Definition of relative error

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Hi all,

I have a general question about relative error. Suppose that we have a vector of measurements [itex]\hat{b}=\left(\hat{b_{1}},\hat{b_{2}},...,\hat{b_{n}}\right)[/itex]. Furthermore, suppose that these measurements are accurate to 10%.

My natural interpretation of this statement is that there is a "true" vector [itex]b=\left(b_{1},b_{2},...,b_{n}\right)[/itex] such that [itex]\frac{\left|b_{1}-\hat{b_{1}}\right|}{\left|b_{1}\right|}[/itex], [itex]\frac{\left|b_{2}-\hat{b_{2}}\right|}{\left|b_{2}\right|}[/itex], ..., [itex]\frac{\left|b_{n}-\hat{b_{n}}\right|}{\left|b_{n}\right|}≤0.1[/itex].

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 [itex]\hat{b}[/itex] could be defined as [itex]\frac{\left\|b-\hat{b}\right\|}{\left\|b\right\|}[/itex] where [itex]\left\|v\right\|=\max\limits_{i} \left|v_i\right|[/itex].

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 [itex]b=\left(1,2,3\right)[/itex] and [itex]\hat{b}=\left(1.14,1.9,3.15\right)[/itex], then [itex]\frac{\left\|b-\hat{b}\right\|}{\left\|b\right\|}=\frac{0.15}{3}=0.05≤0.1[/itex] which indicates that the relative error in [itex]\hat{b}[/itex] is less than 10%. On the other hand, the relative error in the first entry of [itex]\hat{b}[/itex] is [itex]\frac{0.14}{1}=0.14≥0.1[/itex].

Now, suppose we solve the systems [itex]A\hat{x}=\hat{b}[/itex] and [itex]Ax=b[/itex] where [itex]A[/itex] is invertible. According to the literature,

[itex]\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\|}[/itex]

Where the norm of a matrix [itex]A[/itex] is defined to be [itex]\max\limits_{i} \sum\limits_{j} \left|a_{ij}\right|[/itex].

If we know that the relative error in [itex]\hat{b}[/itex] is less than 10%, then we can put a bound on the relative error in [itex]\hat{x}[/itex]:

[itex]\frac{\left\|\hat{x}-x\right\|}{\left\|x\right\|}≤0.1\left\|A^{-1}\right\|\left\|A\right\|[/itex]

But as shown above, this does not put a bound on the relative error in the individual entries of [itex]\hat{x}[/itex]. 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!
 

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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.
 

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