diff options
Diffstat (limited to 'numpy/linalg')
| -rw-r--r-- | numpy/linalg/linalg.py | 42 |
1 files changed, 23 insertions, 19 deletions
diff --git a/numpy/linalg/linalg.py b/numpy/linalg/linalg.py index 0b7aa132e..ce319c17a 100644 --- a/numpy/linalg/linalg.py +++ b/numpy/linalg/linalg.py @@ -231,12 +231,13 @@ def solve(a, b): ---------- a : array_like, shape (M, M) Coefficient matrix. - b : array_like, shape (M,) + b : array_like, shape (M,) or (M, N) Ordinate or "dependent variable" values. Returns ------- - x : ndarray, shape (M,) + x : ndarray, shape (M,) or (M, N) depending on b + Solution to the system a x = b Raises ------ @@ -531,27 +532,28 @@ def qr(a, mode='full'): Returns ------- - mode = 'full' - q : double or complex array, shape (M, K) + * If mode = 'full': - r : double or complex array, shape (K, N) + * q : ndarray of float or complex, shape (M, K) + * r : ndarray of float or complex, shape (K, N) - Size K = min(M, N) + Size K = min(M, N) - mode = 'r' - r : double or complex array, shape (K, N) + * If mode = 'r': - mode = 'economic' - a2 : double or complex array, shape (M, N) + * r : ndarray of float or complex, shape (K, N) - The diagonal and the upper triangle of a2 contains r, - while the rest of the matrix is undefined. + * If mode = 'economic': - If `a` is a matrix, so are all the return values. + * a2 : ndarray of float or complex, shape (M, N) + + The diagonal and the upper triangle of a2 contains r, + while the rest of the matrix is undefined. Raises ------ - LinAlgError : if factoring fails + LinAlgError + If factoring fails. Notes ----- @@ -561,16 +563,18 @@ def qr(a, mode='full'): For more information on the qr factorization, see for example: http://en.wikipedia.org/wiki/QR_factorization + Subclasses of `ndarray` are preserved, so if `a` is of type `matrix`, + all the return values will be matrices too. Examples -------- >>> a = np.random.randn(9, 6) >>> q, r = np.linalg.qr(a) - >>> np.allclose(a, np.dot(q, r)) # a does equal qr + >>> np.allclose(a, np.dot(q, r)) # a does equal qr True >>> r2 = np.linalg.qr(a, mode='r') >>> r3 = np.linalg.qr(a, mode='economic') - >>> np.allclose(r, r2) # mode='r' returns the same r as mode='full' + >>> np.allclose(r, r2) # mode='r' returns the same r as mode='full' True >>> # But only triu parts are guaranteed equal when mode='economic' >>> np.allclose(r, np.triu(r3[:6,:6], k=0)) @@ -582,11 +586,11 @@ def qr(a, mode='full'): What are the least-squares-best `m` and `y0` in ``y = y0 + mx`` for the following data: {(0,1), (1,0), (1,2), (2,1)}. (Graph the points and you'll see that it should be y0 = 0, m = 1.) The answer is provided - by solving the over-determined matrix equation ``Ax = b``, where: + by solving the over-determined matrix equation ``Ax = b``, where:: A = array([[0, 1], [1, 1], [1, 1], [2, 1]]) - - x = array([[y0], [m]]) b = array([[1], [0], [2], [1]]) + x = array([[y0], [m]]) + b = array([[1], [0], [2], [1]]) If A = qr such that q is orthonormal (which is always possible via Gram-Schmidt), then ``x = inv(r) * (q.T) * b``. (In numpy practice, |