 My Math Forum Need help for some Numerical Analysis questions.
 User Name Remember Me? Password

 Linear Algebra Linear Algebra Math Forum

 October 30th, 2018, 06:17 PM #1 Member   Joined: Feb 2018 From: Canada Posts: 46 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: 2,529
Thanks: 1389

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

$\begin{pmatrix}s\\t\end{pmatrix} = x = R^{-1}Q^T b$ Attached Images Clipboard01.jpg (21.4 KB, 25 views) October 31st, 2018, 01:09 AM #3 Senior Member   Joined: Sep 2015 From: USA Posts: 2,529 Thanks: 1389 for #2 have a look at this http://faculty.nps.edu/rgera/ma3042/2009/ch7.4.pdf Thanks from Shanonhaliwell October 31st, 2018, 06:09 PM   #4
Member

Joined: Feb 2018
From: Canada

Posts: 46
Thanks: 2

Quote:
 Originally Posted by romsek for #2 have a look at this http://faculty.nps.edu/rgera/ma3042/2009/ch7.4.pdf
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:
$cond_2(A) = \|A\|_2\|A^{-1}\|_2 \leq \|A\|_F\|A^{-1}\|_F$ Tags analysis, numerical, questions Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post Faith Real Analysis 1 July 7th, 2014 04:33 AM evinda Applied Math 9 November 3rd, 2013 08:22 AM saugata bose Computer Science 0 March 8th, 2013 11:03 AM wale06 Applied Math 0 October 24th, 2011 04:44 PM

 Contact - Home - Forums - Cryptocurrency Forum - Top

Copyright © 2019 My Math Forum. All rights reserved.      