For the system of equations Ax=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.
- Calculate LU factors of A
- Ax = (LU)x = L(Ux) = Ly = b
- Solve Ly = b by forward substitution
- y = L-1b
- Solve Ux=y by backward substitution
- x = U-1y