Question

Write a function in Python that solves the linear system ??=? using Gaussian Elimination, taking ?,? as input. The function should have two phases: the elimination phase, and the back substitution phase. You can use numpy library.

Answer 1

The solution to the above problem is:-

import numpy as np

def f_elimination(A, b, n):
"""
Here we are calculating the forward part of Gaussian elimination.
"""
for row in range(0, n-1):
for i in range(row+1, n):
factor = A[i,row] / A[row,row]
for j in range(row, n):
A[i,j] = A[i,j] - factor * A[row,j]

b[i] = b[i] - factor * b[row]

print('A = \n%s and b = %s' % (A,b))
return A, b

def back_substitute(a, b, n):
""""
Here we are doing back substitution, returns the Gauss result.
"""
x = np.zeros((n,1))
x[n-1] = b[n-1] / a[n-1, n-1]
for row in range(n-2, -1, -1):
sum = b[row]
for j in range(row+1, n):
sum = sum - a[row,j] * x[j]
x[row] = sum / a[row,row]
return x

def gaus(A, b):
"""
This function performs Gauss elimination without pivoting.
"""
n = A.shape[0]

# Check for zero diagonal elements
if any(np.diag(A)==0):
raise ZeroDivisionError(('Division by zero will occur; '
'pivoting currently not supported'))

A, b = f_elimination(A, b, n) # calling f_elimination and storing the result in A,b.
return back_substitute(A, b, n) # Here we are returning the back_substitute result.

# Main program starts here
if __name__ == '__main__':
A = np.array([[9, 3, 5],
[4, -3, -24, ],
[1, 4, -4, ]])
b = np.array([0, 4, 2])
x = gaus(A, b) # calling guassian function and storing the above result in x.
print('Gauss result is x = \n %s' % x)

The output for the above code is:-

- O X 5. CA WINDOWS\SYSTEM32\cmd.exe A = [[ 93 5] [ 0 -4 -26] i 3 -4]] and b = [@ 4 2] A = [[ 935] I -4 -26] [ -23]] and b = [@ 4 5] Gauss result is x = [[-0.01690821] [ 0.41304348] [-0.2173913 ]] Press any key to continue ...

  

Thank you.