A student is given a small system of linear equations representing the balance of flows in a simple network.  

The system is expressed as:

$$
A \vec{x} = \vec{b}
$$

Where:

- $$A$$ is a $$3×3$$ matrix;  
- $$\vec{b}$$ is the vector of known quantities.  

The student's goal is to solve for $$\vec{x}$$ by performing LU decomposition on matrix $$A$$, then using forward and backward substitution to find the solution.  

Compare your solution with `numpy`'s built-in solver to verify correctness.  

```python


import unittest
import numpy as np
import user_code  # ÐºÐ¾Ð´ ÑÑÑÐ´ÐµÐ½ÑÐ°

def _dynamic(tc, ok, ok_msg, fail_msg):
    tc._testMethodName = ok_msg if ok else fail_msg
    (tc.assertTrue if ok else tc.fail)(tc._testMethodName)

ATOL = 1e-8
RTOL = 1e-7

class TestLUChallenge(unittest.TestCase):

    def test_variables_exist_and_shapes(self):
        reasons = []
        A = getattr(user_code, "A", None)
        b = getattr(user_code, "b", None)
        L = getattr(user_code, "L", None)
        U = getattr(user_code, "U", None)
        y = getattr(user_code, "y", None)
        x = getattr(user_code, "x", None)

        if not (isinstance(A, np.ndarray) and A.shape == (3,3)):
            reasons.append(f"A must be 3x3 ndarray; got {None if A is None else A.shape}")
        if not (isinstance(b, np.ndarray) and b.shape == (3,)):
            reasons.append(f"b must be shape (3,); got {None if b is None else b.shape}")
        if not (isinstance(L, np.ndarray) and L.shape == (3,3)):
            reasons.append(f"L must be 3x3 ndarray; got {None if L is None else L.shape}")
        if not (isinstance(U, np.ndarray) and U.shape == (3,3)):
            reasons.append(f"U must be 3x3 ndarray; got {None if U is None else U.shape}")
        if not (isinstance(y, np.ndarray) and y.shape == (3,)):
            reasons.append(f"y must be shape (3,); got {None if y is None else y.shape}")
        if not (isinstance(x, np.ndarray) and x.shape == (3,)):
            reasons.append(f"x must be shape (3,); got {None if x is None else x.shape}")

        _dynamic(self,
                 len(reasons) == 0,
                 "PASS: variables exist with correct shapes",
                 "FAIL: " + " | ".join(reasons) if reasons else "FAIL")

    def test_LU_structure_and_reconstruction(self):
        reasons = []
        A, L, U = user_code.A, user_code.L, user_code.U

        # L: ones on diagonal, zeros above
        if not np.allclose(np.diag(L), np.ones(3), atol=ATOL, rtol=RTOL):
            reasons.append("L diagonal must be all ones")
        if not np.allclose(np.triu(L, k=1), np.zeros_like(np.triu(L, k=1)), atol=ATOL, rtol=RTOL):
            reasons.append("L must be lower triangular (zeros above diagonal)")

        # U: zeros below diagonal
        if not np.allclose(np.tril(U, k=-1), np.zeros_like(np.tril(U, k=-1)), atol=ATOL, rtol=RTOL):
            reasons.append("U must be upper triangular (zeros below diagonal)")

        # Reconstruction
        if not np.allclose(L @ U, A, atol=1e-7, rtol=1e-6):
            reasons.append("L @ U does not reconstruct A within tolerance")

        _dynamic(self,
                 len(reasons) == 0,
                 "PASS: L/U structure and A â L@U",
                 "FAIL: " + " | ".join(reasons) if reasons else "FAIL")

    def test_forward_and_backward(self):
        reasons = []
        A, b, L, U, y, x = user_code.A, user_code.b, user_code.L, user_code.U, user_code.y, user_code.x

        # Forward: Ly â b
        if not np.allclose(L @ y, b, atol=1e-7, rtol=1e-6):
            reasons.append("Forward substitution incorrect: L @ y != b")

        # Backward: Ux â y
        if not np.allclose(U @ x, y, atol=1e-7, rtol=1e-6):
            reasons.append("Backward substitution incorrect: U @ x != y")

        _dynamic(self,
                 len(reasons) == 0,
                 "PASS: forward (Ly=b) & backward (Ux=y) checks",
                 "FAIL: " + " | ".join(reasons) if reasons else "FAIL")

    def test_solution_matches_numpy(self):
        reasons = []
        A, b, x = user_code.A, user_code.b, user_code.x
        x_np = np.linalg.solve(A, b)

        if not np.allclose(x, x_np, atol=1e-7, rtol=1e-6):
            reasons.append(f"x does not match np.linalg.solve: expected {x_np}, got {x}")

        _dynamic(self,
                 len(reasons) == 0,
                 "PASS: x matches NumPy solution",
                 "FAIL: " + " | ".join(reasons) if reasons else "FAIL")


test_code.py

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Challenge: Solving a Linear System with LU Decomposition

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Challenge: Solving a Linear System with LU Decomposition

Oplossing

Awesome!