Abstract：The coefficient matrices are called the Lagrange matrix，and associated with the system of linear equations. The system of linear equations is referred to the Lagrange equation set in this study，which is deduced by the Lagrange multiplier method，and is in general symmetric indefinite matrices. Solving such a system would encounter some intricacies if its leading principal submatrix，i.e. the stiffness matrix，is rank deficient. This is believed to be one of the main reasons that many programmers would unwillingly give up the Lagrange multiplier method but select the penalty function method. Based on the Sherman-Morrison formula and the conventional LDLT decomposition for symmetric positive definite matrices，a robust direct solution is proposed，which is efficient and particularly suitable for parallel computation. As a paradigm，the proposed procedure is used to solve the set of linear equations derived by the element-free Galerkin method(EFGM) with the moving least squares interpolation.
