Consider the following system of equations a Rearrange the e

Consider the following system of equations:

(a) Rearrange the equations to guarantee convergence.

(b) Perform 3 iterations of the Jacobi method to solve the system of equations. Calculate ?_a for each variable after each iteration. Start at ⁰x₁=⁰x₂=⁰x₃=0

Solution

(a) To guarantee convergence we need to set the eqution 1st which has highest coefficient of x1 and equation 2nd which has highest coefficient of x2 then third one is remaining equation-

5x1 + 2x2 - 2x3 = 3

-x1 + 6x2 + 2x3 = -3

2x1 - 3x2 + 7x3 = 5

Solution (b)

Upto 3 iterative steps-

Assuming x1 = x2 = x3 = 0

Step : 1

x1 = 3/5- (2 x 0)/5- (-2 x 0)/5 = 0.6
x2 = -3/6- (-1 x 0)/6- (2 x 0)/6 = -0.5
x3 = 5/7- (2 x 0)/7- (-3 x 0)/7 = 0.714

Step : 2

x1 = 3/5- (2 x -0.5)/5- (-2 x 0.714)/5 = 1.086
x2 = -3/6- (-1 x 0.6)/6- (2 x 0.714)/6 = -0.638
x3 = 5/7- (2 x 0.6)/7- (-3 x -0.5)/7 = 0.329

Step : 3

x1 = 3/5- (2 x -0.638)/5- (-2 x 0.329)/5 = 0.987
x2 = -3/6- (-1 x 1.086)/6- (2 x 0.329)/6 = -0.429
x3 = 5/7- (2 x 1.086)/7- (-3 x -0.638)/7 = 0.131