How many elements of $LU$ decomposition of a symmetric matrix change after adding a sparse symmetric matrix? Is it more efficient to recompute $LU$ decomposition after adding a sparse matrix comparing to computing it from scratch for the sum?
More formally, let $M$ be a symmetric matrix, $L_M\cdot U_M$ its pivoted $LU$ decomposition, and $N$ a sparse symmetric matrix of the same dimension.
The 1st question was: Is it true that $M+N$ has $LU$ decomposition $L_{M+N}\cdot U_{M+N}$ such that both $L_{M+N}-L_M$ and $U_{M+N}-U_M$ are sparse?
The 2nd question: Provided that the answer to the 1st question is positive, is it possible to compute $L_{M+N}-L_M$ and $U_{M+N}-U_M$ asymptotically faster than computing $L_{M+N}$ and $U_{M+N}$ without knowledge of $L_M$ and $U_M$ would require?