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: S.JEGADEESH
RegisterNumber: 22004355
'''
import numpy as np 
arr=np.array(eval(input()))
n,m=arr.shape
u=np.empty((n,m))
e=np.empty((n,m))
u[:,0]=arr[:,0]
e[:,0]=u[:,0]/np.linalg.norm(u[:,0])
for i in range(n):
    u[:,i]=arr[:,i]
    for j in range(i):
        u[:,i] -= (arr[:,i]@e[:,j])*e[:,j]
        e[:,i] =u[:,i]/np.linalg.norm(u[:,i])
r=np.zeros((n,m))
for i in range(n):
    for j in range(i,m):
        r[i,j]=arr[:,j]@e[:,i]
print(e)
print(r)

Output

Result

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