MATH 4260 -数值分析代考| Numerical Analysis代写| 线性代数代写

以下是MATH 4260 – Numerical Analysis: Linear and Nonlinear Problems的考试案例的答案解析
问题 1.

Let $A \in \mathbb{R}^{6 \times 6}$ be a matrix with characteristic polynomial
$$
p_{A}(\lambda):=\operatorname{det}(A-\lambda I)=(\lambda+4)(\lambda-1)(\lambda-2)(\lambda-3)\left(\lambda^{2}+1\right)
$$
(a) [2 Points] Determine all eigenvalues of $A$.

(b) [3 Points] We apply the power method to $A$. Which eigenvalue does the method converge to and what is the the ratio of the convergence rate?

(c) [4 Points] We apply inverse iteration with shift $s=\frac{7}{4}$ to $A$. Which eigenvalue does the method converge to and what is the ratio of the convergence rate?
Solution: The eigenvalues of the shifted and inverted matrix

证明 .

Solution: The eigenvalues of $A$ are the zeros of $p_{A}(\lambda):$
$$
-4,1,2,3, i,-i
$$
where $i=\sqrt{-1}$

 The dominant eigenvalue of $A$ is $-4,$ and the second largest in absolute value is $3 .$ Thus the power method converges to the eigenvalue -4 and
$$
\text { ratio }=\frac{3}{4}
$$

The eigenvalues of the shifted and inverted matrix
$$
(A-s I)^{-1}=\left(A-\frac{7}{4} I\right)^{-1}
$$
are
$$
\frac{1}{-4-\frac{7}{4}}, \quad \frac{1}{1-\frac{7}{4}}, \quad \frac{1}{2-\frac{7}{4}}, \quad \frac{1}{3-\frac{7}{4}}, \quad \frac{1}{1-\frac{7}{4}}, \quad \frac{1}{-\mathrm{i}-\frac{7}{4}}
$$
The dominant eigenvalue of $(A-s I)^{-1}$ is
$$
\frac{1}{2-\frac{7}{4}}=4
$$
and the second largest in absolute value is
$$
\frac{1}{1-\frac{7}{4}}=-\frac{4}{3}
$$
Thus inverse iteration converges to the eigenvalue 2 of $A$ and
ratio $=\frac{1}{3}$.

问题 2.

Let
$$
A=\left[\begin{array}{rc}
1 & -1+\delta \\
-1 & 1
\end{array}\right]
$$
where $\delta \in \mathbb{R}$ is a parameter.
(a) [2 Points] Show that $A$ is nonsingular if, and only if, $\delta \neq 0$. Determine $A^{-1}$ for $\delta \neq 0$.

(b) [4 Points] For $\delta \neq 0$, determine the condition number $\kappa_{1}(A)$ of $A$ with respect to the matrix norm $\|\cdot\|_{1}$

(c) [3 Points] Discuss the accuracy of the computed solution that you can expect when you solve linear systems $A x=b$ numerically.

证明 .

Solution: Since
$$
\operatorname{det} A=1 \cdot 1-(-1+\delta) \cdot(-1)=\delta
$$
the matrix $A$ is nonsingular if, and only if, $\delta \neq 0$.
For $\delta \neq 0$
$$
A^{-1}=\frac{1}{\operatorname{det} A} \operatorname{Adj}(A)=\frac{1}{\delta}\left[\begin{array}{cc}
1 & 1-\delta \\
1 & 1
\end{array}\right]
$$

 Recall that $\|\cdot\|_{1}$ is the maximumabsolute column sum of the matrix. Thus, we have
$$
\|A\|_{1}=\max \{2,|-1+\delta|+1\}=\left\{\begin{array}{ll}
2 & \text { if } 0<\delta \leq 2 \\
1+|\delta-1| & \text { otherwise }
\end{array}\right.
$$
and
$$
\left\|A^{-1}\right\|_{1}=\frac{1}{|\delta|} \max \{2,|1-\delta|+1\}=\left\{\begin{array}{ll}
\frac{2}{|\delta|} & \text { if } 0<\delta \leq 2, \\
\frac{1+|\delta-1|}{|\delta|} & \text { otherwise. }
\end{array}\right.
$$
It follows that
$$
\kappa_{1}(A)=\|A\|_{1} \cdot\left\|A^{-1}\right\|_{1}=\left\{\begin{array}{ll}
\frac{4}{|\delta|} & \text { if } 0<\delta \leq 2, \\
\frac{(1+|\delta-1|)^{2}}{|\delta|} & \text { otherwise }
\end{array}\right.
$$

 If $\kappa_{1}(A) \gg 1,$ the problem of solving linear systems $A x=b$ is ill-conditioned. In this case, we cannot expect to solve $A x=b$ numerically with any reasonable accuracy, no matter what algorithm we employ. The expression for $\kappa_{1}(A)$ obtained in (b) shows that $A x=b$ is ill-conditioned for $\delta \approx 0$ and for $|\delta| \gg 1$.

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