Remove separate directory for gaussian_elimination_pivoting.py (#11445)

* updating DIRECTORY.md

* Remove separate directory for gaussian_elimination_pivoting.py

Delete the directory linear_algebra/src/gaussian_elimination_pivoting/
and move its algorithm file, gaussian_elimination_pivoting.py, into the
parent src/ directory. The gaussian_elimination_pivoting/ directory only
exists because gaussian_elimination_pivoting.py reads an example numpy
array from matrix.txt, but this input file and IO operation is entirely
unnecessary because gaussian_elimination_pivoting.py already has the
exact same array hard-coded into a variable.

* updating DIRECTORY.md

---------

Co-authored-by: tianyizheng02 <tianyizheng02@users.noreply.github.com>

linear_algebra/src/gaussian_elimination_pivoting.py

@@ -0,0 +1,94 @@
+import numpy as np
+
+
+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
+
+    testmod()
+
+    example_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,
+    )
+
+    print(f"Matrix:\n{example_matrix}")
+    print(f"{solve_linear_system(example_matrix) = }")