。 \end{aligned} ​(1) cond(A)≥1;(2) ∀c∈R，c=0，cond(cA)=∣∣cA∣∣⋅∣∣c1​A−1∣∣=cond(A);(3) 正交矩阵的谱条件数为1，即cond(Q)2​=λmin​(QQT)λmax​(QTQ)​ ​=1(4) 若A为非奇异矩阵，Q为正交矩阵，则 cond(A)2​=cond(QA)2​=cond(AQ)2​cond(QA)2​=λmin​(QAATQT)λmax​(ATQTQA)​ ​=λmin​(AAT)λmax​(ATA)​ ​=cond(A)2​cond(AQ)2​=λmin​(AQQTAT)λmax​(QTATAQ)​ ​=λmin​(AAT)λmax​(ATA)​ ​=cond(A)2​注：正交变换不改变矩阵的特征值。​

3. 误差与残差

设线性方程
A x ⃗ = b \bold A\vec{x}=b Ax =b
的准确解为 x ⃗ = u ⃗ \vec{x}=\vec{u} x =u ，而某一实际解得的向量为 v ⃗ \vec{v} v 。定义残差为：
r ⃗ ≜ b ⃗ − A v ⃗ ( 1 ) \vec{r}\triangleq\vec{b}-\bold A\vec{v}\quad(1) r ≜b −Av (1)
定义绝对误差为：
e ⃗ ≜ v ⃗ − u ⃗ ( 2 ) \vec{e}\triangleq\vec{v}-\vec{u}\quad(2) e ≜v −u (2)

残差与误差满足关系式：
A e ⃗ = − r ⃗ ( 3 ) \bold A\vec{e}=-\vec{r}\quad(3) Ae =−r (3)
证明： A e ⃗ = A ( v ⃗ − u ⃗ ) = A v ⃗ − b ⃗ = − r ⃗ ( 证毕 ) \bold A\vec{e}=\bold{A}(\vec{v}-\vec{u})=\bold{A}\vec{v}-\vec{b}=-\vec{r}\quad(证毕) Ae =A(v −u )=Av −b =−r (证毕)

根据式（3）便可通过残差对误差进行估计：
∣ ∣ e ⃗ ∣ ∣ = ∣ ∣ − A − 1 r ⃗ ∣ ∣ ≤ ∣ ∣ A − 1 ∣ ∣ ⋅ ∣ ∣ r ⃗ ∣ ∣ ||\vec{e}||=||-\bold A^{-1}\vec{r}||\le||\bold A^{-1}||\cdot||\vec{r}|| ∣∣e ∣∣=∣∣−A−1r ∣∣≤∣∣A−1∣∣⋅∣∣r ∣∣
不过，绝对误差的大小往往没有太大的实际意义，更有意义的是相对误差 ∣ ∣ e ⃗ ∣ ∣ ∣ ∣ u ⃗ ∣ ∣ \dfrac{||\vec{e}||}{||\vec{u}||} ∣∣u ∣∣∣∣e ∣∣​ 的大小。

