This article develops the preconditioning technique as a method to address the accuracy issue caused by ill‐conditioning. Given a preconditioner M for an ill‐conditioned linear system Ax=b, we show that, if the inverse of the preconditioner M⁻¹ can be applied to vectors accurately, then the linear system can be solved accurately. A stability concept called inverse‐equivalent accuracy is introduced to describe the high accuracy that is achieved and an error analysis will be presented. Numerical examples are presented to illustrate the error analysis and the performance of the methods.

中文翻译：

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预处理，以获取病态线性系统的精确解

本文开发了预处理技术，作为解决由不良条件引起的精度问题的方法。给定一个预处理器中号为病态线性系统斧= b，我们证明了，如果预处理器的逆中号^-1可以应用于载体准确，则线性系统可以解决准确。引入了一种称为反等效精度的稳定性概念，以描述所达到的高精度，并进行误差分析。数值例子说明了误差分析和方法的性能。