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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?

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    $\begingroup$ 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). $\endgroup$
    – Federico Poloni
    Mar 21, 2019 at 8:05
  • $\begingroup$ @FedericoPoloni You are right the question was unclear, I edited it. $\endgroup$
    – pinpon
    Mar 22, 2019 at 8:18

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I think most solvers will give you a residual error. If you flip your equation to read:

$Ax - b = r\approx 0$

The solver will usually provide you with the residual error like: $|r|$.

Now if you want to compare different solvers you might use that as a measurement of the error. As a rule of thumb: you will not find a solver which is optimal for every System A. Usually there are also speed/stability tradeofs or such.

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