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    $\begingroup$ Have you tried LeastSquares[a, b]? $\endgroup$ Commented Jun 8, 2022 at 14:40
  • $\begingroup$ @MarcoB, thanks. After about 10 mins it spits out a solution that when plugged back in has a small error for roughly 250~ equations and enormous errors for a lot of the others. I guess the initial numerical errors in a and b are too large despite using 200 digits precision. So this probably indicates that there is little to salvage here. $\endgroup$ Commented Jun 8, 2022 at 15:05
  • $\begingroup$ Also I was hoping this example could serve as a practice for a similar problem with additional positive definiteness constraints. Is there an equivalent of LeastSquares that allows positive definiteness constraints (on 2x2 matrices consisting of the x[i]). SemidefiniteOptimization only accepts a linear f which is why I thought I should be solving a truncated set of equations exactly rather than minimizing an error. Minimizing the error will involve squaring the terms in the objective. Can such a LeastSquares problem with these types of constraints still be solved in Mathematica. $\endgroup$ Commented Jun 8, 2022 at 15:13
  • $\begingroup$ (I think the problem is still supposed to be convex by the same logic that LeastSquares is convex despite using a quadratic objective.) $\endgroup$ Commented Jun 8, 2022 at 15:14