Arnoldi Iteration¶
Copyright (C) 2020 Andreas Kloeckner
MIT License
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import numpy as np
import numpy.linalg as la
import matplotlib.pyplot as pt
Let us make a matrix with a defined set of eigenvalues and eigenvectors, given by eigvals and eigvecs.
np.random.seed(40)
# Generate matrix with eigenvalues 1...25
n = 25
eigvals = np.linspace(1., n, n)
eigvecs = np.random.randn(n, n)
print(eigvals)
A = la.solve(eigvecs, np.dot(np.diag(eigvals), eigvecs))
print(la.eig(A)[0])
[ 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.] [25. 24. 23. 1. 2. 3. 22. 4. 21. 20. 5. 6. 7. 19. 18. 8. 9. 17. 16. 10. 11. 12. 15. 14. 13.]
Initialization¶
Set up $Q$ and $H$:
Q = np.zeros((n, n))
H = np.zeros((n, n))
k = 0
Pick a starting vector, normalize it
x0 = np.random.randn(n)
x0 = x0/la.norm(x0)
# Poke it into the first column of Q
Q[:, k] = x0
del x0
Make a list to save arrays of Ritz values:
ritz_values = []
Algorithm¶
Carry out one iteration of Arnoldi iteration.
Run this cell in-place (Ctrl-Enter) until H is filled.
print(k)
u = A @ Q[:, k]
# Carry out Gram-Schmidt on u against Q
for j in range(k+1):
qj = Q[:, j]
H[j,k] = qj @ u
u = u - H[j,k]*qj
if k+1 < n:
H[k+1, k] = la.norm(u)
Q[:, k+1] = u/H[k+1, k]
k += 1
pt.spy(H)
ritz_values.append(la.eig(H)[0])
24
Check that $Q^T A Q =H$:
la.norm(Q.T @ A @ Q - H)/ la.norm(A)
1.992040740292051e-08
Check that Q is orthogonal:
la.norm(Q.T @ Q - np.eye(n))
7.675050192387198e-08
Plot convergence of Ritz values¶
Enable the Ritz value collection above to make this work.
for i, rv in enumerate(ritz_values):
pt.plot([i] * len(rv), rv, "x")
/home/andreas/src/env-3.8/lib/python3.8/site-packages/numpy/core/_asarray.py:85: ComplexWarning: Casting complex values to real discards the imaginary part return array(a, dtype, copy=False, order=order) /home/andreas/src/env-3.8/lib/python3.8/site-packages/numpy/core/_asarray.py:85: ComplexWarning: Casting complex values to real discards the imaginary part return array(a, dtype, copy=False, order=order) /home/andreas/src/env-3.8/lib/python3.8/site-packages/numpy/core/_asarray.py:85: ComplexWarning: Casting complex values to real discards the imaginary part return array(a, dtype, copy=False, order=order) /home/andreas/src/env-3.8/lib/python3.8/site-packages/numpy/core/_asarray.py:85: ComplexWarning: Casting complex values to real discards the imaginary part return array(a, dtype, copy=False, order=order) /home/andreas/src/env-3.8/lib/python3.8/site-packages/numpy/core/_asarray.py:85: ComplexWarning: Casting complex values to real discards the imaginary part return array(a, dtype, copy=False, order=order) /home/andreas/src/env-3.8/lib/python3.8/site-packages/numpy/core/_asarray.py:85: ComplexWarning: Casting complex values to real discards the imaginary part return array(a, dtype, copy=False, order=order) /home/andreas/src/env-3.8/lib/python3.8/site-packages/numpy/core/_asarray.py:85: ComplexWarning: Casting complex values to real discards the imaginary part return array(a, dtype, copy=False, order=order)
