Motivated by a recent work on a preconditioned MINRES for flipped linear systems in imaging, in this note we extend the scope of that research for including more precise boundary conditions such as reflective and anti-reflective ones. We prove spectral results for the matrix-sequences associated to the original problem, which justify the use of the MINRES in the current setting. The theoretical spectral analysis is supported by a wide variety of numerical experiments, concerning the visualization of the spectra of the original matrices in various ways. We also report numerical tests regarding the convergence speed and regularization features of the associated GMRES and MINRES methods. Conclusions and open problems end the present study.

翻译：受近期关于成像中翻转线性系统的预条件MINRES研究的启发，本文将该研究范围扩展至包含更精确的边界条件（如反射和反反射条件）。我们证明了原始问题相关矩阵序列的谱性质，从而论证了在当前设定下使用MINRES方法的合理性。该理论谱分析得到了大量数值实验的支持，这些实验以多种方式可视化了原始矩阵的谱分布。我们还报告了关于相关GMRES和MINRES方法收敛速度与正则化特征的数值测试结果。最后，本文以结论与未解决问题作为结束。