import numpy as np

def solveEqns(A, v):
    """This function uses Gaussian Elimination with partial pivoting to solve
        systems of linear equations of the form Ax = v"""

    n = len(A)
    L = len(v)

    Pivot = A[0][0] # Set the (1,1) element of A as our Pivot

    count = -1 #The count will tell us the column in which our pivot lives.
    for j in range(0, n):
        Pivot = A[0][j]
        count += 1 #Count has the same value as j, i.e. the column of the pivot.
        if not all (A[i][j] == 0 for i in range(0, n)):
            #check if all elements in column are 0
            break

    maxEl = abs(Pivot)


    #Now we ensure the element in the column with highest absolute value
    #is at the top
    for k in range(0, n):
           if abs(A[k][count]) >= maxEl:
               maxEl = abs(A[k][count])
               maxRow = k
           else:
               maxRow = 0


    #Swap row with maximum element with current top row, column by column.
    for i in range(0, n):
        tmp = A[maxRow][i]
        A[maxRow][i] = A[0][i]
        A[0][i] = tmp

    #Swap the corresponding elements in v to ensure our equations still hold.
    tmp2 = v[maxRow]
    v[maxRow] = v[0]
    v[0] = tmp2

    print(A)
    print(v)

    for k in range(L):
        if A[k][k] == 0:
            div = 1 #prevents us dividing by 0 if we have a column of 0s
        else:
            div = A[k][k]

        A[k, :] /= div
        v[k] /= div

        for i in range(k+1, L):
            mult = A[i][k]
            A[i, :] -= mult*A[k, :]
            v[i] -= mult*v[k]


    #Substitute back in
    x = np.empty(L, float)
    for k in range(L-1, -1, -1):
        x[k] = v[k]
        for i in range(k+1, L):
            x[k] -= A[k, i]*x[i]

    print(x)