Question

For the following system of linear equations: −2x​₁−x₂+10x₃=9 10x₁​+2x₂−x₃=7x₁​+8x₂​+3x₃​=−4​ Solve this system using both Jacobi and Guess Seidel methods also compare between the two methods. initial values [x]t=[0,0,0], Perform only 3 iterations. b) What does it mean by diagonally dominant matrix, explain it and be specific.

Answer 1

To solve the given system of linear equations, we can use both the Jacobi and Gauss-Seidel methods. The Jacobi method involves splitting the coefficient matrix and setting up an iteration equation, while the Gauss-Seidel method involves splitting the coefficient matrix into three parts. By performing 3 iterations using both methods, we can compare the solutions obtained.

Step-by-step explanation:

The given system of linear equations can be solved using both the Jacobi and Gauss-Seidel methods. Let's start with the Jacobi method.

Jacobi Method:

1. Write the given system of equations in matrix form, where A is the coefficient matrix, X is the column vector of variables, and B is the column vector of constants.
2. Split matrix A into its diagonal matrix D and the remaining matrix R (A = D + R), where D contains only the diagonal elements of A.
3. Set up an iteration equation: X(t+1) = D^(-1) * (B - RX(t)), where X(t) is the column vector of variables at iteration t.
4. Starting with the initial guess X(0) = [0, 0, 0], perform 3 iterations to solve for X(3).

Now, let's move on to the Gauss-Seidel method.

Gauss-Seidel Method:

1. Write the given system of equations in matrix form, where A is the coefficient matrix, X is the column vector of variables, and B is the column vector of constants.
2. Split matrix A into its lower triangular matrix L, diagonal matrix D, and upper triangular matrix U (A = L + D + U).
3. Set up an iteration equation: X(t+1) = (D + L)^(-1) * (B - UX(t)), where X(t) is the column vector of variables at iteration t.
4. Starting with the initial guess X(0) = [0, 0, 0], perform 3 iterations to solve for X(3).

After performing both methods, compare the solutions obtained in terms of accuracy, number of iterations, and convergence rate.