From ad9948d5d4d65d1457b58d278e780a1b9470a715 Mon Sep 17 00:00:00 2001 From: Mohammad Esfandiyar Date: Mon, 30 Oct 2023 16:50:47 +0330 Subject: [PATCH] implementation of Gaussian Elimination pivoting as a numerical linear algebra algorithm (#10457) * Adding new implementation Adding my python implementation of Gaussian Elimination pivoting as a numerical linear algebra algorithm * Delete linear_algebra/src/GaussianEliminationpivoting.py * Adding new implementation Adding my python implementation of Gaussian Elimination pivoting as a numerical linear algebra algorithm * Delete linear_algebra/src/gaussianeliminationpivoting.py * Adding new implementation Adding my python implementation of Gaussian Elimination pivoting as a numerical linear algebra algorithm for the third time because the last two times had conflict with the rules in PR * [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci * Delete linear_algebra/src/gaussianeliminationpivoting.py * Adding gaussianeliminationpivoting.py Adding my python implementation of Gaussian Elimination pivoting as a numerical linear algebra algorithm for the fourth time because the last three times had conflict with the rules in PR and bots * [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci * Update gaussianeliminationpivoting.py * [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci * Update gaussianeliminationpivoting.py * [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci * Update gaussianeliminationpivoting.py I changed a to matrix and coeff_matrix for better clarity * [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci * Update gaussianeliminationpivoting.py * [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see 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.../gaussian_elimination_pivoting.py | 101 ++++++++++++++++++ .../gaussian_elimination_pivoting/matrix.txt | 4 + 2 files changed, 105 insertions(+) create mode 100644 linear_algebra/src/gaussian_elimination_pivoting/gaussian_elimination_pivoting.py create mode 100644 linear_algebra/src/gaussian_elimination_pivoting/matrix.txt diff --git a/linear_algebra/src/gaussian_elimination_pivoting/gaussian_elimination_pivoting.py b/linear_algebra/src/gaussian_elimination_pivoting/gaussian_elimination_pivoting.py new file mode 100644 index 000000000..2a86350e9 --- /dev/null +++ b/linear_algebra/src/gaussian_elimination_pivoting/gaussian_elimination_pivoting.py @@ -0,0 +1,101 @@ +import numpy as np + +matrix = np.array( + [ + [5.0, -5.0, -3.0, 4.0, -11.0], + [1.0, -4.0, 6.0, -4.0, -10.0], + [-2.0, -5.0, 4.0, -5.0, -12.0], + [-3.0, -3.0, 5.0, -5.0, 8.0], + ], + dtype=float, +) + + +def solve_linear_system(matrix: np.ndarray) -> np.ndarray: + """ + Solve a linear system of equations using Gaussian elimination with partial pivoting + + Args: + - matrix: Coefficient matrix with the last column representing the constants. + + Returns: + - Solution vector. + + Raises: + - ValueError: If the matrix is not correct (i.e., singular). + + https://courses.engr.illinois.edu/cs357/su2013/lect.htm Lecture 7 + + Example: + >>> A = np.array([[2, 1, -1], [-3, -1, 2], [-2, 1, 2]], dtype=float) + >>> B = np.array([8, -11, -3], dtype=float) + >>> solution = solve_linear_system(np.column_stack((A, B))) + >>> np.allclose(solution, np.array([2., 3., -1.])) + True + >>> solve_linear_system(np.array([[0, 0], [0, 0]], dtype=float)) + array([nan, nan]) + """ + ab = np.copy(matrix) + num_of_rows = ab.shape[0] + num_of_columns = ab.shape[1] - 1 + x_lst: list[float] = [] + + # Lead element search + for column_num in range(num_of_rows): + for i in range(column_num, num_of_columns): + if abs(ab[i][column_num]) > abs(ab[column_num][column_num]): + ab[[column_num, i]] = ab[[i, column_num]] + if ab[column_num, column_num] == 0.0: + raise ValueError("Matrix is not correct") + else: + pass + if column_num != 0: + for i in range(column_num, num_of_rows): + ab[i, :] -= ( + ab[i, column_num - 1] + / ab[column_num - 1, column_num - 1] + * ab[column_num - 1, :] + ) + + # Upper triangular matrix + for column_num in range(num_of_rows): + for i in range(column_num, num_of_columns): + if abs(ab[i][column_num]) > abs(ab[column_num][column_num]): + ab[[column_num, i]] = ab[[i, column_num]] + if ab[column_num, column_num] == 0.0: + raise ValueError("Matrix is not correct") + else: + pass + if column_num != 0: + for i in range(column_num, num_of_rows): + ab[i, :] -= ( + ab[i, column_num - 1] + / ab[column_num - 1, column_num - 1] + * ab[column_num - 1, :] + ) + + # Find x vector (Back Substitution) + for column_num in range(num_of_rows - 1, -1, -1): + x = ab[column_num, -1] / ab[column_num, column_num] + x_lst.insert(0, x) + for i in range(column_num - 1, -1, -1): + ab[i, -1] -= ab[i, column_num] * x + + # Return the solution vector + return np.asarray(x_lst) + + +if __name__ == "__main__": + from doctest import testmod + from pathlib import Path + + testmod() + file_path = Path(__file__).parent / "matrix.txt" + try: + matrix = np.loadtxt(file_path) + except FileNotFoundError: + print(f"Error: {file_path} not found. Using default matrix instead.") + + # Example usage: + print(f"Matrix:\n{matrix}") + print(f"{solve_linear_system(matrix) = }") diff --git a/linear_algebra/src/gaussian_elimination_pivoting/matrix.txt b/linear_algebra/src/gaussian_elimination_pivoting/matrix.txt new file mode 100644 index 000000000..dd895ad85 --- /dev/null +++ b/linear_algebra/src/gaussian_elimination_pivoting/matrix.txt @@ -0,0 +1,4 @@ +5.0 -5.0 -3.0 4.0 -11.0 +1.0 -4.0 6.0 -4.0 -10.0 +-2.0 -5.0 4.0 -5.0 -12.0 +-3.0 -3.0 5.0 -5.0 8.0 \ No newline at end of file