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.

中文翻译：

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多鞍点系统的对称正定预条件子

我们考虑块三对角形式的多个鞍点系统的对称正定预条件子，它可以在 Minres 算法中应用。我们描述了这样一个预处理器，当精确应用预处理器时，预处理矩阵只有两个不同的特征值：$1$ 和 $-1$。与更广泛研究的块对角预处理器相比，我们讨论了这种方法的相对优点，指定了与不精确应用新预处理器相关的计算工作，并调查了块对角线情况的许多理论结果。数值结果验证了我们的理论发现。