In this work, a new algorithm for solving symmetric indefinite systems of linear equations is presented. It factorizes the matrix into the form LDLt using Jacobi rotations in order to increase the pivot's absolute value. Furthermore, Rook's pivoting strategy is also adapted and implemented. In determinate compatible systems, the computational cost of the algorithm was similar to the cost of the Bunch-Kaufman method, but the error was approximately 50 % smaller for intermediate and large matrices, regardless of the condition number of the coefficient matrix. Furthermore, unlike Bunch-Kaufman, the new algorithm calculates with little additional cost the fundamental basis of the null space, and obtains the minimal least squares and minimum norm solutions. In minimal least squares with minimum norm problems, the new algorithm was compared with the LAPACK Complete Orthogonal Decomposition algorithm, among others. The obtained error with both algorithms was similar but the computational cost was at least 20 % smaller with the new algorithm, even though the Complete Orthogonal Decomposition is implemented in a blocked form.

翻译：本文提出了一种求解对称不定线性方程组的新算法。该算法采用Jacobi旋转变换将矩阵分解为LDLt形式，以增大主元的绝对值。此外，本文还改进并实现了Rook选主元策略。对于确定相容方程组，该算法的计算成本与Bunch-Kaufman方法相近，但对于中大型矩阵，无论系数矩阵的条件数如何，其误差均减小约50%。与Bunch-Kaufman方法不同，新算法能以极小的额外代价计算零空间的基础基，并获得最小二乘最小范数解。在最小二乘最小范数问题中，新算法与LAPACK完全正交分解算法等进行了对比。两种算法获得的误差相近，但新算法的计算成本至少降低20%，尽管完全正交分解算法采用了分块实现形式。