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Fix sphinx/build_docs warnings for linear_algebra (#12483)

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* Fix sphinx/build_docs warnings for linear_algebra/

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Maxim Smolskiy
2024-12-30 00:35:34 +03:00
committed by GitHub
parent 3622e940c9
commit 94b3777936
6 changed files with 54 additions and 34 deletions
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20
linear_algebra/lu_decomposition.py
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@ -2,13 +2,14 @@
Lower-upper (LU) decomposition factors a matrix as a product of a lower
triangular matrix and an upper triangular matrix. A square matrix has an LU
decomposition under the following conditions:
- If the matrix is invertible, then it has an LU decomposition if and only
if all of its leading principal minors are non-zero (see
https://en.wikipedia.org/wiki/Minor_(linear_algebra) for an explanation of
leading principal minors of a matrix).
if all of its leading principal minors are non-zero (see
https://en.wikipedia.org/wiki/Minor_(linear_algebra) for an explanation of
leading principal minors of a matrix).
- If the matrix is singular (i.e., not invertible) and it has a rank of k
(i.e., it has k linearly independent columns), then it has an LU
decomposition if its first k leading principal minors are non-zero.
(i.e., it has k linearly independent columns), then it has an LU
decomposition if its first k leading principal minors are non-zero.
This algorithm will simply attempt to perform LU decomposition on any square
matrix and raise an error if no such decomposition exists.
@ -25,6 +26,7 @@ def lower_upper_decomposition(table: np.ndarray) -> tuple[np.ndarray, np.ndarray
"""
Perform LU decomposition on a given matrix and raises an error if the matrix
isn't square or if no such decomposition exists
>>> matrix = np.array([[2, -2, 1], [0, 1, 2], [5, 3, 1]])
>>> lower_mat, upper_mat = lower_upper_decomposition(matrix)
>>> lower_mat
@ -45,7 +47,7 @@ def lower_upper_decomposition(table: np.ndarray) -> tuple[np.ndarray, np.ndarray
array([[ 4. , 3. ],
[ 0. , -1.5]])
# Matrix is not square
>>> # Matrix is not square
>>> matrix = np.array([[2, -2, 1], [0, 1, 2]])
>>> lower_mat, upper_mat = lower_upper_decomposition(matrix)
Traceback (most recent call last):
@ -54,14 +56,14 @@ def lower_upper_decomposition(table: np.ndarray) -> tuple[np.ndarray, np.ndarray
[[ 2 -2 1]
[ 0 1 2]]
# Matrix is invertible, but its first leading principal minor is 0
>>> # Matrix is invertible, but its first leading principal minor is 0
>>> matrix = np.array([[0, 1], [1, 0]])
>>> lower_mat, upper_mat = lower_upper_decomposition(matrix)
Traceback (most recent call last):
...
ArithmeticError: No LU decomposition exists
# Matrix is singular, but its first leading principal minor is 1
>>> # Matrix is singular, but its first leading principal minor is 1
>>> matrix = np.array([[1, 0], [1, 0]])
>>> lower_mat, upper_mat = lower_upper_decomposition(matrix)
>>> lower_mat
@ -71,7 +73,7 @@ def lower_upper_decomposition(table: np.ndarray) -> tuple[np.ndarray, np.ndarray
array([[1., 0.],
[0., 0.]])
# Matrix is singular, but its first leading principal minor is 0
>>> # Matrix is singular, but its first leading principal minor is 0
>>> matrix = np.array([[0, 1], [0, 1]])
>>> lower_mat, upper_mat = lower_upper_decomposition(matrix)
Traceback (most recent call last):