这是一份umass麻省大学 MATH 545作业代写的成功案例
问题 1.
$$
A^{(n-1)}=U
$$
Define
$$
Q=Q_{1}, \cdots Q_{n-1}, P=P_{n-1} P_{n-2} \ldots P_{1},
$$
证明 .
and
$$
L=P\left(M_{n-1} P_{n-1} \ldots M_{1} P_{\mathrm{i}}\right)^{-1} .
$$
Then it can he shown that
$$
P A Q=L U,
$$
where $P$ and $Q$ are both permutution matrices and $L$ is unit triangular and $U$ is upper triangular.
MATH 545 COURSE NOTES ：
$$
A=\left(\begin{array}{cc}
0.00011 & 1 \
1 & 1
\end{array}\right) \text {. }
$$
- Gaussian elimination without pivoting gives
$$
A^{(1)}=U=\left(\begin{array}{cc}
0.0001 & 1 \
0 & -10^{4}
\end{array}\right) \text {, }
$$$\hat{\rho}=$ the growth factor $=10^{4}$. - Gaussian elimination with partial pivoting yields
$$
\begin{gathered}
A^{(1)}=U=\left(\begin{array}{ll}
1 & 1 \
0 & 1
\end{array}\right), \
\max \left|a_{i j}^{(1)}\right|=1, \max \left|a_{i j}\right|=1, \
p=\text { the growth factor }=1 .
\end{gathered}
$$