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Commit 94b3777

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MaximSmolskiypre-commit-ci[bot]
andauthored
Fix sphinx/build_docs warnings for linear_algebra (TheAlgorithms#12483)
* Fix sphinx/build_docs warnings for linear_algebra/ * [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci --------- Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com>
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‎linear_algebra/gaussian_elimination.py‎

Lines changed: 19 additions & 9 deletions
Original file line numberDiff line numberDiff line change
@@ -1,6 +1,6 @@
11
"""
2-
Gaussian elimination method for solving a system of linear equations.
3-
Gaussian elimination - https://en.wikipedia.org/wiki/Gaussian_elimination
2+
| Gaussian elimination method for solving a system of linear equations.
3+
| Gaussian elimination - https://en.wikipedia.org/wiki/Gaussian_elimination
44
"""
55

66
import numpy as np
@@ -13,12 +13,17 @@ def retroactive_resolution(
1313
) -> NDArray[float64]:
1414
"""
1515
This function performs a retroactive linear system resolution
16-
for triangular matrix
16+
for triangular matrix
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1818
Examples:
19-
2x1 + 2x2 - 1x3 = 5 2x1 + 2x2 = -1
20-
0x1 - 2x2 - 1x3 = -7 0x1 - 2x2 = -1
21-
0x1 + 0x2 + 5x3 = 15
19+
1.
20+
* 2x1 + 2x2 - 1x3 = 5
21+
* 0x1 - 2x2 - 1x3 = -7
22+
* 0x1 + 0x2 + 5x3 = 15
23+
2.
24+
* 2x1 + 2x2 = -1
25+
* 0x1 - 2x2 = -1
26+
2227
>>> gaussian_elimination([[2, 2, -1], [0, -2, -1], [0, 0, 5]], [[5], [-7], [15]])
2328
array([[2.],
2429
[2.],
@@ -45,9 +50,14 @@ def gaussian_elimination(
4550
This function performs Gaussian elimination method
4651
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Examples:
48-
1x1 - 4x2 - 2x3 = -2 1x1 + 2x2 = 5
49-
5x1 + 2x2 - 2x3 = -3 5x1 + 2x2 = 5
50-
1x1 - 1x2 + 0x3 = 4
53+
1.
54+
* 1x1 - 4x2 - 2x3 = -2
55+
* 5x1 + 2x2 - 2x3 = -3
56+
* 1x1 - 1x2 + 0x3 = 4
57+
2.
58+
* 1x1 + 2x2 = 5
59+
* 5x1 + 2x2 = 5
60+
5161
>>> gaussian_elimination([[1, -4, -2], [5, 2, -2], [1, -1, 0]], [[-2], [-3], [4]])
5262
array([[ 2.3 ],
5363
[-1.7 ],

‎linear_algebra/lu_decomposition.py‎

Lines changed: 11 additions & 9 deletions
Original file line numberDiff line numberDiff line change
@@ -2,13 +2,14 @@
22
Lower-upper (LU) decomposition factors a matrix as a product of a lower
33
triangular matrix and an upper triangular matrix. A square matrix has an LU
44
decomposition under the following conditions:
5+
56
- If the matrix is invertible, then it has an LU decomposition if and only
6-
if all of its leading principal minors are non-zero (see
7-
https://en.wikipedia.org/wiki/Minor_(linear_algebra) for an explanation of
8-
leading principal minors of a matrix).
7+
if all of its leading principal minors are non-zero (see
8+
https://en.wikipedia.org/wiki/Minor_(linear_algebra) for an explanation of
9+
leading principal minors of a matrix).
910
- If the matrix is singular (i.e., not invertible) and it has a rank of k
10-
(i.e., it has k linearly independent columns), then it has an LU
11-
decomposition if its first k leading principal minors are non-zero.
11+
(i.e., it has k linearly independent columns), then it has an LU
12+
decomposition if its first k leading principal minors are non-zero.
1213
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This algorithm will simply attempt to perform LU decomposition on any square
1415
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
2526
"""
2627
Perform LU decomposition on a given matrix and raises an error if the matrix
2728
isn't square or if no such decomposition exists
29+
2830
>>> matrix = np.array([[2, -2, 1], [0, 1, 2], [5, 3, 1]])
2931
>>> lower_mat, upper_mat = lower_upper_decomposition(matrix)
3032
>>> lower_mat
@@ -45,7 +47,7 @@ def lower_upper_decomposition(table: np.ndarray) -> tuple[np.ndarray, np.ndarray
4547
array([[ 4. , 3. ],
4648
[ 0. , -1.5]])
4749
48-
# Matrix is not square
50+
>>> # Matrix is not square
4951
>>> matrix = np.array([[2, -2, 1], [0, 1, 2]])
5052
>>> lower_mat, upper_mat = lower_upper_decomposition(matrix)
5153
Traceback (most recent call last):
@@ -54,14 +56,14 @@ def lower_upper_decomposition(table: np.ndarray) -> tuple[np.ndarray, np.ndarray
5456
[[ 2 -2 1]
5557
[ 0 1 2]]
5658
57-
# Matrix is invertible, but its first leading principal minor is 0
59+
>>> # Matrix is invertible, but its first leading principal minor is 0
5860
>>> matrix = np.array([[0, 1], [1, 0]])
5961
>>> lower_mat, upper_mat = lower_upper_decomposition(matrix)
6062
Traceback (most recent call last):
6163
...
6264
ArithmeticError: No LU decomposition exists
6365
64-
# Matrix is singular, but its first leading principal minor is 1
66+
>>> # Matrix is singular, but its first leading principal minor is 1
6567
>>> matrix = np.array([[1, 0], [1, 0]])
6668
>>> lower_mat, upper_mat = lower_upper_decomposition(matrix)
6769
>>> lower_mat
@@ -71,7 +73,7 @@ def lower_upper_decomposition(table: np.ndarray) -> tuple[np.ndarray, np.ndarray
7173
array([[1., 0.],
7274
[0., 0.]])
7375
74-
# Matrix is singular, but its first leading principal minor is 0
76+
>>> # Matrix is singular, but its first leading principal minor is 0
7577
>>> matrix = np.array([[0, 1], [0, 1]])
7678
>>> lower_mat, upper_mat = lower_upper_decomposition(matrix)
7779
Traceback (most recent call last):

