maths.cholesky_decomposition

Functions

cholesky_decomposition(→ numpy.ndarray)	Return a Cholesky decomposition of the matrix A.
solve_cholesky(→ numpy.ndarray)	Given a Cholesky decomposition L L^T = A of a matrix A, solve the

Module Contents

maths.cholesky_decomposition.cholesky_decomposition(matrix: numpy.ndarray) → numpy.ndarray

Return a Cholesky decomposition of the matrix A.

The Cholesky decomposition decomposes the square, positive definite matrix A into a lower triangular matrix L such that A = L L^T.

https://en.wikipedia.org/wiki/Cholesky_decomposition

Arguments: A – a numpy.ndarray of shape (n, n)

>>> A = np.array([[4, 12, -16], [12, 37, -43], [-16, -43, 98]], dtype=float)
>>> L = cholesky_decomposition(A)
>>> np.allclose(L, np.array([[2, 0, 0], [6, 1, 0], [-8, 5, 3]]))
True

>>> # check that the decomposition is correct
>>> np.allclose(L @ L.T, A)
True

>>> # check that L is lower triangular
>>> np.allclose(np.tril(L), L)
True

The Cholesky decomposition can be used to solve the linear system A x = y.

>>> x_true = np.array([1, 2, 3], dtype=float)
>>> y = A @ x_true
>>> x = solve_cholesky(L, y)
>>> np.allclose(x, x_true)
True

It can also be used to solve multiple equations A X = Y simultaneously.

>>> X_true = np.random.rand(3, 3)
>>> Y = A @ X_true
>>> X = solve_cholesky(L, Y)
>>> np.allclose(X, X_true)
True

maths.cholesky_decomposition.solve_cholesky(lower_triangle: numpy.ndarray, right_hand_side: numpy.ndarray) → numpy.ndarray

Given a Cholesky decomposition L L^T = A of a matrix A, solve the system of equations A X = Y where the right-hand side Y is either a matrix or a vector.

>>> L = np.array([[2, 0], [3, 4]], dtype=float)
>>> Y = np.array([[22, 54], [81, 193]], dtype=float)
>>> X = solve_cholesky(L, Y)
>>> np.allclose(X, np.array([[1, 3], [3, 7]], dtype=float))
True