added Schur complement to linear algebra (TheAlgorithms#4793)

* added schur complement and tests to linear algebra

* updated according to checklist

* updated variable names and typing

* added two testcases for input validation

* fixed import order

Co-authored-by: Ivan Radonov <ivan.radonov@ad.mentormate.bg>

Author: iradonov

linear_algebra/src/schur_complement.py

@@ -0,0 +1,94 @@
+import unittest
+
+import numpy as np
+
+
+def schur_complement(
+    mat_a: np.ndarray,
+    mat_b: np.ndarray,
+    mat_c: np.ndarray,
+    pseudo_inv: np.ndarray = None,
+) -> np.ndarray:
+    """
+    Schur complement of a symmetric matrix X given as a 2x2 block matrix
+    consisting of matrices A, B and C.
+    Matrix A must be quadratic and non-singular.
+    In case A is singular, a pseudo-inverse may be provided using
+    the pseudo_inv argument.
+
+    Link to Wiki: https://en.wikipedia.org/wiki/Schur_complement
+    See also Convex Optimization – Boyd and Vandenberghe, A.5.5
+    >>> import numpy as np
+    >>> a = np.array([[1, 2], [2, 1]])
+    >>> b = np.array([[0, 3], [3, 0]])
+    >>> c = np.array([[2, 1], [6, 3]])
+    >>> schur_complement(a, b, c)
+    array([[ 5., -5.],
+           [ 0.,  6.]])
+    """
+    shape_a = np.shape(mat_a)
+    shape_b = np.shape(mat_b)
+    shape_c = np.shape(mat_c)
+
+    if shape_a[0] != shape_b[0]:
+        raise ValueError(
+            f"Expected the same number of rows for A and B. \
+            Instead found A of size {shape_a} and B of size {shape_b}"
+        )
+
+    if shape_b[1] != shape_c[1]:
+        raise ValueError(
+            f"Expected the same number of columns for B and C. \
+            Instead found B of size {shape_b} and C of size {shape_c}"
+        )
+
+    a_inv = pseudo_inv
+    if a_inv is None:
+        try:
+            a_inv = np.linalg.inv(mat_a)
+        except np.linalg.LinAlgError:
+            raise ValueError(
+                "Input matrix A is not invertible. Cannot compute Schur complement."
+            )
+
+    return mat_c - mat_b.T @ a_inv @ mat_b
+
+
+class TestSchurComplement(unittest.TestCase):
+    def test_schur_complement(self) -> None:
+        a = np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]])
+        b = np.array([[0, 3], [3, 0], [2, 3]])
+        c = np.array([[2, 1], [6, 3]])
+
+        s = schur_complement(a, b, c)
+
+        input_matrix = np.block([[a, b], [b.T, c]])
+
+        det_x = np.linalg.det(input_matrix)
+        det_a = np.linalg.det(a)
+        det_s = np.linalg.det(s)
+
+        self.assertAlmostEqual(det_x, det_a * det_s)
+
+    def test_improper_a_b_dimensions(self) -> None:
+        a = np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]])
+        b = np.array([[0, 3], [3, 0], [2, 3]])
+        c = np.array([[2, 1], [6, 3]])
+
+        with self.assertRaises(ValueError):
+            schur_complement(a, b, c)
+
+    def test_improper_b_c_dimensions(self) -> None:
+        a = np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]])
+        b = np.array([[0, 3], [3, 0], [2, 3]])
+        c = np.array([[2, 1, 3], [6, 3, 5]])
+
+        with self.assertRaises(ValueError):
+            schur_complement(a, b, c)
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()
+    unittest.main()