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I encountered the following algorithm (Algorithm 1 in a paper from Stroubolis on generalized finite element methods) for the solution of a the system $Ax = b$ where $A$ is positive semi-definite or nearly singular positive-definite matrix:

Define $A_\epsilon := A + \epsilon I$ for $I$ identity and $\epsilon > 0$. Then do

$$ c_0 = A_\epsilon^{-1}b, \quad r_0 = b - A c_0, \\ z_i = A_\epsilon^{-1}r_i, \quad v_i = A z_i, \quad c_{i+1} = c_i + z_i \\ r_{i+1} = r_i -v_i = b - Ac_i = (I - AA_\epsilon)^{-1}r_{i-1} $$

until $z_i^\top A z_i / c_i^\top A c_i $ is small enough. Then $c_i$ is taken as the result.

The authors did not provide any source or further information about the properties of this algorithm. Has anybody seen this algorithm elsewhere and could point me to a reference where I could find out more?

Thanks!

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  • $\begingroup$ I have edited the question. I would now only like to know if anybody has seen this algorithm somewhere and can point me to a reference. I do not ask anymore if anyone can explain properties of the algorithm. $\endgroup$ Commented May 16 at 11:18
  • $\begingroup$ It seems to be just a classical iterative method with a splitting $A=M-N$, where $M=A_{\epsilon}$. Any theory about these methods applied to solving singular systems (which I'm not familiar with) would then apply. $\endgroup$ Commented May 20 at 4:02
  • $\begingroup$ Thank you @AlgebraicPavel ! I will refresh my memory on these basics and apply it to this algorithm. You are probably right and this might also be why the authors did not provide any further information. $\endgroup$ Commented May 20 at 9:55

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