Create a Simultaneous Equation Solver Algorithm (#8773)

* Added simultaneous_linear_equation_solver.py

* Removed Augment class, replaced with recursive functions

* [pre-commit.ci] auto fixes from pre-commit.com hooks

for more information, see https://pre-commit.ci

* fixed edge cases

* Update settings.json

---------

Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com>
Co-authored-by: Christian Clauss <cclauss@me.com>

.vscode/settings.json

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+{
+    "githubPullRequests.ignoredPullRequestBranches": [
+        "master"
+    ]
+}

maths/simultaneous_linear_equation_solver.py

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+"""
+https://en.wikipedia.org/wiki/Augmented_matrix
+
+This algorithm solves simultaneous linear equations of the form
+λa + λb + λc + λd + ... = γ as [λ, λ, λ, λ, ..., γ]
+Where λ & γ are individual coefficients, the no. of equations = no. of coefficients - 1
+
+Note in order to work there must exist 1 equation where all instances of λ and γ != 0
+"""
+
+
+def simplify(current_set: list[list]) -> list[list]:
+    """
+    >>> simplify([[1, 2, 3], [4, 5, 6]])
+    [[1.0, 2.0, 3.0], [0.0, 0.75, 1.5]]
+    >>> simplify([[5, 2, 5], [5, 1, 10]])
+    [[1.0, 0.4, 1.0], [0.0, 0.2, -1.0]]
+    """
+    # Divide each row by magnitude of first term --> creates 'unit' matrix
+    duplicate_set = current_set.copy()
+    for row_index, row in enumerate(duplicate_set):
+        magnitude = row[0]
+        for column_index, column in enumerate(row):
+            if magnitude == 0:
+                current_set[row_index][column_index] = column
+                continue
+            current_set[row_index][column_index] = column / magnitude
+    # Subtract to cancel term
+    first_row = current_set[0]
+    final_set = [first_row]
+    current_set = current_set[1::]
+    for row in current_set:
+        temp_row = []
+        # If first term is 0, it is already in form we want, so we preserve it
+        if row[0] == 0:
+            final_set.append(row)
+            continue
+        for column_index in range(len(row)):
+            temp_row.append(first_row[column_index] - row[column_index])
+        final_set.append(temp_row)
+    # Create next recursion iteration set
+    if len(final_set[0]) != 3:
+        current_first_row = final_set[0]
+        current_first_column = []
+        next_iteration = []
+        for row in final_set[1::]:
+            current_first_column.append(row[0])
+            next_iteration.append(row[1::])
+        resultant = simplify(next_iteration)
+        for i in range(len(resultant)):
+            resultant[i].insert(0, current_first_column[i])
+        resultant.insert(0, current_first_row)
+        final_set = resultant
+    return final_set
+
+
+def solve_simultaneous(equations: list[list]) -> list:
+    """
+    >>> solve_simultaneous([[1, 2, 3],[4, 5, 6]])
+    [-1.0, 2.0]
+    >>> solve_simultaneous([[0, -3, 1, 7],[3, 2, -1, 11],[5, 1, -2, 12]])
+    [6.4, 1.2, 10.6]
+    >>> solve_simultaneous([])
+    Traceback (most recent call last):
+        ...
+    IndexError: solve_simultaneous() requires n lists of length n+1
+    >>> solve_simultaneous([[1, 2, 3],[1, 2]])
+    Traceback (most recent call last):
+        ...
+    IndexError: solve_simultaneous() requires n lists of length n+1
+    >>> solve_simultaneous([[1, 2, 3],["a", 7, 8]])
+    Traceback (most recent call last):
+        ...
+    ValueError: solve_simultaneous() requires lists of integers
+    >>> solve_simultaneous([[0, 2, 3],[4, 0, 6]])
+    Traceback (most recent call last):
+        ...
+    ValueError: solve_simultaneous() requires at least 1 full equation
+    """
+    if len(equations) == 0:
+        raise IndexError("solve_simultaneous() requires n lists of length n+1")
+    _length = len(equations) + 1
+    if any(len(item) != _length for item in equations):
+        raise IndexError("solve_simultaneous() requires n lists of length n+1")
+    for row in equations:
+        if any(not isinstance(column, (int, float)) for column in row):
+            raise ValueError("solve_simultaneous() requires lists of integers")
+    if len(equations) == 1:
+        return [equations[0][-1] / equations[0][0]]
+    data_set = equations.copy()
+    if any(0 in row for row in data_set):
+        temp_data = data_set.copy()
+        full_row = []
+        for row_index, row in enumerate(temp_data):
+            if 0 not in row:
+                full_row = data_set.pop(row_index)
+                break
+        if not full_row:
+            raise ValueError("solve_simultaneous() requires at least 1 full equation")
+        data_set.insert(0, full_row)
+    useable_form = data_set.copy()
+    simplified = simplify(useable_form)
+    simplified = simplified[::-1]
+    solutions: list = []
+    for row in simplified:
+        current_solution = row[-1]
+        if not solutions:
+            if row[-2] == 0:
+                solutions.append(0)
+                continue
+            solutions.append(current_solution / row[-2])
+            continue
+        temp_row = row.copy()[: len(row) - 1 :]
+        while temp_row[0] == 0:
+            temp_row.pop(0)
+        if len(temp_row) == 0:
+            solutions.append(0)
+            continue
+        temp_row = temp_row[1::]
+        temp_row = temp_row[::-1]
+        for column_index, column in enumerate(temp_row):
+            current_solution -= column * solutions[column_index]
+        solutions.append(current_solution)
+    final = []
+    for item in solutions:
+        final.append(float(round(item, 5)))
+    return final[::-1]
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()
+    eq = [
+        [2, 1, 1, 1, 1, 4],
+        [1, 2, 1, 1, 1, 5],
+        [1, 1, 2, 1, 1, 6],
+        [1, 1, 1, 2, 1, 7],
+        [1, 1, 1, 1, 2, 8],
+    ]
+    print(solve_simultaneous(eq))
+    print(solve_simultaneous([[4, 2]]))