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METHODOLOGY
Suppose the measured data is given by:
where is the true data, is the error in the data.
If a vector was the true solution then . Using we can get an approximate solution for given by:
Substituting in , we get:
This gives us: which we can use to find the upper bound of the relative error .
To do this we will start with:
...and then multiply both sides by to get:
We know that and we can substitute this into the above equation to get:
It can be proven that that for any matrix A and any vector x, the equality will always hold true. We can use this in solving for the relative error. Using this equality, we get:
Divide both sides by :
Since :
We have finished finding an equation which tells us an upper bound on the relative error . This says the larger the condition number, the less control we have on the relative error.
ANSWER
To solve this question we have to substitute in and . The question gives , but we want the largest possible relative error . Since is proportional to , we want the largest value of which is 0.1, so we will use . From the part (c) of question 1, we got that , and we have both values needed to solve for the relative error.
FINAL ANSWER
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