We present a polynomial preconditioner for solving large systems of linear equations. The polynomial is derived from the minimum residual polynomial and is straightforward to compute and implement. It this paper, we study the polynomial preconditioner applied to GMRES; however it could be used with any Krylov solver. Stability control using added roots allows for high degree polynomials. We discuss the effectiveness and challenges of root-adding and give an additional check for stability. This polynomial preconditioning algorithm can dramatically improve convergence for difficult problems and can reduce dot products by an even greater margin.
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