This report shows on real data that the direct methods such as LDL decomposition and Gaussian elimination for solving linear systems with ill-conditioned matrices provide inaccurate results due to divisions by very small numbers, which in turn results in peaking phenomena and large estimation errors. Richardson iteration provides accurate results without peaking phenomena since division by small numbers is absent in the Richardson approach. In addition, two preconditioners are considered and compared in the Richardson iteration: 1) the simplest and robust preconditioner based on the maximum row sum matrix norm and 2) the optimal one based on calculation of the eigenvalues. It is shown that the simplest preconditioner is more robust for ill-conditioned case and therefore it is recommended for many applications.

翻译：本报告基于实际数据表明，在求解病态矩阵线性系统时，LDL分解和高斯消元等直接方法会因除以极小数值导致结果不准确，进而引发峰值现象和较大估计误差。Richardson迭代因避免了除以小数值的环节，在无峰值现象的情况下提供了精确结果。此外，Richardson迭代中考虑并比较了两种预处理器：1）基于最大行和矩阵范数的最简鲁棒预处理器；2）基于特征值计算的最优预处理器。研究表明，最简预处理器在病态情况下具有更强的鲁棒性，因此推荐在多数应用场景中使用。