6.2: Jacobi Method for solving Linear Equations

During class today we will write an iterative method (named after Carl Gustav Jacob Jacobi) to solve the following system of equations:

\[ 6x + 2y - ~z = 4~ \nonumber \]

\[~ x + 5y + ~z = 3~ \nonumber \]

\[ 2x +~ y + 4z = 27 \nonumber \]

Here is a basic outline of the Jacobi method algorithm:

1. Initialize each of the variables as zero \( x_0 = 0, y_0 = 0, z_0 = 0 \)
2. Calculate the next iteration using the above equations and the values from the previous iterations. For example here is the formula for calculating \(x_i\) from \(y_{(i-1)}\) and \(z_{(i-1)}\) based on the first equation: \(x_i = \dfrac{4-2y_{(i-1)} + z_{(i-1)}}{6} \). Similarly, we can obtain the update for \(y_i\) and \(z_i\) from the second and third equations, respectively.
3. Increment the iteration counter \(i=i+1\) and repeat Step 2.
4. Stop when the answer “converges” or a maximum number of iterations has been reached. (ex. \(i = 100\))

In other words, the Jacobi Methid will not work an all problems.

%matplotlib inline
import matplotlib.pylab as plt

x = []
y = []
z = []

#step 1: inicialize to zero
x.append(0)
y.append(0)
z.append(0)

for i in range(1,100):
    xi = (4 - 2*y[i-1]+ z[i-1])/6
#####Start your code here #####
    yi = 0 #Change this line
    zi = 0 #Change this line
#####End of your code here#####        
    #Add latest value to history
    x.append(xi)
    y.append(yi)
    z.append(zi)

#Plot History of values
plt.plot(x, label='x')
plt.plot(y, label='y')
plt.plot(z, label='z')
plt.xlabel('Iteration')
plt.ylabel('Value')
plt.legend(loc=1);

\[ x = \nonumber \]

\[ y = \nonumber \]

\[ z = \nonumber \]

# Put your code here