implementation of Gaussian Elimination pivoting as a numerical linear algebra algorithm

linear_algebra/src/gaussianeliminationpivoting.py

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+import numpy as np
+from typing import List
+
+def custom_pivoting(a: np.ndarray, n: int, i: int) -> int:
+    """
+    Selects the index of the minimum absolute value in the i-th column of a matrix.
+
+    Parameters:
+    - a (np.ndarray): The input matrix.
+    - n (int): The size of the matrix.
+    - i (int): The column index.
+
+    Returns:
+    - int: The index of the minimum absolute value in the i-th column.
+
+    Example:
+    >>> a_matrix = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]], dtype=float)
+    >>> custom_pivoting(a_matrix, 3, 1)
+    0
+    """
+    min_index = i
+    for index in range(i + 1, n):
+        if abs(a[index][i]) < abs(a[min_index][i]):
+            min_index = index
+    return min_index
+
+
+def custom_gauss_elimination_pivoting(a: List[List[float]], b: List[float], n: int) -> List[float]:
+    """
+    Solves a system of linear equations using Gaussian elimination with partial pivoting.
+
+    Parameters:
+    - a (List[List[float]]): The coefficient matrix.
+    - b (List[float]): The constant vector.
+    - n (int): The size of the system.
+
+    Returns:
+    - List[float]: The solution vector.
+
+    Example:
+    >>> a_matrix = [[2, 3, 4], [1, -2, 3], [3, 4, 5]]
+    >>> b_vector = [20, 9, 11]
+    >>> custom_gauss_elimination_pivoting(a_matrix, b_vector, 3)
+    [1.0, 2.0, 3.0]
+    """
+    result = []
+    for i in range(n - 1):
+        new_index = custom_pivoting(a, n, i)
+        a[i], a[new_index] = a[new_index], a[i]
+        b[i], b[new_index] = b[new_index], b[i]
+        pivot = a[i][i]
+        for j in range(i + 1, n):
+            m = -1 * a[j][i] / pivot
+            for k in range(0, n):
+                a[j][k] += m * a[i][k]
+            b[j] += m * b[i]
+
+    for p in range(n - 1, -1, -1):
+        result.append(b[p] / a[p][p])
+        for q in range(p - 1, -1, -1):
+            b[q] = b[q] - result[n - p - 1] * a[q][p]
+    return result
+
+
+# Example usage:
+# n_size = 3
+# a_matrix = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]], dtype=float)
+# b_vector = np.array([10, 11, 12], dtype=float)
+
+# solution = custom_gauss_elimination_pivoting(a_matrix, b_vector, n_size)
+# print("Solution:", solution)
+
+
+#URL that points to Wikipedia or another similar explanation.
+#>>>>>>URL:https://courses.engr.illinois.edu/cs357/su2013/lectures/lecture07.pdf<<<<<#