- #1
andresc889
- 5
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Hi all,
I have a general question about relative error. Suppose that we have a vector of measurements \hat{b}=\left(\hat{b_{1}},\hat{b_{2}},...,\hat{b_{n}}\right). Furthermore, suppose that these measurements are accurate to 10%.
My natural interpretation of this statement is that there is a "true" vector b=\left(b_{1},b_{2},...,b_{n}\right) such that \frac{\left|b_{1}-\hat{b_{1}}\right|}{\left|b_{1}\right|}, \frac{\left|b_{2}-\hat{b_{2}}\right|}{\left|b_{2}\right|}, ..., \frac{\left|b_{n}-\hat{b_{n}}\right|}{\left|b_{n}\right|}≤0.1.
I have seen in the literature that we can use the maximum norm of a vector to define the relative error. So, the relative error in \hat{b} could be defined as \frac{\left\|b-\hat{b}\right\|}{\left\|b\right\|} where \left\|v\right\|=\max\limits_{i} \left|v_i\right|.
The problem that I find with this is the fact that we can't conclude anything about the individual entries from this definition. For example, if b=\left(1,2,3\right) and \hat{b}=\left(1.14,1.9,3.15\right), then \frac{\left\|b-\hat{b}\right\|}{\left\|b\right\|}=\frac{0.15}{3}=0.05≤0.1 which indicates that the relative error in \hat{b} is less than 10%. On the other hand, the relative error in the first entry of \hat{b} is \frac{0.14}{1}=0.14≥0.1.
Now, suppose we solve the systems A\hat{x}=\hat{b} and Ax=b where A is invertible. According to the literature,
\frac{\left\|\hat{x}-x\right\|}{\left\|x\right\|}≤\left\|A^{-1}\right\|\left\|A\right\|\frac{\left\|\hat{b}-b\right\|}{\left\|b\right\|}
Where the norm of a matrix A is defined to be \max\limits_{i} \sum\limits_{j} \left|a_{ij}\right|.
If we know that the relative error in \hat{b} is less than 10%, then we can put a bound on the relative error in \hat{x}:
\frac{\left\|\hat{x}-x\right\|}{\left\|x\right\|}≤0.1\left\|A^{-1}\right\|\left\|A\right\|
But as shown above, this does not put a bound on the relative error in the individual entries of \hat{x}. So my question is, what is the point of finding the relative error in the vector if we cannot use that to put a bound on the relative error of the individual entries? Maybe I'm misinterpreting something here?
Thanks!
I have a general question about relative error. Suppose that we have a vector of measurements \hat{b}=\left(\hat{b_{1}},\hat{b_{2}},...,\hat{b_{n}}\right). Furthermore, suppose that these measurements are accurate to 10%.
My natural interpretation of this statement is that there is a "true" vector b=\left(b_{1},b_{2},...,b_{n}\right) such that \frac{\left|b_{1}-\hat{b_{1}}\right|}{\left|b_{1}\right|}, \frac{\left|b_{2}-\hat{b_{2}}\right|}{\left|b_{2}\right|}, ..., \frac{\left|b_{n}-\hat{b_{n}}\right|}{\left|b_{n}\right|}≤0.1.
I have seen in the literature that we can use the maximum norm of a vector to define the relative error. So, the relative error in \hat{b} could be defined as \frac{\left\|b-\hat{b}\right\|}{\left\|b\right\|} where \left\|v\right\|=\max\limits_{i} \left|v_i\right|.
The problem that I find with this is the fact that we can't conclude anything about the individual entries from this definition. For example, if b=\left(1,2,3\right) and \hat{b}=\left(1.14,1.9,3.15\right), then \frac{\left\|b-\hat{b}\right\|}{\left\|b\right\|}=\frac{0.15}{3}=0.05≤0.1 which indicates that the relative error in \hat{b} is less than 10%. On the other hand, the relative error in the first entry of \hat{b} is \frac{0.14}{1}=0.14≥0.1.
Now, suppose we solve the systems A\hat{x}=\hat{b} and Ax=b where A is invertible. According to the literature,
\frac{\left\|\hat{x}-x\right\|}{\left\|x\right\|}≤\left\|A^{-1}\right\|\left\|A\right\|\frac{\left\|\hat{b}-b\right\|}{\left\|b\right\|}
Where the norm of a matrix A is defined to be \max\limits_{i} \sum\limits_{j} \left|a_{ij}\right|.
If we know that the relative error in \hat{b} is less than 10%, then we can put a bound on the relative error in \hat{x}:
\frac{\left\|\hat{x}-x\right\|}{\left\|x\right\|}≤0.1\left\|A^{-1}\right\|\left\|A\right\|
But as shown above, this does not put a bound on the relative error in the individual entries of \hat{x}. So my question is, what is the point of finding the relative error in the vector if we cannot use that to put a bound on the relative error of the individual entries? Maybe I'm misinterpreting something here?
Thanks!