Decomposition of singular large sparse matrix by adding dummy links and dummy degrees
Tsung‐Wu Lin, Hsing‐Tai Shiau, Jin‐Ten Huang · Journal of the Chinese Institute of Engineers · 1992
This paper proposes a simple scheme to decompose an n×n nonpositive definite matrix, A, associated with simultaneous equations, A X = B, into a triple‐factors (lower triangular, diagonal, and upper triangular matrices), i.e., Å = L D U, without interchanging rows or columns of A, but with A expanded with new rows and new columns to an m×m matrix Å. Whenever a near‐zero diagonal element, say āii , is encountered and used as a pivoting element, an appropriate positive real number, say p, is added to this diagonal element, and a new term —pxk is also added to the i‐th equation, where xk is a new variable called “dummy variable'’. If we also add a new equation —pxi + pxk = 0 to enforce the new added variable xk equal to xi then the modified i‐th equation has the same effect as the original equation. Therefore, the original solution X can be found directly from the expanded solution of the modified expanded equation. The method is very useful in solving the following problems: (1) nonlinear problems near the limit state, (2) postbuckling analysis, (3) system equations with constraint conditions, and (4) getting eigenvectors from eigenvalues.