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* Remove eval from arithmetic_analysis/newton_raphson.py
* Relocate contents of arithmetic_analysis/
Delete the arithmetic_analysis/ directory and relocate its files because
the purpose of the directory was always ill-defined. "Arithmetic
analysis" isn't a field of math, and the directory's files contained
algorithms for linear algebra, numerical analysis, and physics.
Relocated the directory's linear algebra algorithms to linear_algebra/,
its numerical analysis algorithms to a new subdirectory called
maths/numerical_analysis/, and its single physics algorithm to physics/.
* updating DIRECTORY.md
---------
Co-authored-by: github-actions <${GITHUB_ACTOR}@users.noreply.github.com>
112 lines
3.8 KiB
Python
112 lines
3.8 KiB
Python
"""
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Lower–upper (LU) decomposition factors a matrix as a product of a lower
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triangular matrix and an upper triangular matrix. A square matrix has an LU
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decomposition under the following conditions:
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- If the matrix is invertible, then it has an LU decomposition if and only
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if all of its leading principal minors are non-zero (see
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https://en.wikipedia.org/wiki/Minor_(linear_algebra) for an explanation of
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leading principal minors of a matrix).
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- If the matrix is singular (i.e., not invertible) and it has a rank of k
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(i.e., it has k linearly independent columns), then it has an LU
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decomposition if its first k leading principal minors are non-zero.
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This algorithm will simply attempt to perform LU decomposition on any square
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matrix and raise an error if no such decomposition exists.
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Reference: https://en.wikipedia.org/wiki/LU_decomposition
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"""
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from __future__ import annotations
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import numpy as np
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def lower_upper_decomposition(table: np.ndarray) -> tuple[np.ndarray, np.ndarray]:
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"""
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Perform LU decomposition on a given matrix and raises an error if the matrix
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isn't square or if no such decomposition exists
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>>> matrix = np.array([[2, -2, 1], [0, 1, 2], [5, 3, 1]])
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>>> lower_mat, upper_mat = lower_upper_decomposition(matrix)
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>>> lower_mat
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array([[1. , 0. , 0. ],
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[0. , 1. , 0. ],
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[2.5, 8. , 1. ]])
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>>> upper_mat
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array([[ 2. , -2. , 1. ],
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[ 0. , 1. , 2. ],
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[ 0. , 0. , -17.5]])
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>>> matrix = np.array([[4, 3], [6, 3]])
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>>> lower_mat, upper_mat = lower_upper_decomposition(matrix)
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>>> lower_mat
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array([[1. , 0. ],
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[1.5, 1. ]])
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>>> upper_mat
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array([[ 4. , 3. ],
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[ 0. , -1.5]])
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# Matrix is not square
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>>> matrix = np.array([[2, -2, 1], [0, 1, 2]])
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>>> lower_mat, upper_mat = lower_upper_decomposition(matrix)
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Traceback (most recent call last):
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...
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ValueError: 'table' has to be of square shaped array but got a 2x3 array:
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[[ 2 -2 1]
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[ 0 1 2]]
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# Matrix is invertible, but its first leading principal minor is 0
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>>> matrix = np.array([[0, 1], [1, 0]])
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>>> lower_mat, upper_mat = lower_upper_decomposition(matrix)
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Traceback (most recent call last):
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...
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ArithmeticError: No LU decomposition exists
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# Matrix is singular, but its first leading principal minor is 1
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>>> matrix = np.array([[1, 0], [1, 0]])
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>>> lower_mat, upper_mat = lower_upper_decomposition(matrix)
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>>> lower_mat
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array([[1., 0.],
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[1., 1.]])
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>>> upper_mat
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array([[1., 0.],
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[0., 0.]])
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# Matrix is singular, but its first leading principal minor is 0
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>>> matrix = np.array([[0, 1], [0, 1]])
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>>> lower_mat, upper_mat = lower_upper_decomposition(matrix)
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Traceback (most recent call last):
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...
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ArithmeticError: No LU decomposition exists
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"""
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# Ensure that table is a square array
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rows, columns = np.shape(table)
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if rows != columns:
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msg = (
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"'table' has to be of square shaped array but got a "
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f"{rows}x{columns} array:\n{table}"
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)
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raise ValueError(msg)
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lower = np.zeros((rows, columns))
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upper = np.zeros((rows, columns))
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# in 'total', the necessary data is extracted through slices
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# and the sum of the products is obtained.
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for i in range(columns):
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for j in range(i):
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total = np.sum(lower[i, :i] * upper[:i, j])
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if upper[j][j] == 0:
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raise ArithmeticError("No LU decomposition exists")
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lower[i][j] = (table[i][j] - total) / upper[j][j]
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lower[i][i] = 1
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for j in range(i, columns):
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total = np.sum(lower[i, :i] * upper[:i, j])
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upper[i][j] = table[i][j] - total
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return lower, upper
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if __name__ == "__main__":
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import doctest
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doctest.testmod()
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