A is a large ill-conditioned matrix that is available only as a function performing matrix-vector products, and a diagonal (weight) matrix W that is full rank. The eigenvalues of A'A are widely spread whereas the eigenvalues of A'WA are almost flat, at least for the first 1000 or so. This is convenient for solving least squares problems but makes it difficult to get an accurate max eigenvalue by power iteration.
Is there a way to estimate the largest eigenvalue of A'WA from that of A'A and W plus knowledge that the spectrum of A'WA is flat. I considered the svd (A = USV'):
A'WA = VSU'WUSV' = QTQ' (the svd of A'WA) ~ σI
This leaves σI ~ SU'WUS which is still a mess. Is it possible to estimate σ somehow from W and the largest eigenvalue of A'A.