QR Iteration

Copyright (C) 2023 Andreas Kloeckner

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In [6]:
import numpy as np
import numpy.linalg as la

np.set_printoptions(linewidth=120)

Make a matrix with given eigenvalues:

In [7]:
n = 5

np.random.seed(70)
eigvecs = np.random.randn(n, n)
eigvals = np.sort(np.random.randn(n))

A = np.dot(la.solve(eigvecs, np.diag(eigvals)), eigvecs)
print(eigvals)

[-1.3657822  -0.78460489 -0.08829521  0.30824369  0.52110266]

Unshifted QR

In [121]:
X = A
Qall = np.eye(n)
In [142]:
Q, R = la.qr(X)
X = R @ Q

Qall = Qall @ Q

print(X)

[[-1.36578253e+00  8.75139040e-02  3.03786794e+00  2.62459350e-01  2.97606186e-01]
 [-2.24915745e-06 -7.84620834e-01 -4.58283345e-01  1.38868353e+00 -2.45154950e-01]
 [ 1.07650630e-09  4.64043042e-05  5.21117944e-01  1.51976500e-01 -7.37805721e-02]
 [ 1.84131018e-14  8.27020225e-10  1.39850164e-06  3.08244689e-01  1.76856522e-01]
 [-4.46854645e-26  4.39962637e-21 -5.92936598e-17  3.37890623e-12 -8.82952134e-02]]
In [169]:
np.tril(X, -1)
Out[169]:
array([[ 0.00000000e+000,  0.00000000e+000,  0.00000000e+000,  0.00000000e+000,  0.00000000e+000],
       [-3.53477485e-033,  0.00000000e+000,  0.00000000e+000,  0.00000000e+000,  0.00000000e+000],
       [-3.72649019e-043,  8.68015875e-012,  0.00000000e+000,  0.00000000e+000,  0.00000000e+000],
       [ 1.80007802e-085,  1.09679232e-053, -1.15248643e-043,  0.00000000e+000,  0.00000000e+000],
       [ 3.51450476e-152, -2.20139581e-120,  8.11711939e-110,  5.61178424e-068,  0.00000000e+000]])
In [170]:
la.norm(np.tril(X, -1))
Out[170]:
8.680158751671073e-12
In [143]:
la.norm(A - Qall @ X @ Qall.T, 2) / la.norm(A, 2)
Out[143]:
1.572363044898993e-15

Shifted QR

In [145]:
X = A
Qall = np.eye(n)
In [164]:
i = -4
sigma = X[i,i]
Q, R = la.qr(X - sigma*np.eye(n))
X = R @ Q + sigma*np.eye(n)

Qall = Qall @ Q

print(X)

[[-1.36578220e+000  8.74081894e-002 -7.91894537e-001  1.33958935e+000 -2.63903577e+000]
 [-3.53477485e-033 -7.84604886e-001  8.67481345e-001 -1.12831918e+000 -4.15900506e-001]
 [-3.72649019e-043  8.68015875e-012 -8.82952134e-002 -1.96984053e-001  2.33745979e-002]
 [ 1.80007802e-085  1.09679232e-053 -1.15248643e-043  3.08243691e-001 -1.43127333e-001]
 [ 3.51450476e-152 -2.20139581e-120  8.11711939e-110  5.61178424e-068  5.21102657e-001]]

To compare convergence speed, count iterations until left-of-diagonal entries decay below $10^{-10}$.

In [172]:
la.norm(np.tril(X, -1))
Out[172]:
8.680158751671073e-12
In [173]:
la.norm(A - Qall @ X @ Qall.T, 2) / la.norm(A, 2)
Out[173]:
1.5498995078019794e-15
In [ ]: