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Hint for a problem on condition number

kalish

Member
Oct 7, 2013
99
I would like to know if the second part of this question is asking something different.

**Problem:** Consider the linear system $19x_1+20x_2=b_1, 20x_1+21x_2=b_2$. Compute the condition number of the coefficient matrix. Is the system well-conditioned with respect to perturbations of the right-handside constants ${b_1,b_2}$?

Do I need to introduce a $\delta$ into the right-handside, or is computing the coefficient number enough to conjecture about the condition of the right-handside constants?

Thanks.
 

Ackbach

Indicium Physicus
Staff member
Jan 26, 2012
4,197
The second part of the problem as asking if the condition number you just computed is highly dependent on $b_{1}$ and $b_{2}$. What happens if you change the RHS's just a little? Does the condition number change a lot when you do that? I don't think you need to introduce another variable, at least not yet. What do you get for the condition number? And how are you computing it?
 

kalish

Member
Oct 7, 2013
99
I get 1601 = 41*41 for the condition number, and I got it by computing the norm of the matrix A and the norm of the matrix A^(-1), an then multiplying them together. The norms of the matrices are both 41.

Isn't this enough for the purposes of this problem?
 

Ackbach

Indicium Physicus
Staff member
Jan 26, 2012
4,197
Did you use the formula
$$\|A\|= \max \{ \|Ax \|:x \in \mathbb{R}^{2}, \|x \|=1 \}?$$
If so, what vector norm did you use? Euclidean?
 

kalish

Member
Oct 7, 2013
99
I used the following norm:

$$\|A_{n\mathbb x n}|=\max_{1\leq i \leq n}\sum_{j=1}^{n}\|a_{ij}\|$$