# Assessing numerical error in solving a least squares problem

I have a linear system of the type

$$Ax = b$$

I want to minimise $$|b - Ax|^2$$. I know there are different approaches to directly solve the system (Normal equation + Cholesky, QR decomposition, SVD decomposition) that have different numerical stability.

I would like to ask how it is possible to estimate the impact of the numerical error on the estimates of each parameter $$x$$.

• Can it be done solely on the base of the condition number?

• Should it be done using Monte Carlo simulation?

• How should such simulation be designed?

• Is your problem a linear system (square?), or a least squares problem $\min \|Ax-b\|$? Because the algorithms you list are the classical choices to solve LS problems, not to solve linear systems (even though some of those decompositions can do both jobs). – Federico Poloni Mar 21 '19 at 8:05
• @FedericoPoloni You are right the question was unclear, I edited it. – pinpon Mar 22 '19 at 8:18

$$Ax - b = r\approx 0$$
The solver will usually provide you with the residual error like: $$|r|$$.