Direct, iterative methods; Gauss-Seidel numerical

Direct and iterative methods

• Direct methods give the solution in a finite, known number of operations (exact apart from round-off). Examples: Gauss elimination, Gauss-Jordan, LU and Cholesky decomposition, matrix inverse (Cramer's rule). They are best for small or dense systems.
• Iterative methods start from a trial vector and improve it repeatedly until the change is below a tolerance. Examples: Jacobi, Gauss-Seidel, SOR, conjugate gradient. They are best for large sparse systems and need convergence conditions such as diagonal dominance.

Gauss-Seidel solution

The system is diagonally dominant ($12>8$, $5>4$, $13>10$), so it converges. Rearranging:

$$x=\frac{1-3y+5z}{12},\quad y=\frac{28-x-3z}{5},\quad z=\frac{76-3x-7y}{13}$$

Start $(x,y,z)=(1,2,3)$.

Example of iteration 1: $x=(1-3(2)+5(3))/12=0.8333$; $y=(28-0.8333-9)/5=3.6333$; $z=(76-2.5-25.4333)/13=3.6974$.

Answer: $x\approx1,\ y\approx3,\ z\approx4$ (exact solution $x=1,y=3,z=4$).