Add Jacobi Iteration Method (#5113)

* Added Jacobi Iteration Method

Added this method in arithmetic_analysis folder. This method is used to solve system of linear equations.

* Added comments

* Added reference link

* Update jacobi_iteration_method.py

* Changes for codespell test

* Update jacobi_iteration_method.py

* Update jacobi_iteration_method.py

* Update arithmetic_analysis/jacobi_iteration_method.py

Co-authored-by: Christian Clauss <[email protected]>

* updating DIRECTORY.md

* Update arithmetic_analysis/jacobi_iteration_method.py

Co-authored-by: Christian Clauss <[email protected]>

* Update arithmetic_analysis/jacobi_iteration_method.py

Co-authored-by: Christian Clauss <[email protected]>

* Update arithmetic_analysis/jacobi_iteration_method.py

Co-authored-by: Christian Clauss <[email protected]>

* Update arithmetic_analysis/jacobi_iteration_method.py

Co-authored-by: Christian Clauss <[email protected]>

* Update arithmetic_analysis/jacobi_iteration_method.py

Co-authored-by: Christian Clauss <[email protected]>

* Update arithmetic_analysis/jacobi_iteration_method.py

Co-authored-by: Christian Clauss <[email protected]>

* Update jacobi_iteration_method.py

* Update jacobi_iteration_method.py

* Update jacobi_iteration_method.py

* fix styles

Co-authored-by: Christian Clauss <[email protected]>
Co-authored-by: github-actions <${GITHUB_ACTOR}@users.noreply.github.com>
Co-authored-by: John Law <[email protected]>

Author: 4 people

Diff for: DIRECTORY.md

@@ -4,6 +4,7 @@
   * [Gaussian Elimination](https://github.com/TheAlgorithms/Python/blob/master/arithmetic_analysis/gaussian_elimination.py)
   * [In Static Equilibrium](https://github.com/TheAlgorithms/Python/blob/master/arithmetic_analysis/in_static_equilibrium.py)
   * [Intersection](https://github.com/TheAlgorithms/Python/blob/master/arithmetic_analysis/intersection.py)
+  * [Jacobi Iteration Method](https://github.com/TheAlgorithms/Python/blob/master/arithmetic_analysis/jacobi_iteration_method.py)
   * [Lu Decomposition](https://github.com/TheAlgorithms/Python/blob/master/arithmetic_analysis/lu_decomposition.py)
   * [Newton Forward Interpolation](https://github.com/TheAlgorithms/Python/blob/master/arithmetic_analysis/newton_forward_interpolation.py)
   * [Newton Method](https://github.com/TheAlgorithms/Python/blob/master/arithmetic_analysis/newton_method.py)

