Large linear systems are often solved by generating a sequence of approximations instead of eliminating every unknown directly.

Split

$$A=D+L+U,$$

where $D$ is the diagonal and $L,U$ contain the lower and upper off-diagonal parts.

The Jacobi method updates using values from the previous iterate:

$$\mathbf x^{(k+1)}=D^{-1}\bigl(\mathbf b-(L+U)\mathbf x^{(k)}\bigr).$$

The Gauss-Seidel method reuses newly computed components immediately during the sweep.

Convergence depends on the matrix; neither method converges for every nonsingular system. Conditions such as strict diagonal dominance provide useful sufficient guarantees.

Iterative solvers are attractive for large sparse systems because they can exploit matrix structure and stop once the residual or estimated error is sufficiently small rather than computing an exact symbolic solution.