Abstract The problem of solving large-scale systems of linear algebraic equations arises in a wide range of applications. In many cases the preconditioned iterative method is a method of choice. This paper deals with the approximate inverse preconditioning AINV/SAINV based on the incomplete generalized Gram–Schmidt process. This type of the approximate inverse preconditioning has been repeatedly used for matrix diagonalization in computation of electronic structures but approximating inverses is of an interest in parallel computations in general. Our approach uses adaptive dropping of the matrix entries with the control based on the computed intermediate quantities. Strategy has been introduced as a way to solve di cult application problems and it is motivated by recent theoretical results on the loss of orthogonality in the generalized Gram– Schmidt process. Nevertheless, there are more aspects of the approach that need to be better understood. The diagonal pivoting based on a rough estimation of condition numbers of leading principal submatrices can sometimes provide inefficient preconditioners. This short study proposes another type of pivoting, namely the pivoting that exploits incremental condition estimation based on monitoring both direct and inverse factors of the approximate factorization. Such pivoting remains rather cheap and it can provide in many cases more reliable preconditioner. Numerical examples from real-world problems, small enough to enable a full analysis, are used to illustrate the potential gains of the new approach.

中文翻译：

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关于因式近似逆预处理中​​的适应性的注释

摘要 求解大规模线性代数方程组的问题在广泛的应用中出现。在许多情况下，预处理迭代方法是一种选择方法。本文讨论了基于不完全广义 Gram-Schmidt 过程的近似逆预处理 AINV/SAINV。这种类型的近似逆预处理已反复用于电子结构计算中的矩阵对角化，但近似逆通常在并行计算中很有趣。我们的方法使用基于计算的中间量的控制矩阵条目的自适应删除。策略已被引入作为解决困难应用问题的一种方法，它的动机是最近关于广义 Gram-Schmidt 过程中正交性损失的理论结果。尽管如此，该方法还有更多方面需要更好地理解。基于对主要主子矩阵的条件数的粗略估计的对角线旋转有时会提供低效的预处理器。这项简短的研究提出了另一种类型的旋转，即利用增量条件估计的旋转，该旋转基于监视近似因式分解的直接和逆向因素。这种旋转仍然相当便宜，并且在许多情况下可以提供更可靠的预处理器。来自现实世界问题的数值示例，小到足以进行全面分析，