# LU decomposition

## LU Solvers

For the system of equations $A\cdot\phi=B$.
The solvers based on factorization are widely popular. The factorization of a non singular matrix A in two matrices L and U, called lower and upper matrices, leads to a direct procedure for the evaluation of the inverse of the matrix. Thus making it possible to calculate the solution vector with given source vector. It now remains a two step process: a forward substitution followed by a backward substitution.

## Algorithm

Calculate LU factors of A
$A\cdot \phi = (LU)\cdot \phi = L(U\cdot\phi)=LY=B$
Solve $LY=B$ by forward substitution
$Y = L^{-1} B$
Solve $U\phi =Y$ by backward substitution
$\phi = U^{-1}Y$