madhanbabu2004 / qrdecomposition

QR Decomposition

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:

import numpy as np
def QR(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,m):
            R[i, j] =A[:, j] @ Q[:, i]
    print(Q)
    print(R)
a = np.array(eval(input()))
QR(a)
        

Output

Result

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