Definition of relative error

[andresc889]

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!

 

[Mark44]

For your example, with $b = <1, 2, 3>$ and $\hat b = <1.14, 1.9, 3.15>$ a vector of the absolute values of the relative errors would be $<\frac {.14} 1, \frac {.1}2, \frac {.15} 3> = < .14, .05, .05>$. The mean of these values is .24/3 = .08 which is less than .1.