create qr_decomposition (#1363)

maths/qr_decomposition.py

@@ -0,0 +1,71 @@
+import numpy as np
+
+
+def qr_householder(A):
+    """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
+    method of decomposing A uses the Householder reflection, which is
+    numerically stable and of complexity O(n^3).
+
+    https://en.wikipedia.org/wiki/QR_decomposition#Using_Householder_reflections
+
+    Arguments:
+    A -- a numpy.ndarray of shape (m, n)
+
+    Note: several optimizations can be made for numeric efficiency, but this is
+    intended to demonstrate how it would be represented in a mathematics
+    textbook.  In cases where efficiency is particularly important, an optimized
+    version from BLAS should be used.
+
+    >>> A = np.array([[12, -51, 4], [6, 167, -68], [-4, 24, -41]], dtype=float)
+    >>> Q, R = qr_householder(A)
+
+    >>> # check that the decomposition is correct
+    >>> np.allclose(Q@R, A)
+    True
+
+    >>> # check that Q is orthogonal
+    >>> np.allclose(Q@Q.T, np.eye(A.shape[0]))
+    True
+    >>> np.allclose(Q.T@Q, np.eye(A.shape[0]))
+    True
+
+    >>> # check that R is upper triangular
+    >>> np.allclose(np.triu(R), R)
+    True
+    """
+    m, n = A.shape
+    t = min(m, n)
+    Q = np.eye(m)
+    R = A.copy()
+
+    for k in range(t - 1):
+        # select a column of modified matrix A':
+        x = R[k:, [k]]
+        # construct first basis vector
+        e1 = np.zeros_like(x)
+        e1[0] = 1.0
+        # determine scaling factor
+        alpha = np.linalg.norm(x)
+        # construct vector v for Householder reflection
+        v = x + np.sign(x[0])*alpha*e1
+        v /= np.linalg.norm(v)
+
+        # construct the Householder matrix
+        Q_k = np.eye(m - k) - 2.0*v@v.T
+        # pad with ones and zeros as necessary
+        Q_k = np.block([[np.eye(k),            np.zeros((k, m - k))],
+                        [np.zeros((m - k, k)), Q_k                 ]])
+
+        Q = Q@Q_k.T
+        R = Q_k@R
+
+    return Q, R
+
+
+if __name__ == "__main__":
+    import doctest
+    doctest.testmod()