In quantum mechanics, the wavefunction of N electrons is given by a determinant. I am working on a Monte Carlo algorithm. At each Monte Carlo step, I need to add or remove an electron, which means adding or removing a line and a column to my matrix.
I use Gauss elimination to compute an upper triangle matrix. Adding a line and a column is simple and can be done in a time growing as $O(n^2)$. Removing the last line and column is also feasible simply in $O(n^2)$. To the ith line, I saw no other possibilty than recomputing the pivots from i to n-1 which requires a time $O(n^3)$.
Does anybody see another possibily?