/NumPy 1.17


numpy.linalg.qr(a, mode='reduced') [source]

Compute the qr factorization of a matrix.

Factor the matrix a as qr, where q is orthonormal and r is upper-triangular.

a : array_like, shape (M, N)

Matrix to be factored.

mode : {‘reduced’, ‘complete’, ‘r’, ‘raw’, ‘full’, ‘economic’}, optional

If K = min(M, N), then

  • ‘reduced’ : returns q, r with dimensions (M, K), (K, N) (default)
  • ‘complete’ : returns q, r with dimensions (M, M), (M, N)
  • ‘r’ : returns r only with dimensions (K, N)
  • ‘raw’ : returns h, tau with dimensions (N, M), (K,)
  • ‘full’ : alias of ‘reduced’, deprecated
  • ‘economic’ : returns h from ‘raw’, deprecated.

The options ‘reduced’, ‘complete, and ‘raw’ are new in numpy 1.8, see the notes for more information. The default is ‘reduced’, and to maintain backward compatibility with earlier versions of numpy both it and the old default ‘full’ can be omitted. Note that array h returned in ‘raw’ mode is transposed for calling Fortran. The ‘economic’ mode is deprecated. The modes ‘full’ and ‘economic’ may be passed using only the first letter for backwards compatibility, but all others must be spelled out. See the Notes for more explanation.

q : ndarray of float or complex, optional

A matrix with orthonormal columns. When mode = ‘complete’ the result is an orthogonal/unitary matrix depending on whether or not a is real/complex. The determinant may be either +/- 1 in that case.

r : ndarray of float or complex, optional

The upper-triangular matrix.

(h, tau) : ndarrays of np.double or np.cdouble, optional

The array h contains the Householder reflectors that generate q along with r. The tau array contains scaling factors for the reflectors. In the deprecated ‘economic’ mode only h is returned.


If factoring fails.


This is an interface to the LAPACK routines dgeqrf, zgeqrf, dorgqr, and zungqr.

For more information on the qr factorization, see for example: https://en.wikipedia.org/wiki/QR_factorization

Subclasses of ndarray are preserved except for the ‘raw’ mode. So if a is of type matrix, all the return values will be matrices too.

New ‘reduced’, ‘complete’, and ‘raw’ options for mode were added in NumPy 1.8.0 and the old option ‘full’ was made an alias of ‘reduced’. In addition the options ‘full’ and ‘economic’ were deprecated. Because ‘full’ was the previous default and ‘reduced’ is the new default, backward compatibility can be maintained by letting mode default. The ‘raw’ option was added so that LAPACK routines that can multiply arrays by q using the Householder reflectors can be used. Note that in this case the returned arrays are of type np.double or np.cdouble and the h array is transposed to be FORTRAN compatible. No routines using the ‘raw’ return are currently exposed by numpy, but some are available in lapack_lite and just await the necessary work.


>>> a = np.random.randn(9, 6)
>>> q, r = np.linalg.qr(a)
>>> np.allclose(a, np.dot(q, r))  # a does equal qr
>>> 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'
>>> # But only triu parts are guaranteed equal when mode='economic'
>>> np.allclose(r, np.triu(r3[:6,:6], k=0))

Example illustrating a common use of qr: solving of least squares problems

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:

A = array([[0, 1], [1, 1], [1, 1], [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, however, we simply use lstsq.)

>>> A = np.array([[0, 1], [1, 1], [1, 1], [2, 1]])
>>> A
array([[0, 1],
       [1, 1],
       [1, 1],
       [2, 1]])
>>> b = np.array([1, 0, 2, 1])
>>> q, r = np.linalg.qr(A)
>>> p = np.dot(q.T, b)
>>> np.dot(np.linalg.inv(r), p)
array([  1.1e-16,   1.0e+00])

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