To implement QR decomposition algorithm using the Gram-Schmidt method.
- Hardware – PCs
- Anaconda – Python 3.7 Installation / Moodle-Code Runner
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Intialize the matrix Q and u
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The vector u and e is given by
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,m):
R[i,j] = A[:,j] @ Q[:,i]
print (Q)
print (R)
a = np.array(eval(input()))
QR_Decomposition(a)
## Result
Thus the QR decomposition algorithm using the Gram-Schmidt process is written and verified the result.