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In this paper, the new pivoting strategy named boundedly partial pivoting is presented in the process of the incomplete LDLT factorization, based on the maximum weighted matchings in the preprocessing of solving a symmetric indefinite linear system. This new pivoting method is acquired by changing the tri-diagonal pivoting algorithm, and none of permutations of rows and columns are needed. Meanwhile, the lower triangular factor L is bounded. Numerical results demonstrate that the newly proposed pivoting strategy can extend the scope of applications of the LDLT preconditioner, compared with the tri-diagonal pivoting. Moreover, almost the same results are obtained or our boundedly partial pivoting is more efficient with respect to the iteration speed, when both the tri-diagonal pivoting and the boundedly partial pivoting are applicable to incomplete LDLT factorization. [ABSTRACT FROM AUTHOR]