Numerical Method - Old Questions

9. Compare and contrast between Jacobi iterative method and Gauss Seidal method. 

Both the Jacobi and Gauss-Seidel methods are iterative methods for solving the linear system Ax = b.

In the Jacobi method the updated vector x is used for the computations only after all the variables (i.e. all components of the vector x) have been updated. On the other hand in the Gauss-Seidel method, the updated variables are used in the computations as soon as they are updated.

Thus in the Jacobi method, during the computations for a particular iteration, the “known” values are all from the previous iteration. However in the Gauss-Seidel method, the “known” values are a mix of variable values from the previous iteration (whose values have not yet been evaluated in the current iteration), as well as variable values that have already been updated in the current iteration.

Even though the Gauss-Seidel’s method uses the improved values as soon as they are computed, this does not ensure that the Gauss-Seidel’s method would converge faster than Jacobi iterations.

Consider a system of linear equation;

Then, these equation can be written as;

From Gauss Jacobi approximation; we have, (with initial approximation x₁ = a, x₂ = b & x₃ = c)

And, from Gauss Seidel approximation;