linear-algebra-extra.ipynb

Kernel: Python 3 (ipykernel)

Extra material about Linear Algebra

import numpy as np
xrange=range

Custom implementation of general gaussian method

#Gaussian Elimination
def row_lamb( i, lamb, A ):
    B = np.copy(A)
    B[i] = lamb*B[i]
    return B

#Combination function
def row_comb( i, j, lamb, A ):
    B = np.copy(A)
    B[i] = lamb*B[j] + B[i]
    return B

#Swap function
def row_swap( i, j, A ):
    B = np.copy(A)
    B[[i,j]] = B[[j,i]]
    return B

def Gaussian_Elimination( A0 ):
    #Making local copy of matrix
    A = np.copy(A0)
    #Detecting size of matrix
    n = len(A)
    
    #Sweeping all the columns in order to eliminate coefficients of the i-th variable
    for i in xrange( 0, n ):
        
        #Sweeping all the rows for the i-th column in order to find the first non-zero coefficient
        for j in xrange( i, n ):
            if A[i,j] != 0:
                #Normalization coefficient
                Norm = 1.0*A[i,j]
                break
                
        #Applying swap operation to put the non-zero coefficient in the i-th row
        A = row_swap( i, j, A )
        
        #Eliminating the coefficient associated to the i-th variable
        for j in xrange( i+1, n ):
            A = row_comb( j, i, -A[j,i]/Norm, A )
            
    #Normalizing n-th variable
    A = row_lamb( n-1, 1.0/A[n-1,n-1], A )
    
    #Finding solution
    x = np.zeros( n )
    x[n-1] = A[n-1,n]
    for i in xrange( n-1, -1, -1 ):
        x[i] = ( A[i,n] - sum(A[i,i+1:n]*x[i+1:n]) )/A[i,i]
    
    #Upper diagonal matrix and solutions x
    return A, x

np.random.seed(3)
M =  np.random.random( (4,5) )
print ("Augmented Matrix M:\n", M, "\n")
#Solving using Gaussian Elimination
D, x = Gaussian_Elimination(M)
print ("Augmented Diagonal Matrix D:\n", D, "\n")
print ("Obtained solution:\n", x, "\n")
print ("Exact solution:\n", (M[:,:4]**-1*M[:,4]).T, "\n")

Augmented Matrix M:
 [[0.5507979  0.70814782 0.29090474 0.51082761 0.89294695]
 [0.89629309 0.12558531 0.20724288 0.0514672  0.44080984]
 [0.02987621 0.45683322 0.64914405 0.27848728 0.6762549 ]
 [0.59086282 0.02398188 0.55885409 0.25925245 0.4151012 ]] 

Augmented Diagonal Matrix D:
 [[ 5.50797903e-01  7.08147823e-01  2.90904739e-01  5.10827605e-01
   8.92946954e-01]
 [ 0.00000000e+00 -1.02675749e+00 -2.66135662e-01 -7.79783697e-01
  -1.01224976e+00]
 [ 0.00000000e+00  0.00000000e+00  5.24909828e-01 -6.69967089e-02
   2.15310020e-01]
 [ 0.00000000e+00  0.00000000e+00 -1.70372042e-16  1.00000000e+00
   9.32073626e-03]] 

Obtained solution:
 [0.27395594 0.8721633  0.41137442 0.00932074] 

Exact solution:
 [[ 1.62118801  0.9962667  29.88822639  1.51125934]
 [ 0.62248281  3.51004303  0.9649251  18.38095262]
 [ 2.324661    3.26310322  1.041764    1.21007418]
 [ 0.81260526  8.06535367  1.4905571   1.60114669]]