A preconditioned variant of the Golub and Kahan (1965) bidiagonalization process recently proposed by Arioli (2013) and Arioli and Orban (2013) allows us to establish that SYMMLQ and MINRES applied to least-squares problems in symmetric saddle-point form perform redundant work and are combinations of methods such as LSQR and LSMR. A well-chosen preconditioner allows us to formulate a projected variant of the Golub-Kahan process that forms the basis of specialized numerical methods for linear least-squares problems with linear equality constraints. As before, full-space methods such as SYMMLQ and MINRES applied to the symmetric saddle-point system defining the optimality conditions of such problems perform redundant work and are combinations of projected variants of methods such as LSQR and LSMR. We establish connections between numerical methods for least-squares problems, full-space methods and the projected and constraint-preconditioned Krylov methods of Gould, Orban, and Rees (2013).
Published March 2014 , 23 pages