S. T. Choi, M. A. Saunders, "Algorithm & Documentation: MINRES-QLP for Singular Symmetric and Hermitian Linear Equations and Least-Squares Problems," Preprint ANL/MCS-P3027-0812, August 2012. [pdf]
We describe algorithm MINRES-QLP and its FORTRAN 90 implementation for solving symmetric or Hermitian linear systems or least-squares problems. If the system is singular, MINRES-QLP computes the unique minimum-length solution (also known as the pseudoinverse solution), which generally eludes MINRES. In all cases, it overcomes a potential instability in the original MINRES algorithm. A positive-definite preconditioner may be supplied. Our FORTRAN 90 implementation illustrates a design pattern that allows users to make problem data known to the solver, but hidden and secure from other program units. In particular, we circumvent the need for reverse communication. While we focus here on a FORTRAN 90 implementation, we also provide and maintain MATLAB versions of MINRES and MINRES-QLP.