```
"""
Lowerâ€“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 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.
This algorithm will simply attempt to perform LU decomposition on any square
matrix and raise an error if no such decomposition exists.
Reference: https://en.wikipedia.org/wiki/LU_decomposition
"""
from __future__ import annotations
import numpy as np
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
array([[1. , 0. , 0. ],
[0. , 1. , 0. ],
[2.5, 8. , 1. ]])
>>> upper_mat
array([[ 2. , -2. , 1. ],
[ 0. , 1. , 2. ],
[ 0. , 0. , -17.5]])
>>> matrix = np.array([[4, 3], [6, 3]])
>>> lower_mat, upper_mat = lower_upper_decomposition(matrix)
>>> lower_mat
array([[1. , 0. ],
[1.5, 1. ]])
>>> upper_mat
array([[ 4. , 3. ],
[ 0. , -1.5]])
# 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):
...
ValueError: 'table' has to be of square shaped array but got a 2x3 array:
[[ 2 -2 1]
[ 0 1 2]]
# 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 = np.array([[1, 0], [1, 0]])
>>> lower_mat, upper_mat = lower_upper_decomposition(matrix)
>>> lower_mat
array([[1., 0.],
[1., 1.]])
>>> upper_mat
array([[1., 0.],
[0., 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):
...
ArithmeticError: No LU decomposition exists
"""
# Ensure that table is a square array
rows, columns = np.shape(table)
if rows != columns:
msg = (
"'table' has to be of square shaped array but got a "
f"{rows}x{columns} array:\n{table}"
)
raise ValueError(msg)
lower = np.zeros((rows, columns))
upper = np.zeros((rows, columns))
# in 'total', the necessary data is extracted through slices
# and the sum of the products is obtained.
for i in range(columns):
for j in range(i):
total = np.sum(lower[i, :i] * upper[:i, j])
if upper[j][j] == 0:
raise ArithmeticError("No LU decomposition exists")
lower[i][j] = (table[i][j] - total) / upper[j][j]
lower[i][i] = 1
for j in range(i, columns):
total = np.sum(lower[i, :i] * upper[:i, j])
upper[i][j] = table[i][j] - total
return lower, upper
if __name__ == "__main__":
import doctest
doctest.testmod()
```