MHB Upper bound of the relative error

AI Thread Summary
The discussion revolves around finding the upper bound of the relative error in a linear system defined by the matrix A and vector b. The initial claim of a relative error of 2 is confirmed as correct, while an alternative upper bound of 2.001 is also suggested. Participants discuss using the condition number of the matrix to derive the upper bound without directly using the values of x and y. The formula relating the error in x to the error in b is clarified, leading to a refined estimate of the relative error as less than or equal to 2.0005. The conversation concludes with participants confirming their findings and expressing gratitude for the assistance.
evinda
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Hello! :)
I am looking at the following exercise:
Let the linear system $Ax=b$ with $\begin{pmatrix}
2.001 & 2\\
2& 2
\end{pmatrix}$
,$b=\begin{bmatrix}
2.001 &2
\end{bmatrix}^T$ and y an approximate solution,so that $Ay-b=\begin{bmatrix}
0.001 &0
\end{bmatrix}^T$ .Find an upper bound of the relative error $\frac{||x-y||_{1}}{||x||_{1}}$ .

I found that it is equal to 2,could you tell me if it is right?
 
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evinda said:
Hello! :)
I am looking at the following exercise:
Let the linear system $Ax=b$ with $\begin{pmatrix}
2.001 & 2\\
2& 2
\end{pmatrix}$
,$b=\begin{bmatrix}
2.001 &2
\end{bmatrix}^T$ and y an approximate solution,so that $Ay-b=\begin{bmatrix}
0.001 &0
\end{bmatrix}^T$ .Find an upper bound of the relative error $\frac{||x-y||_{1}}{||x||_{1}}$ .

I found that it is equal to 2,could you tell me if it is right?

Hi! ;)

The actual result is $\frac{||x-y||_{1}}{||x||_{1}} = 2$.
But assuming we're not supposed to use the solution for $x$ and $y$, I get an upper bound of $2.001$.
 
I like Serena said:
Hi! ;)

The actual result is $\frac{||x-y||_{1}}{||x||_{1}} = 2$.
But assuming we're not supposed to use the solution for $x$ and $y$, I get an upper bound of $2.001$.

How can we find the upper bound,without using the values of $x$ and $y$ ? :confused:
 
evinda said:
How can we find the upper bound,without using the values of $x$ and $y$ ? :confused:

By using the condition number of the matrix.
$$\text{cond}_1(A) = ||A||_1 \cdot ||A^{-1}||_1$$
Can it be that it is in your notes? :rolleyes:
 
I like Serena said:
By using the condition number of the matrix.
$$\text{cond}_1(A) = ||A||_1 \cdot ||A^{-1}||_1$$
Can it be that it is in your notes? :rolleyes:

To elaborate:
$$\frac{||\Delta x||_1}{||x||_1} \le \text{cond}_1(A) \frac{||\Delta b||_1}{||b||_1}$$
 
I like Serena said:
To elaborate:
$$\frac{||\Delta x||_1}{||x||_1} \le \text{cond}_1(A) \frac{||\Delta b||_1}{||b||_1}$$

We use $\frac{||\Delta x||_1}{||x||_1} \le \text{cond}_1(A) \frac{||\Delta b||_1}{||b||_1}$,if we have a derangment of $x$ and $b$.Right?But...is there a derangment of $b$ in this case?? :confused:
 
evinda said:
We use $\frac{||\Delta x||_1}{||x||_1} \le \text{cond}_1(A) \frac{||\Delta b||_1}{||b||_1}$,if we have a derangment of $x$ and $b$.Right?But...is there a derangment of $b$ in this case?? :confused:

Huh? How are derangements involved?? :confused:
What do you think a derangement is?

When $Ax=b$, then $\frac{||\Delta x||_1}{||x||_1} \le \text{cond}_1(A) \frac{||\Delta b||_1}{||b||_1}$ applies.
 
I like Serena said:
Huh? How are derangements involved?? :confused:
What do you think a derangement is?

When $Ax=b$, then $\frac{||\Delta x||_1}{||x||_1} \le \text{cond}_1(A) \frac{||\Delta b||_1}{||b||_1}$ applies.

Could I also do this like that: $\frac{||x-y||}{||x||}=\frac{||A^{-1}Ay-A^{-1}b||}{||x||}=\frac{||A^{-1}r||}{||x||}$ or would this be wrong?

- - - Updated - - -

I like Serena said:
Huh? How are derangements involved?? :confused:
What do you think a derangement is?

When $Ax=b$, then $\frac{||\Delta x||_1}{||x||_1} \le \text{cond}_1(A) \frac{||\Delta b||_1}{||b||_1}$ applies.

Don't we ue this formula if b and x change?? :confused:
 
evinda said:
Could I also do this like that: $\frac{||x-y||}{||x||}=\frac{||A^{-1}Ay-A^{-1}b||}{||x||}=\frac{||A^{-1}r||}{||x||}$ or would this be wrong?

That is correct... but it doesn't get us where we need to go...

Note that $||\Delta x|| = ||x - y||$, which is the error in $x$.
And that $||\Delta b|| = || Ay - b ||$, the error in $b$.
- - - Updated - - -
Don't we ue this formula if b and x change?? :confused:

Huh?? :confused:
 
  • #10
I like Serena said:
That is correct... but it doesn't get us where we need to go...

Note that $||\Delta x|| = ||x - y||$, which is the error in $x$.
And that $||\Delta b|| = || Ay - b ||$, the error in $b$.
I understand now! :) I applied the formula you said me and I found that the relative error is $\leq 2.0005$.Have you found the same??
 
  • #11
evinda said:
I understand now! :) I applied the formula you said me and I found that the relative error is $\leq 2.0005$.Have you found the same??

Yes! (Nod)
 
  • #12
I like Serena said:
Yes! (Nod)

Great!Thank you very much! :)
 
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