def lstsq(A, b):
"""
Solves the linear system of equations Ax = b by
computing a vector x that minimizes the Euclidean
norm ||b - Ax||^2.
Concretely, uses the QR decomposition of A to deal
with an over or well determined system.
Params
------
- A: a numpy array of shape (M, N).
- b: a numpy array of shape (M,).
Returns:
- x: a numpy array of shape (N, ).
"""
M, N = A.shape
# if well-determined, use PLU
if (M == N):
solver = LU(A, pivoting='partial')
else:
solver = QR(A)
# solve for x
x = solver.solve(b)
return x
def test_solve_multi_reduce(self):
A = np.random.randn(100, 60)
b = np.random.randn(100, 10)
actual = QR(A, reduce=True).solve(b)
expected = LA.lstsq(A, b)[0]
self.assertTrue(np.allclose(actual, expected))
def test_solve_single_complete(self):
A = np.random.randn(100, 60)
b = np.random.randn(100)
actual = QR(A, reduce=False).solve(b)
expected = LA.lstsq(A, b)[0]
self.assertTrue(np.allclose(actual, expected))
def test_householder_reduce(self):
T = np.random.randn(100, 60)
actual = QR(T, reduce=True).householder()
expected = LA.qr(T, mode='reduced')
self.assertTrue(
all(np.allclose(a, e) for a, e in zip(actual, expected))
)
def test_gram_schmidt(self):
T = np.random.randn(100, 60)
actual = QR(T).gram_schmidt()
Q, R = LA.qr(T)
# enforce uniqueness for numpy version
D = create_diag(np.sign(diag(R)))
Q = np.dot(Q, D)
R = np.dot(D, R)
expected = (Q, R)
self.assertTrue(
all(np.allclose(a, e) for a, e in zip(actual, expected))
)