Return a QR-decomposition of the matrix A using Householder reflection. The QR-decomposition decomposes the matrix A of shape (m, n) into an orthogonal matrix Q of shape (m, m) and an upper triangular matrix R of shape (m, n). Note that the matrix A does not have to be square. This
(a: np.ndarray)
| 2 | |
| 3 | |
| 4 | def qr_householder(a: np.ndarray): |
| 5 | """Return a QR-decomposition of the matrix A using Householder reflection. |
| 6 | |
| 7 | The QR-decomposition decomposes the matrix A of shape (m, n) into an |
| 8 | orthogonal matrix Q of shape (m, m) and an upper triangular matrix R of |
| 9 | shape (m, n). Note that the matrix A does not have to be square. This |
| 10 | method of decomposing A uses the Householder reflection, which is |
| 11 | numerically stable and of complexity O(n^3). |
| 12 | |
| 13 | https://en.wikipedia.org/wiki/QR_decomposition#Using_Householder_reflections |
| 14 | |
| 15 | Arguments: |
| 16 | A -- a numpy.ndarray of shape (m, n) |
| 17 | |
| 18 | Note: several optimizations can be made for numeric efficiency, but this is |
| 19 | intended to demonstrate how it would be represented in a mathematics |
| 20 | textbook. In cases where efficiency is particularly important, an optimized |
| 21 | version from BLAS should be used. |
| 22 | |
| 23 | >>> A = np.array([[12, -51, 4], [6, 167, -68], [-4, 24, -41]], dtype=float) |
| 24 | >>> Q, R = qr_householder(A) |
| 25 | |
| 26 | >>> # check that the decomposition is correct |
| 27 | >>> np.allclose(Q@R, A) |
| 28 | True |
| 29 | |
| 30 | >>> # check that Q is orthogonal |
| 31 | >>> np.allclose(Q@Q.T, np.eye(A.shape[0])) |
| 32 | True |
| 33 | >>> np.allclose(Q.T@Q, np.eye(A.shape[0])) |
| 34 | True |
| 35 | |
| 36 | >>> # check that R is upper triangular |
| 37 | >>> np.allclose(np.triu(R), R) |
| 38 | True |
| 39 | """ |
| 40 | m, n = a.shape |
| 41 | t = min(m, n) |
| 42 | q = np.eye(m) |
| 43 | r = a.copy() |
| 44 | |
| 45 | for k in range(t - 1): |
| 46 | # select a column of modified matrix A': |
| 47 | x = r[k:, [k]] |
| 48 | # construct first basis vector |
| 49 | e1 = np.zeros_like(x) |
| 50 | e1[0] = 1.0 |
| 51 | # determine scaling factor |
| 52 | alpha = np.linalg.norm(x) |
| 53 | # construct vector v for Householder reflection |
| 54 | v = x + np.sign(x[0]) * alpha * e1 |
| 55 | v /= np.linalg.norm(v) |
| 56 | |
| 57 | # construct the Householder matrix |
| 58 | q_k = np.eye(m - k) - 2.0 * v @ v.T |
| 59 | # pad with ones and zeros as necessary |
| 60 | q_k = np.block([[np.eye(k), np.zeros((k, m - k))], [np.zeros((m - k, k)), q_k]]) |
| 61 |