mirror of
https://github.com/TheAlgorithms/Python.git
synced 2025-04-07 22:35:54 +00:00
Rename variables
This commit is contained in:
parent
818448b05d
commit
4522258980
@ -1,8 +1,7 @@
|
||||
import numpy as np
|
||||
|
||||
|
||||
# ruff: noqa: N803,N806
|
||||
def cholesky_decomposition(A: np.ndarray) -> np.ndarray:
|
||||
def cholesky_decomposition(a: np.ndarray) -> np.ndarray:
|
||||
"""Return a Cholesky decomposition of the matrix A.
|
||||
|
||||
The Cholesky decomposition decomposes the square, positive definite matrix A
|
||||
@ -26,7 +25,7 @@ def cholesky_decomposition(A: np.ndarray) -> np.ndarray:
|
||||
>>> np.allclose(np.tril(L), L)
|
||||
True
|
||||
|
||||
The Cholesky decomposition can be used to solve the system of equations A x = y.
|
||||
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
|
||||
@ -43,28 +42,30 @@ def cholesky_decomposition(A: np.ndarray) -> np.ndarray:
|
||||
True
|
||||
"""
|
||||
|
||||
assert A.shape[0] == A.shape[1], f"Matrix A is not square, {A.shape=}"
|
||||
assert np.allclose(A, A.T), "Matrix A must be symmetric"
|
||||
assert a.shape[0] == a.shape[1], f"Matrix A is not square, {a.shape=}"
|
||||
assert np.allclose(a, a.T), "Matrix A must be symmetric"
|
||||
|
||||
n = A.shape[0]
|
||||
L = np.tril(A)
|
||||
n = a.shape[0]
|
||||
lower_triangle = np.tril(a)
|
||||
|
||||
for i in range(n):
|
||||
for j in range(i + 1):
|
||||
L[i, j] -= np.sum(L[i, :j] * L[j, :j])
|
||||
lower_triangle[i, j] -= np.sum(
|
||||
lower_triangle[i, :j] * lower_triangle[j, :j]
|
||||
)
|
||||
|
||||
if i == j:
|
||||
if L[i, i] <= 0:
|
||||
if lower_triangle[i, i] <= 0:
|
||||
raise ValueError("Matrix A is not positive definite")
|
||||
|
||||
L[i, i] = np.sqrt(L[i, i])
|
||||
lower_triangle[i, i] = np.sqrt(lower_triangle[i, i])
|
||||
else:
|
||||
L[i, j] /= L[j, j]
|
||||
lower_triangle[i, j] /= lower_triangle[j, j]
|
||||
|
||||
return L
|
||||
return lower_triangle
|
||||
|
||||
|
||||
def solve_cholesky(L: np.ndarray, Y: np.ndarray) -> np.ndarray:
|
||||
def solve_cholesky(lower_triangle: np.ndarray, y: np.ndarray) -> np.ndarray:
|
||||
"""Given a Cholesky decomposition L L^T = A of a matrix A, solve the
|
||||
system of equations A X = Y where Y is either a matrix or a vector.
|
||||
|
||||
@ -75,32 +76,36 @@ def solve_cholesky(L: np.ndarray, Y: np.ndarray) -> np.ndarray:
|
||||
True
|
||||
"""
|
||||
|
||||
assert L.shape[0] == L.shape[1], f"Matrix L is not square, {L.shape=}"
|
||||
assert np.allclose(np.tril(L), L), "Matrix L is not lower triangular"
|
||||
assert (
|
||||
lower_triangle.shape[0] == lower_triangle.shape[1]
|
||||
), f"Matrix L is not square, {lower_triangle.shape=}"
|
||||
assert np.allclose(
|
||||
np.tril(lower_triangle), lower_triangle
|
||||
), "Matrix L is not lower triangular"
|
||||
|
||||
# Handle vector case by reshaping to matrix and then flattening again
|
||||
if len(Y.shape) == 1:
|
||||
return solve_cholesky(L, Y.reshape(-1, 1)).ravel()
|
||||
if len(y.shape) == 1:
|
||||
return solve_cholesky(lower_triangle, y.reshape(-1, 1)).ravel()
|
||||
|
||||
n = Y.shape[0]
|
||||
n = y.shape[0]
|
||||
|
||||
# Solve L W = B for W
|
||||
W = Y.copy()
|
||||
w = y.copy()
|
||||
for i in range(n):
|
||||
for j in range(i):
|
||||
W[i] -= L[i, j] * W[j]
|
||||
w[i] -= lower_triangle[i, j] * w[j]
|
||||
|
||||
W[i] /= L[i, i]
|
||||
w[i] /= lower_triangle[i, i]
|
||||
|
||||
# Solve L^T X = W for X
|
||||
X = W
|
||||
x = w
|
||||
for i in reversed(range(n)):
|
||||
for j in range(i + 1, n):
|
||||
X[i] -= L[j, i] * X[j]
|
||||
x[i] -= lower_triangle[j, i] * x[j]
|
||||
|
||||
X[i] /= L[i, i]
|
||||
x[i] /= lower_triangle[i, i]
|
||||
|
||||
return X
|
||||
return x
|
||||
|
||||
|
||||
if __name__ == "__main__":
|
||||
|
Loading…
x
Reference in New Issue
Block a user