CG, MINRES, and SYMMLQ are Krylov subspace methods for solving large symmetric systems of linear equations. CG (the conjugate-gradient method) is reliable on positive-definite systems, while MINRES and SYMMLQ are designed for indefinite systems. When these methods are applied to an inconsistent system (that is, a singular symmetric least-squares problem), CG could break down and SYMMLQ's solution could explode, while MINRES would give a least-squares solution but not necessarily the minimum-length solution (often called the pseudoinverse solution). This understanding motivates us to design a MINRES-like algorithm to compute minimum-length solutions to singular symmetric systems.; MINRES uses QR factors of the tridiagonal matrix from the Lanczos process (where R is upper-tridiagonal). Our algorithm uses a QLP decomposition (where rotations on the right reduce R to lower-tridiagonal form), and so we call it MINRES-QLP. On singular or nonsingular systems, MINRES-QLP can give more accurate solutions than MINRES or SYMMLQ. We derive preconditioned MINRES-QLP, new stopping rules, and better estimates of the solution and residual norms, the matrix norm and condition number.; For a singular matrix of arbitrary shape, we observe that null vectors can be obtained by solving least-squares problems involving the transpose of the matrix. For sparse rectangular matrices, this suggests an application of the iterative solver LSQR. In the square case, MINRES, MINRES-QLP, or LSQR are applicable. Results are given for solving homogeneous systems, computing the stationary probability vector for Markov Chain models, and finding null vectors for sparse systems arising in helioseismology.
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