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109 lines
2.8 KiB
Python
109 lines
2.8 KiB
Python
import numpy as np
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# ruff: noqa: N803,N806
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def cholesky_decomposition(A: np.ndarray) -> np.ndarray:
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"""Return a Cholesky decomposition of the matrix A.
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The Cholesky decomposition decomposes the square, positive definite matrix A
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into a lower triangular matrix L such that A = L L^T.
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https://en.wikipedia.org/wiki/Cholesky_decomposition
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Arguments:
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A -- a numpy.ndarray of shape (n, n)
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>>> A = np.array([[4, 12, -16], [12, 37, -43], [-16, -43, 98]], dtype=float)
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>>> L = cholesky_decomposition(A)
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>>> np.allclose(L, np.array([[2, 0, 0], [6, 1, 0], [-8, 5, 3]]))
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True
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>>> # check that the decomposition is correct
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>>> np.allclose(L @ L.T, A)
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True
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>>> # check that L is lower triangular
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>>> np.allclose(np.tril(L), L)
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True
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The Cholesky decomposition can be used to solve the system of equations A x = y.
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>>> x_true = np.array([1, 2, 3], dtype=float)
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>>> y = A @ x_true
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>>> x = solve_cholesky(L, y)
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>>> np.allclose(x, x_true)
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True
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It can also be used to solve multiple equations A X = Y simultaneously.
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>>> X_true = np.random.rand(3, 3)
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>>> Y = A @ X_true
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>>> X = solve_cholesky(L, Y)
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>>> np.allclose(X, X_true)
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True
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"""
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assert A.shape[0] == A.shape[1], f"A is not square, {A.shape=}"
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n = A.shape[0]
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L = np.tril(A)
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for i in range(n):
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for j in range(i + 1):
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L[i, j] -= np.sum(L[i, :j] * L[j, :j])
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if i == j:
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if L[i, i] <= 0:
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raise ValueError("Matrix A is not positive definite")
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L[i, i] = np.sqrt(L[i, i])
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else:
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L[i, j] /= L[j, j]
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return L
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def solve_cholesky(L: np.ndarray, Y: np.ndarray) -> np.ndarray:
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"""Given a Cholesky decomposition L L^T = A of a matrix A, solve the
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system of equations A X = Y where B is either a matrix or a vector.
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>>> L = np.array([[2, 0], [3, 4]], dtype=float)
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>>> Y = np.array([[22, 54], [81, 193]], dtype=float)
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>>> X = solve_cholesky(L, Y)
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>>> np.allclose(X, np.array([[1, 3], [3, 7]], dtype=float))
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True
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"""
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assert L.shape[0] == L.shape[1], f"L is not square, {L.shape=}"
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assert np.allclose(np.tril(L), L), "L is not lower triangular"
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# Handle vector case by reshaping to matrix and then flattening again
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if len(Y.shape) == 1:
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return solve_cholesky(L, Y.reshape(-1, 1)).ravel()
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n = Y.shape[0]
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# Solve L W = B for W
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W = Y.copy()
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for i in range(n):
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for j in range(i):
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W[i] -= L[i, j] * W[j]
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W[i] /= L[i, i]
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# Solve L^T X = W for X
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X = W
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for i in reversed(range(n)):
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for j in range(i + 1, n):
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X[i] -= L[j, i] * X[j]
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X[i] /= L[i, i]
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return X
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if __name__ == "__main__":
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import doctest
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doctest.testmod()
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