# Write function gaussElim(A,b) for solving a system of simultaneous ...

## Transcribed Text

Write function gaussElim(A,b) for solving a system of simultaneous equations Ax = b using Gauss elimination with partial pivoting. The input parameters to the function are the square matrix, A, and the column vector b. The function should return the solution to the simultaneous equations. In addition, the function should not change the input matrix in the process of solving the equations. Test your program using the following problems. You can check your work by solving the problems using a NumPy function (e.g. x = np.linalg.solve(A,b)) (2x+y-z=1.0 - = 2x + y - 2z = - 2 x - y + Z = 2 # main # This represents a set of 6 simultaneous equations. Mrx = np.array([ [1.0,3,2,4,3,1], (-4,0,3,2,3,4], [3,-1,3,2,2,5], [3,3,12,2,- - 6,-4], [-1,-2,-3,7,6,4], [7,5,0,0,4,2] ]) Rhs = np.array([[ 4, 5, 6, 10, 6, - -8 ]]) S = gaussElim(Mrx,Rhs.T) print(s) You can use the following code to return only the last columns of the reduced matrix A return A[:, nc-1] # extracts the last column of the matrix or return A[:, nr : nc] # extracts the last columns of the matrix. This works best if you have multiple constant columns

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import numpy as np

def gaussElim(A, b):
"""
This function solves system of linear equations Ax = b using Gauss elimination with
partial pivoting. Input parameters are:

A - square matrix
b - column vector
We will assume that A is singular matrix so that sysyem has unique solution. Output is:
solution - solution to the system, row vector
"""
# number of equations
A_rows = len(A)

# append the column b to the right side of the matrix A so we obtain n x n+1 matrix A_tilde
A_tilde = np.c_[A, b]

"""
Find the index of the entry in each colum with the largest absolute value. This entry is called the pivot.
We assumed that pivot it is not equal to 0, othervise the matrix A is singular.
Bring the row with pivot to the diagonal i.e interchange rows k and P_k
"""
for k in range(A_rows):
for P_k in range(k, A_rows):
if abs(A_tilde[P_k, k]) > abs(A_tilde[k, k]):
A_tilde[[k, P_k]] = A_tilde[[P_k, k]]
"""
Iterate over rows below the pivot and calculate the factor temp_ in order to make the k-th entry zero.
Further, subtract the temp-th multiple of corresponding element in k-th row for every...

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