  We consider the problem of efficiently solving large-scale linear least
squares problems that have one or more linear constraints that must be
satisfied exactly. Whilst some classical approaches are theoretically well
founded, they can face difficulties when the matrix of constraints contains
dense rows or if an algorithmic transformation used in the solution process
results in a modified problem that is much denser than the original one. To
address this, we propose modifications and new ideas, with an emphasis on
requiring the constraints are satisfied with a small residual. We examine
combining the null-space method with our recently developed algorithm for
computing a null space basis matrix for a "wide" matrix. We further show that a
direct elimination approach enhanced by careful pivoting can be effective in
transforming the problem to an unconstrained sparse-dense least squares problem
that can be solved with existing direct or iterative methods. We also present a
number of solution variants that employ an augmented system formulation, which
can be attractive when solving a sequence of related problems. Numerical
experiments using problems coming from practical applications are used
throughout to demonstrate the effectiveness of the different approaches.
