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October 30th, 2018, 06:17 PM  #1 
Member Joined: Feb 2018 From: Canada Posts: 45 Thanks: 2  Need help for some Numerical Analysis questions.
I having two problems and I have no idea where to start. Question 1) Consider the two skew lines in $R^3$ given by: \[ r_1(s) = (2,0,0) + s(1,1,1); \quad\text{and}\quad r_2(t) = (2,3,1) + t(1,2,1) \] To find the pair of points on the linear which minimize the distance, setup and solve the overdetermined linear system $r_1(s) = r_2(t)$ using a QR decomposition. Question 2) Let $A = \begin{bmatrix} 1 & 1 & 3 \\ 2 & 1 & 5 \\ 2&1& 1 \end{bmatrix} $. Find an upper bound for $cond_2(A)$ using the Frobenius norm. I've been thinking for over an hour and I'm pretty frustrated! If anyone could help me, I would really appreciate it! Last edited by Shanonhaliwell; October 30th, 2018 at 06:27 PM. 
October 31st, 2018, 01:03 AM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,382 Thanks: 1281 
for (1) you want to solve $r_1(s) = r_2(t)$ $r_1(s)  r_2(t) = 0$ $(2,0,0)+s(1,1,1)  (2,3,1)  t(1,2,1) = 0$ waves magic wand $\begin{pmatrix}1&1 \\ 1 &2 \\ 1 &1\end{pmatrix}\begin{pmatrix}s\\t\end{pmatrix} = \begin{pmatrix}0\\3\\1\end{pmatrix}$ or $Ax = b$ $A$ as written has no inverse but you can decompose this as $QRx = b$ where $Q$ is orthogonal and $R$ is upper triangular Then $\begin{pmatrix}s\\t\end{pmatrix} = x = R^{1}Q^T b$ 
October 31st, 2018, 01:09 AM  #3 
Senior Member Joined: Sep 2015 From: USA Posts: 2,382 Thanks: 1281  
October 31st, 2018, 06:09 PM  #4  
Member Joined: Feb 2018 From: Canada Posts: 45 Thanks: 2  Quote: For the second problem, I am not so sure about this. This is what I have gotten so far: \[cond_2(A) = \A\_2\A^{1}\_2 \leq \A\_F\A^{1}\_F \]  

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