linear_algebra.src.conjugate_gradient

Resources: - https://en.wikipedia.org/wiki/Conjugate_gradient_method - https://en.wikipedia.org/wiki/Definite_symmetric_matrix

Functions

_create_spd_matrix(→ Any)	Returns a symmetric positive definite matrix given a dimension.
_is_matrix_spd(→ bool)	Returns True if input matrix is symmetric positive definite.
conjugate_gradient(→ Any)	Returns solution to the linear system np.dot(spd_matrix, x) = b.
test_conjugate_gradient(→ None)

Module Contents

linear_algebra.src.conjugate_gradient._create_spd_matrix(dimension: int) → Any

Returns a symmetric positive definite matrix given a dimension.

Input: dimension gives the square matrix dimension.

Output: spd_matrix is an diminesion x dimensions symmetric positive definite (SPD) matrix.

>>> import numpy as np
>>> dimension = 3
>>> spd_matrix = _create_spd_matrix(dimension)
>>> _is_matrix_spd(spd_matrix)
True

linear_algebra.src.conjugate_gradient._is_matrix_spd(matrix: numpy.ndarray) → bool

Returns True if input matrix is symmetric positive definite. Returns False otherwise.

For a matrix to be SPD, all eigenvalues must be positive.

>>> import numpy as np
>>> matrix = np.array([
... [4.12401784, -5.01453636, -0.63865857],
... [-5.01453636, 12.33347422, -3.40493586],
... [-0.63865857, -3.40493586,  5.78591885]])
>>> _is_matrix_spd(matrix)
True
>>> matrix = np.array([
... [0.34634879,  1.96165514,  2.18277744],
... [0.74074469, -1.19648894, -1.34223498],
... [-0.7687067 ,  0.06018373, -1.16315631]])
>>> _is_matrix_spd(matrix)
False

linear_algebra.src.conjugate_gradient.conjugate_gradient(spd_matrix: numpy.ndarray, load_vector: numpy.ndarray, max_iterations: int = 1000, tol: float = 1e-08) → Any

Returns solution to the linear system np.dot(spd_matrix, x) = b.

Input: spd_matrix is an NxN Symmetric Positive Definite (SPD) matrix. load_vector is an Nx1 vector.

Output: x is an Nx1 vector that is the solution vector.

>>> import numpy as np
>>> spd_matrix = np.array([
... [8.73256573, -5.02034289, -2.68709226],
... [-5.02034289,  3.78188322,  0.91980451],
... [-2.68709226,  0.91980451,  1.94746467]])
>>> b = np.array([
... [-5.80872761],
... [ 3.23807431],
... [ 1.95381422]])
>>> conjugate_gradient(spd_matrix, b)
array([[-0.63114139],
       [-0.01561498],
       [ 0.13979294]])

linear_algebra.src.conjugate_gradient.test_conjugate_gradient() → None

>>> test_conjugate_gradient()  # self running tests