We consider symmetric positive definite preconditioners for multiple saddle-point systems of block tridiagonal form, which can be applied within the MINRES algorithm. We describe such a preconditioner for which the preconditioned matrix has only two distinct eigenvalues, 1 and -1, when the preconditioner is applied exactly. We discuss the relative merits of such an approach compared to a more widely studied block diagonal preconditioner, specify the computational work associated with applying the new preconditioner inexactly, and survey a number of theoretical results for the block diagonal case. Numerical results validate our theoretical findings.

翻译：我们研究适用于块三对角形式多重鞍点系统的对称正定预处理子，这些预处理子可嵌入MINRES算法中。本文描述了一种预处理子，当精确应用时，预处理矩阵仅具有两个不同特征值1和-1。我们比较了该方法与更广泛研究的块对角预处理子的相对优势，详细说明了非精确应用新预处理子所需的计算工作量，并综述了块对角情形下的若干理论结果。数值实验验证了我们的理论发现。