I was looking at old exam papers and was stuck on the following problem:
I have hardly any idea how to progress with the problem. Can some give some explanation about how to progress with the problem?
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Is the question to choose which of the numbers 1-4 is a correct statement? – Zach L. Jun 09 '13 at 05:25
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1@learner: Hint: Look at the condition number of A. – Amzoti Jun 09 '13 at 05:41
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I think options 1,2 correct.Am I right? – learner Jun 09 '13 at 05:51
1 Answers
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I am not sure what "large" and "small" mean precisely here, but I would imagine that this is getting at the operator norm. Given a matrix $A$, there is the operator norm $\|A\|$ which satisfies $$\|Ax \| \leq \|A\|\|x\|$$ for any $x$. In the language of the question, we have the equation $$\|r\| = \|A(x - x_c)\| = \|Ae\| \leq \|A\|\|e\|.$$ If we push the vector $r$ to zero, what happens to $e$, given this inequality? If we push $e$ to zero, what happens to $r$?
Zach L.
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From your calculation it appears to me (1) is the correct choice as if I push $e \to 0 \implies r \to 0.$ Am I right? But here,more than one options is possible.Can other options be a possibility? I also think option 2 is possible whereas 3 is not possible..option 4 is clearly false. So,1,2 are correct.Anything wrong? – learner Jun 09 '13 at 05:40
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3 and 4 don't seem correct. 2 does. For instance, if $A(x,y) = x-y$, then there are arbitrarily large errors that will be sent to zero! – Zach L. Jun 09 '13 at 05:53
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