Bottom Science- Bisection Method
- Secant Method
- Regular Falsi (False Position) Method
- Newton Raphson Method
- Gauss Elimination Method
- Gauss Jordan Method
- Gauss-Seidel Method
- Lagrange Interpolation Method
- Newton Divided Difference Interpolation
- Newton Forward Difference Interpolation
- Newton Backward Difference Interpolation
- Trapezoidal Rule
- Simpson 1/3rd Rule
- Simpson 3/8 Rule
- Euler’s Method
- Euler’s Modified Method
- Runge-Kutta 2nd Order Method
- Runge-Kutta 4th Order Method
- Cubic Spline Method
- Bilinear Interpolation Method
- Milne’s Method
- More topics coming soon…
The Gauss-Seidel method is an iterative method for solving a linear system of equations, typically represented by the matrix equation Ax = b, where A is a matrix of coefficients, x is a vector of unknowns, and b is a vector of constants.
In Python, you can implement the Gauss-Seidel method using a for loop to iterate over the elements of the matrix and update the values of the unknowns based on the equations.
The below code defines a function gauss_seidel that takes as input a matrix of coefficients A, a vector of constants b, an initial guess for the solution x0, a tolerance epsilon, and a maximum number of iterations max_iterations.
The function uses a for loop to iterate over the elements of the matrix and update the values of the unknowns based on the equations.
The function stops iterating when the difference between successive approximations is less than the specified tolerance epsilon or when the maximum number of iterations is reached.
Gauss-Seidel Method in Python
import numpy as np
def gauss_seidel(A, b, x0, epsilon, max_iterations):
n = len(A)
x = x0.copy()
#Gauss-Seidal Method [By Bottom Science]
for i in range(max_iterations):
x_new = np.zeros(n)
for j in range(n):
s1 = np.dot(A[j, :j], x_new[:j])
s2 = np.dot(A[j, j + 1:], x[j + 1:])
x_new[j] = (b[j] - s1 - s2) / A[j, j]
if np.allclose(x, x_new, rtol=epsilon):
return x_new
x = x_new
return x
A = np.array([[10, -1, 2, 0],
[-1, 11, -1, 3],
[2, -1, 10, -1],
[0, 3, -1, 8]])
b = np.array([6, 25, -11, 15])
x0 = np.zeros(4)
eps = 1e-5
max_iter = 100
x = gauss_seidel(A, b, x0, eps, max_iter)
print(x)
#OUTPUT - [ 1.00000067 2.00000002 -1.00000021 0.99999996]
