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a8b6bda993
* 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>
87 lines
2.5 KiB
Python
87 lines
2.5 KiB
Python
"""
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Gaussian elimination method for solving a system of linear equations.
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Gaussian elimination - https://en.wikipedia.org/wiki/Gaussian_elimination
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"""
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import numpy as np
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from numpy import float64
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from numpy.typing import NDArray
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def retroactive_resolution(
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coefficients: NDArray[float64], vector: NDArray[float64]
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) -> NDArray[float64]:
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"""
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This function performs a retroactive linear system resolution
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for triangular matrix
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Examples:
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2x1 + 2x2 - 1x3 = 5 2x1 + 2x2 = -1
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0x1 - 2x2 - 1x3 = -7 0x1 - 2x2 = -1
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0x1 + 0x2 + 5x3 = 15
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>>> gaussian_elimination([[2, 2, -1], [0, -2, -1], [0, 0, 5]], [[5], [-7], [15]])
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array([[2.],
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[2.],
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[3.]])
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>>> gaussian_elimination([[2, 2], [0, -2]], [[-1], [-1]])
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array([[-1. ],
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[ 0.5]])
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"""
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rows, columns = np.shape(coefficients)
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x: NDArray[float64] = np.zeros((rows, 1), dtype=float)
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for row in reversed(range(rows)):
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total = np.dot(coefficients[row, row + 1 :], x[row + 1 :])
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x[row, 0] = (vector[row][0] - total[0]) / coefficients[row, row]
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return x
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def gaussian_elimination(
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coefficients: NDArray[float64], vector: NDArray[float64]
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) -> NDArray[float64]:
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"""
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This function performs Gaussian elimination method
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Examples:
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1x1 - 4x2 - 2x3 = -2 1x1 + 2x2 = 5
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5x1 + 2x2 - 2x3 = -3 5x1 + 2x2 = 5
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1x1 - 1x2 + 0x3 = 4
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>>> gaussian_elimination([[1, -4, -2], [5, 2, -2], [1, -1, 0]], [[-2], [-3], [4]])
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array([[ 2.3 ],
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[-1.7 ],
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[ 5.55]])
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>>> gaussian_elimination([[1, 2], [5, 2]], [[5], [5]])
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array([[0. ],
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[2.5]])
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"""
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# coefficients must to be a square matrix so we need to check first
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rows, columns = np.shape(coefficients)
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if rows != columns:
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return np.array((), dtype=float)
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# augmented matrix
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augmented_mat: NDArray[float64] = np.concatenate((coefficients, vector), axis=1)
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augmented_mat = augmented_mat.astype("float64")
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# scale the matrix leaving it triangular
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for row in range(rows - 1):
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pivot = augmented_mat[row, row]
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for col in range(row + 1, columns):
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factor = augmented_mat[col, row] / pivot
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augmented_mat[col, :] -= factor * augmented_mat[row, :]
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x = retroactive_resolution(
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augmented_mat[:, 0:columns], augmented_mat[:, columns : columns + 1]
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)
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return x
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if __name__ == "__main__":
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import doctest
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doctest.testmod()
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