一方面，
∣ ∣ e ⃗ ∣ ∣ ≤ ∣ ∣ A − 1 ∣ ∣ ⋅ ∣ ∣ r ⃗ ∣ ∣ ∣ ∣ b ⃗ ∣ ∣ = ∣ ∣ A u ⃗ ∣ ∣ ≤ ∣ ∣ A ∣ ∣ ⋅ ∣ ∣ u ⃗ ∣ ∣ ⟹ 1 ∣ ∣ u ⃗ ∣ ∣ ≤ ∣ ∣ A ∣ ∣ ⋅ 1 ∣ ∣ b ⃗ ∣ ∣ } ⟹ ∣ ∣ e ⃗ ∣ ∣ ∣ ∣ u ⃗ ∣ ∣ ≤ ∣ ∣ A − 1 ∣ ∣ ⋅ ∣ ∣ A ∣ ∣ ⋅ ∣ ∣ r ⃗ ∣ ∣ ∣ ∣ b ⃗ ∣ ∣ \left.\begin{matrix} ||\vec{e}||\le||\bold A^{-1}||\cdot||\vec{r}|| \\ \\ ||\vec{b}||=||\bold A\vec{u}||\le||\bold A||\cdot||\vec{u}||\Longrightarrow\dfrac{1}{||\vec{u}||}\le||\bold A||\cdot\dfrac{1}{||\vec{b}||} \end {matrix}\right\} \Longrightarrow \dfrac{||\vec{e}||}{||\vec{u}||}\le||\bold A^{-1}||\cdot||\bold A||\cdot\frac{||\vec{r}||}{||\vec{b}||} ∣∣e ∣∣≤∣∣A−1∣∣⋅∣∣r ∣∣∣∣b ∣∣=∣∣Au ∣∣≤∣∣A∣∣⋅∣∣u ∣∣⟹∣∣u ∣∣1​≤∣∣A∣∣⋅∣∣b ∣∣1​​⎭ ⎬ ⎫​⟹∣∣u ∣∣∣∣e ∣∣​≤∣∣A−1∣∣⋅∣∣A∣∣⋅∣∣b ∣∣∣∣r ∣∣​
另一方面，
∣ ∣ r ⃗ ∣ ∣ = ∣ ∣ A e ⃗ ∣ ∣ ≤ ∣ ∣ A ∣ ∣ ⋅ ∣ ∣ e ⃗ ∣ ∣ ⟹ ∣ ∣ e ⃗ ∣ ∣ ≥ ∣ ∣ r ⃗ ∣ ∣ ⋅ 1 ∣ ∣ A ∣ ∣ ∣ ∣ u ⃗ ∣ ∣ = ∣ ∣ A − 1 b ⃗ ∣ ∣ ≤ ∣ ∣ A − 1 ∣ ∣ ⋅ ∣ ∣ b ⃗ ∣ ∣ } ⟹ ∣ ∣ e ⃗ ∣ ∣ ∣ ∣ u ⃗ ∣ ∣ ≥ 1 ∣ ∣ A − 1 ∣ ∣ ⋅ ∣ ∣ A ∣ ∣ ⋅ ∣ ∣ r ⃗ ∣ ∣ ∣ ∣ b ⃗ ∣ ∣ \left.\begin{matrix} ||\vec{r}||=||\bold A\vec{e}||\le||\bold A||\cdot||\vec{e}||\Longrightarrow||\vec{e}||\ge||\vec{r}||\cdot\dfrac{1}{||\bold A||} \\ \\ ||\vec{u}||=||\bold A^{-1}\vec{b}||\le||\bold A^{-1}||\cdot||\vec{b}|| \end {matrix}\right\} \Longrightarrow \dfrac{||\vec{e}||}{||\vec{u}||}\ge\dfrac{1}{||\bold A^{-1}||\cdot||\bold A||}\cdot\frac{||\vec{r}||}{||\vec{b}||} ∣∣r ∣∣=∣∣Ae ∣∣≤∣∣A∣∣⋅∣∣e ∣∣⟹∣∣e ∣∣≥∣∣r ∣∣⋅∣∣A∣∣1​∣∣u ∣∣=∣∣A−1b ∣∣≤∣∣A−1∣∣⋅∣∣b ∣∣​⎭ ⎬ ⎫​⟹∣∣u ∣∣∣∣e ∣∣​≥∣∣A−1∣∣⋅∣∣A∣∣1​⋅∣∣b ∣∣∣∣r ∣∣​
综上，
1 c o n d ( A ) ⋅ ∣ ∣ r ⃗ ∣ ∣ ∣ ∣ b ⃗ ∣ ∣ ≤ ∣ ∣ e ⃗ ∣ ∣ ∣ ∣ u ⃗ ∣ ∣ ≤ c o n d ( A ) ⋅ ∣ ∣ r ⃗ ∣ ∣ ∣ ∣ b ⃗ ∣ ∣ \dfrac{1}{cond(\bold A)}\cdot\frac{||\vec{r}||}{||\vec{b}||}\le \dfrac{||\vec{e}||}{||\vec{u}||}\le cond(\bold A)\cdot\frac{||\vec{r}||}{||\vec{b}||} cond(A)1​⋅∣∣b ∣∣∣∣r ∣∣​≤∣∣u ∣∣∣∣e ∣∣​≤cond(A)⋅∣∣b ∣∣∣∣r ∣∣​

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