MINRES-QLP: A Krylov Subspace Method for Indefinite or Singular Symmetric Systems
SIAM Journal on Scientific Computing
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Iterative methods are developed and studied for near-singular linear systems Cx = b. Our approach, called the transformed minimal residual algorithm (TMRES), is derived from any convergent iterative scheme Sxk+1 = Txk + b associated with a splitting C = S - T. In each step of TMRES, the transformed residual S-1(b- Cx) is minimized over a Krylov space generated by S-1 T. The original iterative scheme typically converges slowly when C is nearly singular, while a Krylov space generated by S-1 T often contains a much better approximation to a solution. TMRES is algebraically equivalent to the generalized minimal residual algorithm (GMRES) preconditioned by S-1, although there are numerical differences since a different matrix S-1 C is used to generate the Krylov space in preconditioned GMRES. Special attention is given to sparsity and convergence issues related to linear systems of the form $({\bf AA}\tr + \sigma{\bf I}){\bf x} =$ ${\bf b}$, where $\sigma \ge 0$.