Python/linear_algebra/src/power_iteration.py
Dhruv Manilawala 8c2986026b
fix(mypy): type annotations for linear algebra algorithms (#4317)
* fix(mypy): type annotations for linear algebra algorithms

* refactor: remove linear algebra directory from mypy exclude
2021-04-05 15:37:38 +02:00

105 lines
3.4 KiB
Python

import numpy as np
def power_iteration(
input_matrix: np.ndarray,
vector: np.ndarray,
error_tol: float = 1e-12,
max_iterations: int = 100,
) -> tuple[float, np.ndarray]:
"""
Power Iteration.
Find the largest eignevalue and corresponding eigenvector
of matrix input_matrix given a random vector in the same space.
Will work so long as vector has component of largest eigenvector.
input_matrix must be symmetric.
Input
input_matrix: input matrix whose largest eigenvalue we will find.
Numpy array. np.shape(input_matrix) == (N,N).
vector: random initial vector in same space as matrix.
Numpy array. np.shape(vector) == (N,) or (N,1)
Output
largest_eigenvalue: largest eigenvalue of the matrix input_matrix.
Float. Scalar.
largest_eigenvector: eigenvector corresponding to largest_eigenvalue.
Numpy array. np.shape(largest_eigenvector) == (N,) or (N,1).
>>> import numpy as np
>>> input_matrix = np.array([
... [41, 4, 20],
... [ 4, 26, 30],
... [20, 30, 50]
... ])
>>> vector = np.array([41,4,20])
>>> power_iteration(input_matrix,vector)
(79.66086378788381, array([0.44472726, 0.46209842, 0.76725662]))
"""
# Ensure matrix is square.
assert np.shape(input_matrix)[0] == np.shape(input_matrix)[1]
# Ensure proper dimensionality.
assert np.shape(input_matrix)[0] == np.shape(vector)[0]
# Set convergence to False. Will define convergence when we exceed max_iterations
# or when we have small changes from one iteration to next.
convergence = False
lamda_previous = 0
iterations = 0
error = 1e12
while not convergence:
# Multiple matrix by the vector.
w = np.dot(input_matrix, vector)
# Normalize the resulting output vector.
vector = w / np.linalg.norm(w)
# Find rayleigh quotient
# (faster than usual b/c we know vector is normalized already)
lamda = np.dot(vector.T, np.dot(input_matrix, vector))
# Check convergence.
error = np.abs(lamda - lamda_previous) / lamda
iterations += 1
if error <= error_tol or iterations >= max_iterations:
convergence = True
lamda_previous = lamda
return lamda, vector
def test_power_iteration() -> None:
"""
>>> test_power_iteration() # self running tests
"""
# Our implementation.
input_matrix = np.array([[41, 4, 20], [4, 26, 30], [20, 30, 50]])
vector = np.array([41, 4, 20])
eigen_value, eigen_vector = power_iteration(input_matrix, vector)
# Numpy implementation.
# Get eigen values and eigen vectors using built in numpy
# eigh (eigh used for symmetric or hermetian matrices).
eigen_values, eigen_vectors = np.linalg.eigh(input_matrix)
# Last eigen value is the maximum one.
eigen_value_max = eigen_values[-1]
# Last column in this matrix is eigen vector corresponding to largest eigen value.
eigen_vector_max = eigen_vectors[:, -1]
# Check our implementation and numpy gives close answers.
assert np.abs(eigen_value - eigen_value_max) <= 1e-6
# Take absolute values element wise of each eigenvector.
# as they are only unique to a minus sign.
assert np.linalg.norm(np.abs(eigen_vector) - np.abs(eigen_vector_max)) <= 1e-6
if __name__ == "__main__":
import doctest
doctest.testmod()
test_power_iteration()