Polynomial preconditioning is an important tool in solving large linear systems and eigenvalue problems. A polynomial from GMRES can be used to precondition restarted GMRES and restarted Arnoldi. Here we give methods for indefinite matrices that make polynomial preconditioning more generally applicable. The new techniques include balancing the polynomial so that it produces a definite spectrum. Then a stability approach is given that is specialized for the indefinite case. Also, very complex spectra are examined. Then convergence estimates are given for polynomial preconditioning of real, indefinite spectra. Finally, tests are preformed of finding interior eigenvalues.

翻译：多项式预条件处理是求解大规模线性系统和特征值问题的重要工具。来自GMRES的多项式可用于预条件处理重启GMRES和重启Arnoldi方法。本文针对不定矩阵提出了使多项式预条件处理更具普适性的方法。新技术包括通过平衡多项式使其产生定号谱，以及专门针对不定情况设计的稳定性方法。此外，本文还研究了具有高度复杂谱结构的情形，给出了针对实数不定谱的多项式预条件处理收敛性估计。最后通过求解内部特征值的数值实验验证了方法的有效性。