  Golub-Kahan iterative bidiagonalization represents the core algorithm in
several regularization methods for solving large linear noise-polluted
ill-posed problems. We consider a general noise setting and derive explicit
relations between (noise contaminated) bidiagonalization vectors and the
residuals of bidiagonalization-based regularization methods LSQR, LSMR, and
CRAIG. For LSQR and LSMR residuals we prove that the coefficients of the linear
combination of the computed bidiagonalization vectors reflect the amount of
propagated noise in each of these vectors. For CRAIG the residual is only a
multiple of a particular bidiagonalization vector. We show how its size
indicates the regularization effect in each iteration by expressing the CRAIG
solution as the exact solution to a modified compatible problem. Validity of
the results for larger two-dimensional problems and influence of the loss of
orthogonality is also discussed.
