  The weighted nonlinear least-squares problem for low-rank signal estimation
is considered. The problem of constructing a numerical solution that is stable
and fast for long time series is addressed. A modified weighted Gauss-Newton
method, which can be implemented through the direct variable projection onto a
space of low-rank signals, is proposed. For a weight matrix which provides the
maximum likelihood estimator of the signal in the presence of autoregressive
noise of order $p$ the computational cost of iterations is $O(N r^2 + N p^2 + r
N \log N)$ as $N$ tends to infinity, where $N$ is the time-series length, $r$
is the rank of the approximating time series. Moreover, the proposed method can
be applied to data with missing values, without increasing the computational
cost. The method is compared with state-of-the-art methods based on the
variable projection approach in terms of floating-point numerical stability and
computational cost.
