After the algebraic solution has been found the solution has to be written back to the quantities. First, the solution vector has to be re-inserted into the solution quantity, in many cases the elements of the solution vector are added to the quantity values. In case of eigenvalue systems, each eigenvalue is assigned one quantity.
For nonlinear systems using a gradient formulation, line-wise assembly methods offer the advantage that the equations have to be specified only once with respect to the (initially preset) solution vector. The solution vector is used as linearization vector for the next linearization process, while the calculation is performed in the same manner based on the values stored in the solution quantity. This quantity is therefore updated non-linear solution method precedes iteratively.