In this paper, a new preconditioned iterative method is presented to solve a class of nonsymmetric nonsingular or singular saddle point problems. The implementation of the proposed preconditioned Krylov subspace method avoids solving inverse of Schur complement and only needs to solve one linear sub-system at each step, which implies that it may save considerable costs. Theoretical convergence analysis, including the bounds of eigenvalues and eigenvectors, the degree of the minimal polynomial of the preconditioned matrix, are discussed in details. Moreover, a novel algebraic estimation technique for finding a practical iteration parameter is presented, which is very effective and practical even for large scale problems. At last, some numerical examples are carried, showing that the theoretical results are valid and convincing.

在本文中，提出了一种新的预处理迭代方法来解决一类非对称非奇异或奇异鞍点问题。所提出的预处理克雷洛夫子空间方法的实现避免了求解 Schur 补的逆，并且每一步只需要求解一个线性子系统，这意味着它可以节省可观的成本。详细讨论了理论收敛性分析，包括特征值和特征向量的界限，预处理矩阵的最小多项式的次数。此外，提出了一种用于寻找实用迭代参数的新颖代数估计技术，即使对于大规模问题也非常有效和实用。最后给出了一些数值算例，表明理论结果是有效的和令人信服的。