Keeping track of coefficients in Gram-Schmidt

Copyright (C) 2020 Andreas Kloeckner

MIT License

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import numpy as np
import numpy.linalg as la

A = np.random.randn(3, 3)

Let's start from regular old (modified) Gram-Schmidt:

Q = np.zeros(A.shape)

q = A[:, 0]
Q[:, 0] = q/la.norm(q)

# -----------

q = A[:, 1]
coeff = np.dot(Q[:, 0], q)
q = q - coeff*Q[:, 0]
Q[:, 1] = q/la.norm(q)

# -----------

q = A[:, 2]
coeff = np.dot(Q[:, 0], q)
q = q - coeff*Q[:, 0]
coeff = np.dot(Q[:, 1], q)
q = q - coeff*Q[:, 1]
Q[:, 2] = q/la.norm(q)

Q.dot(Q.T)

array([[  1.00000000e+00,   6.15868752e-17,  -5.86239841e-16],
       [  6.15868752e-17,   1.00000000e+00,  -3.18779032e-16],
       [ -5.86239841e-16,  -3.18779032e-16,   1.00000000e+00]])

Now we want to keep track of what vector got added to what other vector, in the style of an elimination matrix.

Let's call that matrix $R$.

• Would it be $A=QR$ or $A=RQ$? Why?
• Where are $R$'s nonzeros?

R = np.zeros((A.shape[0], A.shape[0]))

Q = np.zeros(A.shape)

q = A[:, 0]
Q[:, 0] = q/la.norm(q)

R[0,0] = la.norm(q)

# -----------

q = A[:, 1]
coeff = np.dot(Q[:, 0], q)
R[0,1] = coeff
q = q - coeff*Q[:, 0]
Q[:, 1] = q/la.norm(q)

R[1,1] = la.norm(q)

# -----------

q = A[:, 2]
coeff = np.dot(Q[:, 0], q)
R[0,2] = coeff
q = q - coeff*Q[:, 0]
coeff = np.dot(Q[:, 1], q)
R[1,2] = coeff
q = q- coeff*Q[:, 1]
Q[:, 2] = q/la.norm(q)

R[2,2] = la.norm(q)

R

array([[ 0.37598334,  1.41035995, -2.41234028],
       [ 0.        ,  0.79661434,  1.04638607],
       [ 0.        ,  0.        ,  0.54718387]])

la.norm(Q@R - A)

5.5511151231257827e-17

This is called QR factorization.

---

• When does it break?
• Does it need something like pivoting?
• Can we use it for something?