Computational Mathematics [ Lab Programs ]

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Aim:

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UNIT-III: Linear system of equations:
Jacobi's iteration method and Gauss-Seidal iteration method
Programs:
      • Solution of given system of linear equations using Jacobi's method.
      • Solution of given system of linear equations using Gauss-Seidal method.

Solution :

• Solution of given system of linear equations using Jacobi's method.

PROGRAM: (Jacobis_Method.py)

 
import numpy as np

# Coefficient matrix A
A = np.array([[4, 1, 1],
              [1, 5, 2],
              [1, 2, 4]], dtype=float)

# Constant vector B
B = np.array([6, 8, 7], dtype=float)

# Initial guess
x = np.zeros(len(B))

# Tolerance and max iterations
e = 0.001
max_iter = 50

for k in range(max_iter):
    x_new = np.zeros_like(x)

    for i in range(len(A)):
        s = 0
        for j in range(len(A)):
            if i != j:
                s += A[i][j] * x[j]
        x_new[i] = (B[i] - s) / A[i][i]

    print(f"Iteration {k+1}: {x_new}")

    if np.allclose(x, x_new, atol=e):
        break

    x = x_new

print("\nFinal Solution:", x)

OUTPUT:

 
Iteration 1: [1.5  1.6  1.75]
Iteration 2: [0.6625 0.6    0.575 ]
Iteration 3: [1.20625  1.2375   1.284375]
Iteration 4: [0.86953125 0.845      0.8296875 ]
Iteration 5: [1.08132812 1.09421875 1.11011719]
Iteration 6: [0.94891602 0.9396875  0.93255859]
Iteration 7: [1.03193848 1.03719336 1.04292725]
Iteration 8: [0.97996985 0.97644141 0.9734187 ]
Iteration 9: [1.01253497 1.01463855 1.01678683]
Iteration 10: [0.99214365 0.99077827 0.98954698]
Iteration 11: [1.00491869 1.00575248 1.00657495]
Iteration 12: [0.99691814 0.99638628 0.99589409]
Iteration 13: [1.00192991 1.00225874 1.00257732]
Iteration 14: [0.99879099 0.99858309 0.99838816]
Iteration 15: [1.00075719 1.00088654 1.00101071]
Iteration 16: [0.99952569 0.99944428 0.99936743]
Iteration 17: [1.00029707 1.00034789 1.00039644]
Iteration 18: [0.99981392 0.99978201 0.99975179]

Final Solution: [1.00029707 1.00034789 1.00039644]

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• Solution of system of linear equations using Gauss- Seidal method.

PROGRAM: (Gauss_Seidal_Method.py)

 
import numpy as np

# Coefficient matrix A
A = np.array([[4, 1, 1],
              [1, 5, 2],
              [1, 2, 4]], dtype=float)

# Constant vector B
B = np.array([6, 8, 7], dtype=float)

# Initial guess
x = np.zeros(len(B))

# Tolerance and max iterations
e = 0.001
max_iter = 50

print("Gauss-Seidel Iterations:\n")

for k in range(max_iter):
    x_old = x.copy()

    for i in range(len(A)):
        s1 = 0
        s2 = 0

        for j in range(i):
            s1 += A[i][j] * x[j]      # updated values

        for j in range(i+1, len(A)):
            s2 += A[i][j] * x_old[j]  # old values

        x[i] = (B[i] - s1 - s2) / A[i][i]

    print(f"Iteration {k+1}: {x}")

    if np.allclose(x, x_old, atol=e):
        break

print("\nFinal Solution:", x)

OUTPUT:

 
Gauss-Seidel Iterations:

Iteration 1: [1.5   1.3   0.725]
Iteration 2: [0.99375   1.11125   0.9459375]
Iteration 3: [0.98570313 1.02448438 0.99133203]
Iteration 4: [0.9960459  1.00425801 0.99885952]
Iteration 5: [0.99922062 1.00061207 0.99988881]
Iteration 6: [0.99987478 1.00006952 0.99999655]

Final Solution: [0.99987478 1.00006952 0.99999655]

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Related Content :

1.
UNIT - I: Eigen values and Eigenvectors:
Programs:
      • Finding real and complex Eigen values.
      • Finding Eigen vectors.    View Solution

2.
UNIT - II: Solution of Algebraic and Transcendental Equations:
Bisection method, Newton Raphson Method
Programs:
      • Root of a given equation using Bisection method.
      • Root of a given equation Newton Raphson Method.    View Solution

3.
UNIT-III: Linear system of equations:
Jacobi's iteration method and Gauss-Seidal iteration method
Programs:
      • Solution of given system of linear equations using Jacobi's method.
      • Solution of given system of linear equations using Gauss-Seidal method.    View Solution

4.
UNIT-IV: First-Order ODEs:
Exact and non-exact equations, Applications: exponential growth/decay, Newton's law of cooling.
Programs:
      • Solving exact and non-exact equations.
      • Solving exponential growth/decay and Newton's law of cooling problems.    View Solution

5.
UNIT-V: Higher order linear differential equations with constant coefficients:
Programs:
      • Solving homogeneous ODEs.
      • Solving non-homogeneous ODEs.    View Solution