Question

Use the Gauss-Jacobi method and Gauss-Seidel method to solve the simultaneous linear equations given below. Perform three iterations for each method: 5x+y−z. x+4y+2z , x−2y+5z.

Answer 1

To tackle the simultaneous equations, both the Gauss-Jacobi and Gauss-Seidel methods involve iterative calculations, with the former updating all variables simultaneously and the latter sequentially. Initial approximate values are chosen, and three iterations for each method are needed.

Step-by-step explanation:

The goal is to solve the simultaneous linear equations using both the Gauss-Jacobi and Gauss-Seidel methods. The system of equations given is:

• 5x + y - z = 0
• x + 4y + 2z = 0
• x - 2y + 5z = 0

For the Gauss-Jacobi method, we use the following iterative scheme where the next approximation for each variable is calculated independently using the current iteration values:

1. x(i+1) = (0 - y(i) + z(i))/5
2. y(i+1) = (0 - x(i) - 2z(i))/4
3. z(i+1) = (0 - x(i) + 2y(i))/5

For the Gauss-Seidel method, each variable is updated sequentially and the latest values are used for subsequent calculations:

1. x(i+1) = (0 - y(i) + z(i))/5
2. y(i+1) = (0 - x(i+1) - 2z(i))/4
3. z(i+1) = (0 - x(i+1) + 2y(i+1))/5

For both methods, starting values are needed, and typical choices are x(0) = 0, y(0) = 0, z(0) = 0. Three iterations should be performed for each method. Care should be taken with algebraic steps to ensure accuracy.