Algorithm for QR Decomposition

Aim:

To implement QR decomposition algorithm using the Gram-Schmidt method.

Equipment’s required:

1. Hardware – PCs
2. Anaconda – Python 3.7 Installation / Moodle-Code Runner

Algorithm:

1. Intialize the matrix Q and u

2. The vector u and e is given by

   

   

   

3. Obtain the Q matrix
   

4. Construct the upper triangular matrix R

Program:

Gram-Schmidt Method

''' 
Program to QR decomposition using the Gram-Schmidt method
Developed by: your name
RegisterNumber: 
'''
import numpy as np
def QR_Decomposition(A):
    n,m = A.shape
    Q = np.empty((n,n))
    u = np.empty((n,n))
    u[:,0] = A[:,0]
    Q[:,0] = u[:,0]/np.linalg.norm(u[:,0])
    for i in range(1,n):
        
        u[:,i]=A[:,i]
        for j in range(i):
            u[:,i]-=(A[:,i]@Q[:,j])*Q[:,j]
            
            Q[:,i] = u[:,i]/np.linalg.norm(u[:,i])
            
    R = np.zeros((n,m))
    for i in range(n):
        for j in range(i,n):
            R[i,j] = A[:,j]@Q[:,i]
    print("The Q Matrix is")
    print("",Q)
    print("The R Matrix is")
    print("",R)
a = np.array(eval(input()))
QR_Decomposition(a)

Output

Result

Thus the QR decomposition algorithm using the Gram-Schmidt process is written and verified the result.