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Gauss-Jordan Elimination #12876

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77 changes: 77 additions & 0 deletions linear_algebra/gauss_jordan.py
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Original file line number Diff line number Diff line change
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import numpy as np


def gauss_jordan(
coefficients: np.ndarray, vertices: np.ndarray
) -> tuple[np.ndarray, np.ndarray]:
"""
Performs Gauss-Jordan elimination on the system Ax = b to reduce A to its
Reduced Row Echelon Form (RREF) and transform b accordingly.

Args:
coefficients: A 2D NumPy array representing the coefficient matrix A.
vertices: A column vector (2D NumPy array) representing the RHS b.

Returns:
A tuple containing:
- RREF of matrix A
- Transformed RHS vector b

Raises:
ValueError: If shapes of A and b are incompatible.

See Also:
https://en.wikibooks.org/wiki/Linear_Algebra/Gauss-Jordan_Reduction

Examples:
>>> import numpy as np
>>> A = np.array([[1, 2, -1], [2, 4, -2], [3, 6, -3]])
>>> b = np.array([[1], [2], [3]])
>>> rref_A, rref_b = gauss_jordan(A, b)
>>> np.allclose(rref_A, np.array([[1., 2., -1.], [0., 0., 0.], [0., 0., 0.]]))
True
>>> np.allclose(rref_b, np.array([[1.], [0.], [0.]]))
True

"""
if coefficients.ndim != 2 or vertices.ndim != 2:
raise ValueError("Both inputs must be 2D arrays.")
if coefficients.shape[0] != vertices.shape[0]:
raise ValueError("Number of rows in coefficients and vertices must match.")

coefficients = coefficients.astype(float).copy()
vertices = vertices.astype(float).copy()
rows, cols = coefficients.shape

for col in range(cols):
pivot_row = None
for row in range(col, rows):
if not np.isclose(coefficients[row, col], 0):
pivot_row = row
break

if pivot_row is None:
continue

if pivot_row != col:
coefficients[[col, pivot_row]] = coefficients[[pivot_row, col]]
vertices[[col, pivot_row]] = vertices[[pivot_row, col]]

pivot_val = coefficients[col, col]
coefficients[col] /= pivot_val
vertices[col] /= pivot_val

for row in range(rows):
if row == col:
continue
factor = coefficients[row, col]
coefficients[row] -= factor * coefficients[col]
vertices[row] -= factor * vertices[col]

return coefficients, vertices


if __name__ == "__main__":
import doctest

doctest.testmod()

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