Preconditioned conjugate gradient method for rank deficient least-squares problems

Rank deficient least squares problems appear in obtaining numerical solution of differential equations, computational genetics and other applications. The usual methods to solve the problem are QR decomposition. It is well-known that for large sparse problems, iterative methods are preferable. Miller and Neumann (1987) proposed the 4-block SOR method, and Santos, Silva and Yuan (1997) proposed the 2-block SOR method and the 3-block SOR method for solving the problem. Here some preconditioned conjugate gradient methods are proposed for solving the problem. The error bound and comparison with block SOR methods are studied. We show the best iterative method is the preconditioned conjugate gradient method for solving rank deficient least squares problems.