ITERATIVE TECHNIQUES FOR SOLVING NON-LINEAR SYSTEMS (AND LINEAR SYSTEMS)

Jacobi Iterative Technique Consider the following set of equations.

Convert the set Ax = b in the form of x = Tx + c.

Start with an initial approximation of:

k Results of Jacobi Iteration:

Gauss-Seidel Iterative Technique Consider the following set of equations.

k Results of Gauss-Seidel Iteration: (Blue numbers are for Jacobi iterations.)

It required 15 iterations for Jacobi method and 7 iterations for Gauss-Seidel method to arrive at the solution with a tolerance of The solution is: x 1 = 1, x 2 = 2, x 3 = -1, x 4 = 1

Newton’s Iterative Technique Given:

Jacobian Matrix:

Consider the following set of non-linear equation:

Make an initial guess:

Example of Gauss-Seidel Iteration