The QR decomposition can be used to solve linear systems of the form Ax=b, say of order n, but it induces a considerable increasing relative (or absolute) errors on the unknowns x, specifically the ones with the highest orders when the system dimension is large.
Any explanation for this fact?
Thank you.
The QR method of solving linear equations is normwise backward stable, but not componentwise backward stable. The LU method of solving linear equations is normwise backward stable and componentwise backward stable.
Therefore if $A$ has large standard condition number but small Skeel condition number, then the forward error of solving the problem $Ax = b$ using the QR method can be much larger, than the forward error using the LU method.