Often one is not really interested in solving Ax=b for all possible b but just one particular b. In that case one can iteratively solve for x by algorithms like

Often one is not really interested in solving Ax=b for all possible b but just one particular b. In that case one can iteratively solve for x by algorithms like conjugate gradient which only require multiplying A times the current best guess for x, which can be very fast and avoids numerical instability issues in many cases.