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October 30th, 2018, 07:17 PM   #1
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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 07:27 PM.
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October 31st, 2018, 02:03 AM   #2
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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

$\begin{pmatrix}s\\t\end{pmatrix} = x = R^{-1}Q^T b$

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October 31st, 2018, 02:09 AM   #3
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for #2 have a look at this

http://faculty.nps.edu/rgera/ma3042/2009/ch7.4.pdf
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October 31st, 2018, 07:09 PM   #4
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Quote:
Originally Posted by romsek View Post
Thanks to you, I was able to solve the first problem by using Gram-Schmidt procedure.
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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