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Linear systems in applications are typically well-posed, and yet thecoefficient matrices may be nearly singular in that the condition number$\kappa(\boldsymbol{A})$ may be close to $1/\varepsilon_{w}$, where$\varepsilon_{w}$ denotes the unit roundoff of the working precision. It iswell known that iterative refinement (IR) can make the forward errorindependent of $\kappa(\boldsymbol{A})$ if $\kappa(\boldsymbol{A})$ issufficiently smaller than $1/\varepsilon_{w}$ and the residual is computed inhigher precision. We propose a new iterative method, calledForward-and-Backward Stabilized Minimal Residual or FBSMR, by conceptuallyhybridizing right-preconditioned GMRES (RP-GMRES) with quasi-minimization. Wedevelop FBSMR based on a new theoretical framework ofessential-forward-and-backward stability (EFBS), which extends the backwarderror analysis to consider the intrinsic condition number of a well-posedproblem. We stabilize the forward and backward errors in RP-GMRES to achieveEFBS by evaluating a small portion of the algorithm in higher precision whileevaluating the preconditioner in lower precision. FBSMR can achieve optimalaccuracy in terms of both forward and backward errors for well-posed problemswith unpolluted matrices, independently of $\kappa(\boldsymbol{A})$. Withlow-precision preconditioning, FBSMR can reduce the computational, memory, andenergy requirements over direct methods with or without IR. FBSMR can alsoleverage parallelization-friendly classical Gram-Schmidt in Arnoldi iterationswithout compromising EFBS. We demonstrate the effectiveness of FBSMR using bothrandom and realistic linear systems.

Optimal Solutions of Well-Posed Linear Systems via Low-Precision Right-Preconditioned GMRES with Forward and Backward Stabilization