Consider using the right-preconditioned GMRES (AB-GMRES) for obtaining the minimum-norm solution of inconsistent underdetermined systems of linear equations. Morikuni (Ph.D. thesis, 2013) showed that for some inconsistent and ill-conditioned problems, the iterates may diverge. This is mainly because the Hessenberg matrix in the GMRES method becomes very ill-conditioned so that the backward substitution of the resulting triangular system becomes numerically unstable. We propose a stabilized GMRES based on solving the normal equations corresponding to the above triangular system using the standard Cholesky decomposition. This has the effect of shifting upwards the tiny singular values of the Hessenberg matrix which lead to an inaccurate solution. We analyze why the method works. Numerical experiments show that the proposed method is robust and efficient, not only for applying AB-GMRES to underdetermined systems, but also for applying GMRES to severely ill-conditioned range-symmetric systems of linear equations.
翻译:Morikuni (Ph.D. Thesis, 2013) 显示,对于某些不一致和条件不完善的问题,迭代者可能会出现差异,这主要是因为GMRES方法中的赫森堡矩阵变得非常不成熟,从而导致的三角系统的后向替代在数字上变得不稳定。我们提议一个稳定的 GMRES,其依据是使用标准Cholesky分解法解决与上述三角系统相对应的正常方程式。其效果是将赫森堡矩阵的极小单数值向上移,导致一个不准确的解决方案。我们分析该方法为何起作用。数字实验表明,拟议方法既健全又有效,不仅用于将AB-GMRES应用到定型系统,而且用于将GMRES应用到严重不完善的线性方程对称系统。