Diff for: arithmetic_analysis/jacobi_iteration_method.py

@@ -0,0 +1,163 @@
+"""
+Jacobi Iteration Method - https://en.wikipedia.org/wiki/Jacobi_method
+"""
+from __future__ import annotations
+
+import numpy as np
+
+
+# Method to find solution of system of linear equations
+def jacobi_iteration_method(
+    coefficient_matrix: np.ndarray,
+    constant_matrix: np.ndarray,
+    init_val: list,
+    iterations: int,
+) -> list[float]:
+    """
+    Jacobi Iteration Method:
+    An iterative algorithm to determine the solutions of strictly diagonally dominant
+    system of linear equations
+
+    4x1 +  x2 +  x3 =  2
+     x1 + 5x2 + 2x3 = -6
+     x1 + 2x2 + 4x3 = -4
+
+    x_init = [0.5, -0.5 , -0.5]
+
+    Examples:
+
+    >>> coefficient = np.array([[4, 1, 1], [1, 5, 2], [1, 2, 4]])
+    >>> constant = np.array([[2], [-6], [-4]])
+    >>> init_val = [0.5, -0.5, -0.5]
+    >>> iterations = 3
+    >>> jacobi_iteration_method(coefficient, constant, init_val, iterations)
+    [0.909375, -1.14375, -0.7484375]
+
+
+    >>> coefficient = np.array([[4, 1, 1], [1, 5, 2]])
+    >>> constant = np.array([[2], [-6], [-4]])
+    >>> init_val = [0.5, -0.5, -0.5]
+    >>> iterations = 3
+    >>> jacobi_iteration_method(coefficient, constant, init_val, iterations)
+    Traceback (most recent call last):
+    ...
+    ValueError: Coefficient matrix dimensions must be nxn but received 2x3
+
+    >>> coefficient = np.array([[4, 1, 1], [1, 5, 2], [1, 2, 4]])
+    >>> constant = np.array([[2], [-6]])
+    >>> init_val = [0.5, -0.5, -0.5]
+    >>> iterations = 3
+    >>> jacobi_iteration_method(coefficient, constant, init_val, iterations)
+    Traceback (most recent call last):
+    ...
+    ValueError: Coefficient and constant matrices dimensions must be nxn and nx1 but
+                received 3x3 and 2x1
+
+    >>> coefficient = np.array([[4, 1, 1], [1, 5, 2], [1, 2, 4]])
+    >>> constant = np.array([[2], [-6], [-4]])
+    >>> init_val = [0.5, -0.5]
+    >>> iterations = 3
+    >>> jacobi_iteration_method(coefficient, constant, init_val, iterations)
+    Traceback (most recent call last):
+    ...
+    ValueError: Number of initial values must be equal to number of rows in coefficient
+                matrix but received 2 and 3
+
+    >>> coefficient = np.array([[4, 1, 1], [1, 5, 2], [1, 2, 4]])
+    >>> constant = np.array([[2], [-6], [-4]])
+    >>> init_val = [0.5, -0.5, -0.5]
+    >>> iterations = 0
+    >>> jacobi_iteration_method(coefficient, constant, init_val, iterations)
+    Traceback (most recent call last):
+    ...
+    ValueError: Iterations must be at least 1
+    """
+
+    rows1, cols1 = coefficient_matrix.shape
+    rows2, cols2 = constant_matrix.shape
+
+    if rows1 != cols1:
+        raise ValueError(
+            f"Coefficient matrix dimensions must be nxn but received {rows1}x{cols1}"
+        )
+
+    if cols2 != 1:
+        raise ValueError(f"Constant matrix must be nx1 but received {rows2}x{cols2}")
+
+    if rows1 != rows2:
+        raise ValueError(
+            f"""Coefficient and constant matrices dimensions must be nxn and nx1 but
+            received {rows1}x{cols1} and {rows2}x{cols2}"""
+        )
+
+    if len(init_val) != rows1:
+        raise ValueError(
+            f"""Number of initial values must be equal to number of rows in coefficient
+            matrix but received {len(init_val)} and {rows1}"""
+        )
+
+    if iterations <= 0:
+        raise ValueError("Iterations must be at least 1")
+
+    table = np.concatenate((coefficient_matrix, constant_matrix), axis=1)
+
+    rows, cols = table.shape
+
+    strictly_diagonally_dominant(table)
+
+    # Iterates the whole matrix for given number of times
+    for i in range(iterations):
+        new_val = []
+        for row in range(rows):
+            temp = 0
+            for col in range(cols):
+                if col == row:
+                    denom = table[row][col]
+                elif col == cols - 1:
+                    val = table[row][col]
+                else:
+                    temp += (-1) * table[row][col] * init_val[col]
+            temp = (temp + val) / denom
+            new_val.append(temp)
+        init_val = new_val
+
+    return [float(i) for i in new_val]
+
+
+# Checks if the given matrix is strictly diagonally dominant
+def strictly_diagonally_dominant(table: np.ndarray) -> bool:
+    """
+    >>> table = np.array([[4, 1, 1, 2], [1, 5, 2, -6], [1, 2, 4, -4]])
+    >>> strictly_diagonally_dominant(table)
+    True
+
+    >>> table = np.array([[4, 1, 1, 2], [1, 5, 2, -6], [1, 2, 3, -4]])
+    >>> strictly_diagonally_dominant(table)
+    Traceback (most recent call last):
+    ...
+    ValueError: Coefficient matrix is not strictly diagonally dominant
+    """
+
+    rows, cols = table.shape
+
+    is_diagonally_dominant = True
+
+    for i in range(0, rows):
+        sum = 0
+        for j in range(0, cols - 1):
+            if i == j:
+                continue
+            else:
+                sum += table[i][j]
+
+        if table[i][i] <= sum:
+            raise ValueError("Coefficient matrix is not strictly diagonally dominant")
+
+    return is_diagonally_dominant
+
+
+# Test Cases
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()