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The Lack of Influence of the Right-Hand Side on the Accuracy of Linear System Solution

It is commonly believed that a fortunate right-hand side b can significantly reduce the sensitivity of a system of linear equations Ax=b. We show, both theoretically and experimentally, that this is not true when the system is solved (in floating point arithmetic) with Gaussian elimination or the QR factorization: the error bounds essentially do not depend on b, and the error itself seems to depend only weakly on b. Our error bounds are exact (rather than first-order); they are tight; and they are stronger than the bound of Chan and Foulser. 
We also present computable lower and upper bounds for the relative error. The lower bound gives rise to a stopping criterion for iterative methods that is better than the relative residual. This is because the relative residual can be much larger, and it may be impossible to reduce it to a desired tolerance.