# import libraries
import numpy as np
# create a function to exchange rows
def RwColexchange(A1,b1):
M,N = A1.shape
flag = 0
for i in range(1,M):
# check if 1st element of row is non zero
if A1[i,0] != 0:
# note: list function is used to create new copy of list.
temp_v = list(A1[0,:])
# create temp varibale to store 1st row
temp_u = list(b1[0])
A1[0,:] = A1[i,:]
# row exchange
A1[i,:] = temp_v
# exchange with temp variable
b1[0] = b[i]
b1[i] = temp_u
flag += 1
break
# break the loop after one-row exchange
return A1, b1, flag
# create function for row echelon
def Rwechelon(A1,b1):
(M,N) = A1.shape
for i in range(1,M):
if A1[i,0] != 0:
a = A1[i,0]/A1[0,0]
A1[i,:] = A1[i,:] - a*A1[0,:]
b1[i] = b1[i] - a*b1[0]
# multiplication factor
# row echelon
if A1[0,0] != 1:
b1[0] = b1[0]/A1[0,0] # first divide b otherwise 1st element changes
A1[0,:] = A1[0,:]/A1[0,0]
# make pivot elememt to 1
return A1, b1
# reduced row echelon form
def rref(A,b,P):
M,N = A.shape
index = 0
for j in P:
for i in range(M-1,-1,-1):
if (j!=0) and (i!=0):
if (A[i,j]!=0):
index = i
break
for i in range(index):
a = A[i,j]/A[index,j]
A[i,:] = A[i,:] - a*A[index,:]
b[i] = b[i] - a*b[index]
return A, b
# create a function for solving Ax = b.
def AxbSlv(A,b):
(M,N) = A.shape
# get shape of Matrix "A"
i=0
j=0
fr_var = []
# free variable
pv_var = []
# pivot variable
A2 = A
b2 = b
while (i<M) and (j<N):
A1 = A[i:M, j:N]
b1 = b[i:M]
if A1[0,0] == 0:
# check 1st element of row
A1, b1, flag = RwColexchange(A1,b1)
# row exchange
# if entire row is zero, store j value in free variable
if flag == 0:
fr_var.append(j)
j += 1
continue
else:
pv_var.append(j)
else:
pv_var.append(j)
A1, b1 = Rwechelon(A1,b1)
# row echelon
A2[i:M,j:N] = A1
# change a section of matrix to get reduced matrix
b2[i:M] = b1
i += 1
# go to next diagonal element
j += 1
A2, b2 = rref(A2,b2, pv_var)
# reduced row echelon form
return A2, b2, fr_var, pv_var
# back substitution method
# null solution
def null(A,p,f):
M,N = A.shape
null_matrix = np.zeros((N, len(f)))
for i,c in enumerate(f):
null_vector = np.zeros(N)
null_vector[c] = 1
for j in range(len(p)-1,-1,-1):
p_col = p[j]
for k in range(p_col + 1, N):
null_vector[p_col] -= A[j, k] * null_vector[k]
# Assign the null space vector to the null space matrix
null_matrix[:, i] = null_vector
return null_matrix
# particular solution
def particular(A,b,p,f):
M,N = A.shape
b = b.astype(float)
par_vector = np.zeros(N)
for j in range(len(p)-1,-1,-1):
p_col = p[j]
for k in range(p_col + 1, N):
b[j] -= A[j, k] * par_vector[k]
par_vector[p_col] = b[j]
return par_vector
def Solution(A,b,p,f):
N = null(A,p,f)
P = particular(A,b,p,f)
return N, P
# input matrix A and vector b
A = np.array([[1, 3, 3, 2], [2, 6, 9, 7,], [-1, -3, 3, 4,],])
b = np.array([[1],[5],[5],])
print(A)
print(b)
[[ 1 3
[ 2 6
[-1 -3
[[1]
[5]
[5]]
3
9
3
2]
7]
4]]
# get the Solution of Ax = b
C, d ,Fv ,Pv = AxbSlv(A, b)
Xp, Xn = Solution(C, d, Pv, Fv)
print("A ="+str(C))
print("b ="+str(d))
print("Free variable ="+str(Fv))
print("Pivot variable ="+str(Pv))
print(f"Xp = {Xp}")
print(f"Xn = {Xn}")
A =[[ 1 3 0 -1]
[ 0 0 1 1]
[ 0 0 0 0]]
b =[[-2]
[ 1]
[ 0]]
Free variable =[1, 3]
Pivot variable =[0, 2]
Xp = [[-3. 1.]
[ 1. 0.]
[ 0. -1.]
[ 0. 1.]]
Xn = [-2. 0. 1. 0.]