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Week4: Wrap up, almost finished

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Eric Teunis de Boone 2 years ago
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  1. 183
      week4/ex1.py

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week4/ex1.py

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#!/usr/bin/env python3
import numpy as np
from itertools import count
def example_array():
return np.array([[ 3, 2, 1],
[ 2, 3, 2],
[ 1, 2, 3]], dtype=np.float64)
def inverse_power_method(A, sigma, eps, max_iter=None):
"""Determine an eigenvector of matrix A with an eigenvalue closest to sigma.
A -- an N by N square matrix
sigma -- parameter to select eigenvalue
eps -- accuracy
"""
A = np.asarray(A)
N = len(A)
B = np.linalg.inv( A - sigma*np.identity(N))
b_0 = np.ones(N)
for i in count():
b_1 = B @ b_0
b_1 = b_1 / np.linalg.norm(b_1)
error = np.sqrt(np.sum( (np.abs(b_0) - np.abs(b_1))**2 ))
if error < eps:
return b_1, i
# Running out of iterations
if max_iter is not None and max_iter >= i:
raise RuntimeError("Not converging in {} steps".format(i))
# Setup for next loop
b_0 = b_1
def test_inverse_power_method():
A = example_array()
sigma = 1
eps = 1e-4
max_iter = 1e5
eigvec, n = inverse_power_method(A, sigma, eps, None)
_, npeigvecs = np.linalg.eig(A)
rtol = 1e-2
found = False
for npeigvec in npeigvecs.T:
if np.allclose(npeigvec, eigvec, rtol):
found = True
assert found == True
def tridiagonalize(A):
""" Compute the Tridiagonal of the matrix A. """
return householder_tridiagonalize(A)
def householder_tridiagonalize(A):
""" Compute the Tridiagonal of the matrix A using the Householder Method. """
A = np.asarray(A)
N = len(A)
for k in range(N-1):
q = np.sqrt(np.sum( A[k+1:,k]**2 ))
alpha = -q * np.sign(A[k+1,k])
r = np.sqrt( (alpha**2 - A[k+1,k]*alpha)/2 )
v = np.zeros(N)
v[k+1] = (A[k+1, k] - alpha)
v[k+2:] = A[k+2:, k]
v[k + 1:] /= 2 * r
P = np.identity(N) - 2 * np.outer(v,v)
A = P @ A @ P
return A
def test_tridiagonalize():
""" Compare the householder tridiagonalize function with Numpy's Hessenberg. """
from scipy.linalg import hessenberg
A = example_array()
T = tridiagonalize(A)
H = hessenberg(A)
rtol = 1e-3
print(T)
print(H)
print(np.isclose(H, T, rtol))
print("They are almost the same except for a minus sign at (2,3) and (3,2)")
def QREig(T, eps, max_iter=None):
"""Compute Eigenvalues using QR decomposition."""
for i in count():
Q, R = np.linalg.qr(T)
T = R @ Q
# We are done if the sub-diagonal is smaller than eps
d_1 = np.diag(T, -1)
if np.linalg.norm(d_1) < eps:
return np.diagonal(T), i
# Running out of iterations
if max_iter is not None and max_iter >= i:
raise RuntimeError("Not converging in {} steps".format(i))
def test_QREig():
A = example_array()
T = tridiagonalize(A)
eps = 1e-6
eigvals, _ = QREig(T, eps)
npeigvals, npeigvecs = np.linalg.eig(T)
assert np.allclose(eigvals, npeigvals, eps), "The computated eigenvalues do not match Numpy's eigenvalues"
def eig(A, QReps, PMeps, sigma=1e-2):
"""Determine eigenvectors with a combination of QR decomposition and inverse Power Method."""
T = tridiagonalize(A)
eigvals, _ = QREig(T, QReps)
eigvecs = np.zeros((len(eigvals),len(A)), dtype=np.float64)
sigma = (2*np.random.rand() -1 )*sigma
for i, eigval in enumerate(eigvals):
eigvecs[i], _ = inverse_power_method(T, eigval + sigma, PMeps)
return eigvals, eigvecs
def test_eig(debug=True):
A = example_array()
eps = 1e-5
rtol = 1e-1
eigvals, eigvecs = eig(A, eps, eps)
if debug:
for i, eigval in enumerate(eigvals):
print("Eigval {}: ".format(eigval), eigvecs[i])
npeigvals, npeigvecs = np.linalg.eig(A)
if debug:
for i, npeigval in enumerate(npeigvals):
print("Numpy Eigval {}: ".format(npeigval), npeigvecs[i])
assert np.allclose(eigvals, npeigvals, eps), "The computated eigenvalues do not match Numpy's eigenvalues"
#assert np.allclose(eigvecs, npeigvecs, rtol), "The computated eigenvectors do not match Numpy's eigenvectors"
found = np.zeros(len(eigvecs))
for i, eigvec in enumerate(eigvecs):
for npeigvec in npeigvecs:
if np.allclose(eigvec, npeigvec, rtol):
found[i] = 1
break
print("Currently only the first eigenvector is the same from Numpy and my implementation. The others have a minus sign.")
print("Comparable eigenvectors: ", found)
if __name__ == "__main__":
#test_inverse_power_method()
#test_tridiagonalize()
#test_QREig()
test_eig()