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 uppertriangular.
Parameters: 


Returns: 

Raises: 

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 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' True >>> # But only triu parts are guaranteed equal when mode='economic' >>> np.allclose(r, np.triu(r3[:6,:6], k=0)) True
Example illustrating a common use of qr
: solving of least squares problems
What are the leastsquaresbest 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 overdetermined 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 GramSchmidt), 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([ 1.1e16, 1.0e+00])
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https://docs.scipy.org/doc/numpy1.17.0/reference/generated/numpy.linalg.qr.html