QR factorization of the Jacobian in some structured nonlinear least squares problems

For solving nonlinear least squares problems a Gauss-Newton trust region method is often employed. In the case of orthogonal distance regression it has been believed that solving the resulting linear problem at each trust region step by computing the QR factorization of the full Jacobian matrix would be very inefficient. By taking full advantage of the structure of the sparse blocks in the QR factorization of the Jacobian, we derive here an algorithm with the same overall complexity, which uses a QR factorization of the full Jacobian matrix. The same observation applies also to sparse structured total least squares problems, where similarly structured Jacobian matrices occur.