‎linear_algebra/src/gaussian_elimination_pivoting.py‎

Lines changed: 4 additions & 3 deletions
Original file line numberDiff line numberDiff line change
@@ -6,17 +6,18 @@ def solve_linear_system(matrix: np.ndarray) -> np.ndarray:
66
Solve a linear system of equations using Gaussian elimination with partial pivoting
77
88
Args:
9-
- matrix: Coefficient matrix with the last column representing the constants.
9+
- `matrix`: Coefficient matrix with the last column representing the constants.
1010
1111
Returns:
12-
- Solution vector.
12+
- Solution vector.
1313
1414
Raises:
15-
- ValueError: If the matrix is not correct (i.e., singular).
15+
- ``ValueError``: If the matrix is not correct (i.e., singular).
1616
1717
https://courses.engr.illinois.edu/cs357/su2013/lect.htm Lecture 7
1818
1919
Example:
20+
2021
>>> A = np.array([[2, 1, -1], [-3, -1, 2], [-2, 1, 2]], dtype=float)
2122
>>> B = np.array([8, -11, -3], dtype=float)
2223
>>> solution = solve_linear_system(np.column_stack((A, B)))

‎linear_algebra/src/rank_of_matrix.py‎

Lines changed: 5 additions & 1 deletion
Original file line numberDiff line numberDiff line change
@@ -8,11 +8,15 @@
88
def rank_of_matrix(matrix: list[list[int | float]]) -> int:
99
"""
1010
Finds the rank of a matrix.
11+
1112
Args:
12-
matrix: The matrix as a list of lists.
13+
`matrix`: The matrix as a list of lists.
14+
1315
Returns:
1416
The rank of the matrix.
17+
1518
Example:
19+
1620
>>> matrix1 = [[1, 2, 3],
1721
... [4, 5, 6],
1822
... [7, 8, 9]]

‎linear_algebra/src/schur_complement.py‎

Lines changed: 7 additions & 6 deletions
Original file line numberDiff line numberDiff line change
@@ -12,13 +12,14 @@ def schur_complement(
1212
) -> np.ndarray:
1313
"""
1414
Schur complement of a symmetric matrix X given as a 2x2 block matrix
15-
consisting of matrices A, B and C.
16-
Matrix A must be quadratic and non-singular.
17-
In case A is singular, a pseudo-inverse may be provided using
18-
the pseudo_inv argument.
15+
consisting of matrices `A`, `B` and `C`.
16+
Matrix `A` must be quadratic and non-singular.
17+
In case `A` is singular, a pseudo-inverse may be provided using
18+
the `pseudo_inv` argument.
19+
20+
| Link to Wiki: https://en.wikipedia.org/wiki/Schur_complement
21+
| See also Convex Optimization - Boyd and Vandenberghe, A.5.5
1922
20-
Link to Wiki: https://en.wikipedia.org/wiki/Schur_complement
21-
See also Convex Optimization - Boyd and Vandenberghe, A.5.5
2223
>>> import numpy as np
2324
>>> a = np.array([[1, 2], [2, 1]])
2425
>>> b = np.array([[0, 3], [3, 0]])

‎linear_algebra/src/transformations_2d.py‎

Lines changed: 8 additions & 6 deletions
Original file line numberDiff line numberDiff line change
@@ -3,13 +3,15 @@
33
44
I have added the codes for reflection, projection, scaling and rotation 2D matrices.
55
6+
.. code-block:: python
7+
68
scaling(5) = [[5.0, 0.0], [0.0, 5.0]]
7-
rotation(45) = [[0.5253219888177297, -0.8509035245341184],
8-
[0.8509035245341184, 0.5253219888177297]]
9-
projection(45) = [[0.27596319193541496, 0.446998331800279],
10-
[0.446998331800279, 0.7240368080645851]]
11-
reflection(45) = [[0.05064397763545947, 0.893996663600558],
12-
[0.893996663600558, 0.7018070490682369]]
9+
rotation(45) = [[0.5253219888177297, -0.8509035245341184],
10+
[0.8509035245341184, 0.5253219888177297]]
11+
projection(45) = [[0.27596319193541496, 0.446998331800279],
12+
[0.446998331800279, 0.7240368080645851]]
13+
reflection(45) = [[0.05064397763545947, 0.893996663600558],
14+
[0.893996663600558, 0.7018070490682369]]
1315
"""
1416

1517
from math import cos, sin

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