Thread: jacobian matrix

please help in understanding this problem and any help solving it.

1) Consider the linear system Ax = b where
A = [ 2.2 2] and b = [6.2] and the solution is x = [1]
2 2.2 6.4 2

If we want to apply the Jacobi Iteration to this linear system, what will be the splitting of A, i.e. what are the matrices M and N, corresponding to the Jacobi Iteration? Is the Jacobi Iteration guaranteed to converge to the true solution x? Justify your answer? If convergent, what is the rate of convergence? Apply the Jacobi Iteration with the starting guess x^(0) = [0 0] until
the relative error between the true solution and the kth iterate is less than or equal to 10^(-5), i.e. until (||x - x^(k)||2 ) / (||x||2) <= 10^(-5). Plot how the entries of the kth iterate evolve, i.e. plot x1^(k) and x2^(k) throughout the iteration?

2) Repeat the above steps for the Gauss-Seidel Iteration..

thanks!

Originally Posted by vedicmath

please help in understanding this problem and any help solving it.

1) Consider the linear system Ax = b where
A = [ 2.2 2] and b = [6.2] and the solution is x = [1]
2 2.2 6.4 2

If we want to apply the Jacobi Iteration to this linear system, what will be the splitting of A, i.e. what are the matrices M and N, corresponding to the Jacobi Iteration? Is the Jacobi Iteration guaranteed to converge to the true solution x? Justify your answer? If convergent, what is the rate of convergence? Apply the Jacobi Iteration with the starting guess x^(0) = [0 0] until
the relative error between the true solution and the kth iterate is less than or equal to 10^(-5), i.e. until (||x - x^(k)||2 ) / (||x||2) <= 10^(-5). Plot how the entries of the kth iterate evolve, i.e. plot x1^(k) and x2^(k) throughout the iteration?

2) Repeat the above steps for the Gauss-Seidel Iteration..

thanks!

Straightforward: Jacobi method - Wikipedia, the free encyclopedia