Question

The approximate solution of the system of simultaneous equations

5x - 2y + z = -1

3x + 4y - 2z = 2

4x - y + 3z = 4

by applying Gauss-Jacobi method one time (using initial approximation) as x = 0, y = 0, z = 0) will be:

1. x = 1.25, y = 2.275, z = -3.72
2. x = -1.5, y = 3.25, z = 1.275
3. x = 1.5, y = -2.375, z = 2.234
4. x = -0.2, y = 0.5, z = 1.33

Answer 1

Correct Answer - Option 4 : x = -0.2, y = 0.5, z = 1.33

In the Gauss-Jacobi method, the general scheme for the results of the (n + 1)^th iteration:

\({x_{n + 1}} = \frac{1}{{{a_1}}}\left( {{K_1} - {b_1}{y_n} - {c_1}{z_n}} \right)\)

\({y_{n + 1}} = \frac{1}{{{b_2}}}\left( {{K_2} - {a_2}{x_n} - {c_2}{z_n}} \right)\)

\({z_{n + 1}} = \frac{1}{{{c_3}}}\left( {{K_3} - {a_3}{x_n} - {b_3}{y_n}} \right)\)

Given linear equations are:

5x – 2y + z = -1

3x + 4y – 2z = 2

4x – y + 3z = 4

\({x_{n + 1}} = \frac{1}{5}\left( {1 + 2 \cdot {y_n} - {z_n}} \right) = -0.2 + 0.4{y_n} - 0.2{z_n}\)

\({y_{n + 1}} = \frac{1}{4}\left( {2 - 3{x_n} + 2{z_n}} \right) = 0.5 + 0.75{x_n} + 0.5{z_n}\)

\({z_{n + 1}} = \frac{1}{3}\left( {4 - 4{x_n} + {y_n}} \right) = 1.33 - 1.33{x_n} + 0.33{y_n}\)

Using x₀ = 0, y₀ = 0 and z₀ = 0.

We obtain, x₁ = - 0.2, y₁ = 0.5 and z₁ = 1.33