Are Preconditioned Iterative Linear System Solvers Backward Stable?

Suppose one has very good preconditioners for a class of n by n matrices A – the preconditioned matrix has condition number O(1), independent of n. Could an iterative method replace the standard Gaussian elimination or QR decomposition for solving Ax = b, delivering the same (or better) accuracy? If so, then one would have a practical method for solving a class of linear systems with the same number of operations that it takes to construct the preconditioner and apply the preconditioned matrix (and its transpose if it is nonsymmetric) to a fixed number of vectors, typically O(n 2 ) but possibly O(n).

This is the question that I will explore in this talk, considering station- ary iterative methods (Jacobi, Gauss-Seidel, SOR, etc.), updated residual methods (steepest descent, conjugate gradients, etc.), and the preconditioned Lanczos algorithm, along with iterative refinement.