Add manual determinant calculator using cofactor expansion

maths/optimised_sieve_of_eratosthenes.py

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+# Optimized Sieve of Eratosthenes: An efficient algorithm to compute all prime numbers
+# up to limit. This version skips even numbers after 2, improving both memory and time
+# usage. It is particularly efficient for larger limits (e.g., up to 10**8 on typical
+# hardware).
+# Wikipedia URL - https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes
+
+from math import isqrt
+
+
+def optimized_sieve(limit: int) -> list[int]:
+    """
+    Compute all prime numbers up to and including `limit` using an optimized
+    Sieve of Eratosthenes.
+
+    This implementation skips even numbers after 2 to reduce memory and
+    runtime by about 50%.
+
+    Parameters
+    ----------
+    limit : int
+        Upper bound (inclusive) of the range in which to find prime numbers.
+        Expected to be a non-negative integer. If limit < 2 the function
+        returns an empty list.
+
+    Returns
+    -------
+    list[int]
+        A list of primes in ascending order that are <= limit.
+
+    Examples
+    --------
+    >>> optimized_sieve(10)
+    [2, 3, 5, 7]
+    >>> optimized_sieve(1)
+    []
+    >>> optimized_sieve(2)
+    [2]
+    >>> optimized_sieve(30)
+    [2, 3, 5, 7, 11, 13, 17, 19, 23, 29]
+    """
+    if limit < 2:
+        return []
+
+    # Handle 2 separately, then consider only odd numbers
+    primes = [2] if limit >= 2 else []
+
+    # Only odd numbers from 3 to limit
+    size = (limit - 1) // 2
+    is_prime = [True] * (size + 1)
+    bound = isqrt(limit)
+
+    for i in range((bound - 1) // 2 + 1):
+        if is_prime[i]:
+            p = 2 * i + 3
+            # Start marking from p^2, converted to index
+            start = (p * p - 3) // 2
+            for j in range(start, size + 1, p):
+                is_prime[j] = False
+
+    primes.extend(2 * i + 3 for i in range(size + 1) if is_prime[i])
+    return primes
+
+
+if __name__ == "__main__":
+    print(optimized_sieve(50))

matrix/determinant_calculator.py

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+# Wikipedia URL- https://en.wikipedia.org/wiki/Determinant
+
+def get_minor(matrix: list[list[int] | list[float]], row: int, col: int) -> list[list[int] | list[float]]:
+    """
+    Returns the minor matrix obtained by removing the specified row and column.
+
+    Parameters
+    ----------
+    matrix : list[list[int] | list[float]]
+        The original square matrix.
+    row : int
+        The row to remove.
+    col : int
+        The column to remove.
+
+    Returns
+    -------
+    list[list[int] | list[float]]
+        Minor matrix.
+    """
+    return [r[:col] + r[col+1:] for i, r in enumerate(matrix) if i != row]
+
+
+def determinant_manual(matrix: list[list[int] | list[float]]) -> int | float:
+    """
+    Calculates the determinant of a square matrix using cofactor expansion.
+
+    Parameters
+    ----------
+    matrix : list[list[int] | list[float]]
+        A square matrix.
+
+    Returns
+    -------
+    int | float
+        Determinant of the matrix.
+
+    Examples
+    --------
+    >>> determinant_manual([[1,2],[3,4]])
+    -2
+    >>> determinant_manual([[2]])
+    2
+    >>> determinant_manual([[1,2,3],[4,5,6],[7,8,9]])
+    0
+    """
+    n = len(matrix)
+
+    if n == 1:
+        return matrix[0][0]
+
+    if n == 2:
+        return matrix[0][0]*matrix[1][1] - matrix[0][1]*matrix[1][0]
+
+    det = 0
+    for j in range(n):
+        minor = get_minor(matrix, 0, j)
+        cofactor = (-1) ** j * determinant_manual(minor)
+        det += matrix[0][j] * cofactor
+    return det
+
+
+if __name__ == "__main__":
+    # Simple demo
+    matrix_demo = [
+        [1, 2, 3],
+        [4, 5, 6],
+        [7, 8, 9]
+    ]
+    print(f"The determinant is: {determinant_manual(matrix_demo)}")