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These two techniqies seem closely related:

Are the mathematics the same, just different communities (math or stats), like in Tikhonov regularization or ridge regression?

Or is there a difference, e.g., that IRLS neglects correlations (after weighted least squares) while FGLS does not (after generalized least squares)?

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No, IRLS is a strategy for solving more general p-norm minimization problems by means of a sequence of related 2-norm (least squares) problems.

FGLS may be specialized for a diagonal covariance matrix in Feasible Weighted Least Squares (FWLS).

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