Gaussian elimination corresponds to a sequence of elimination graphs Gi, 0i<N. Let Define from by removing from and all of its G0=G. Gi Gi-1, i>0, vi Gi-1 incident edges from and adding new edges as required to pairwise connect Gi-1, all vertices in . Let denote the set of edges added during the elimination process. Let . F Gaussian elimination applied to produces no new fillin edges. Symbolic G’ factorization computes . Define = = Theorem 3.11: , if and only if , or sequence such that , k1=l1, kp=j, some and i=kq, 2qp1, , 2qp.
Gaussian elimination corresponds to a sequence of elimination graphs Gi, 0i<N. Let G0=G. Define Gi from Gi-1, i>0, by removing vi from Gi-1 and all of its incident edges from Gi-1, and adding new edges as required to pairwise connect all vertices in . Let F denote the set of edges added during the elimination process. Let . Gaussian elimination applied to G’ produces no new fillin edges. Symbolic factorization computes . Define = = Theorem 3.11: , if and only if , or sequence such that k1=l1, kp=j, , i=kq, some 2qp1, and , 2qp.