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scipy.linalg.

# qr

scipy.linalg.qr(a, overwrite_a=False, lwork=None, mode='full', pivoting=False, check_finite=True)

  Compute QR decomposition of a matrix.

  Calculate the decomposition `A = Q R` where Q is unitary/orthogonal and R upper triangular.

  The documentation is written assuming array arguments are of specified “core” shapes. However, array argument(s) of this function may have additional “batch” dimensions prepended to the core shape. In this case, the array is treated as a batch of lower-dimensional slices; see Batched Linear Operations for details.

  Parameters:
    a(M, N) array_like

      Matrix to be decomposed

    overwrite_a bool, optional

      Whether data in a is overwritten (may improve performance if overwrite_a is set to True by reusing the existing input data structure rather than creating a new one.)

    lwork int, optional

      Work array size, lwork >= a.shape[1]. If None or -1, an optimal size is computed.

    mode{‘full’, ‘r’, ‘economic’, ‘raw’}, optional

      Determines what information is to be returned: either both Q and R (‘full’, default), only R (‘r’) or both Q and R but computed in economy-size (‘economic’, see Notes). The final option ‘raw’ (added in SciPy 0.11) makes the function return two matrices (Q, TAU) in the internal format used by LAPACK.

    pivoting bool, optional

      Whether or not factorization should include pivoting for rank-revealing qr decomposition. If pivoting, compute the decomposition `A[:, P] = Q @ R` as above, but where P is chosen such that the diagonal of R is non-increasing. Equivalently, albeit less efficiently, an explicit P matrix may be formed explicitly by permuting the rows or columns (depending on the side of the equation on which it is to be used) of an identity matrix. See Examples.

    check_finite bool, optional

      Whether to check that the input matrix contains only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs.

  Returns:
    Q float or complex ndarray

      Of shape (M, M), or (M, K) for `mode='economic'`. Not returned if `mode='r'`. Replaced by tuple `(Q, TAU)` if `mode='raw'`.

    R float or complex ndarray

      Of shape (M, N), or (K, N) for `mode in ['economic', 'raw']`. `K = min(M, N)`.

    P int ndarray

      Of shape (N,) for `pivoting=True`. Not returned if `pivoting=False`.

  Raises:
    LinAlgError

      Raised if decomposition fails

  Notes

  This is an interface to the LAPACK routines dgeqrf, zgeqrf, dorgqr, zungqr, dgeqp3, and zgeqp3.

  If `mode=economic`, the shapes of Q and R are (M, K) and (K, N) instead of (M,M) and (M,N), with `K=min(M,N)`.

  Examples

      >>> import numpy as np
      >>> from scipy import linalg
      >>> rng = np.random.default_rng()
      >>> a = rng.standard_normal((9, 6))

      >>> q, r = linalg.qr(a)
      >>> np.allclose(a, np.dot(q, r))
      True
      >>> q.shape, r.shape
      ((9, 9), (9, 6))

      >>> r2 = linalg.qr(a, mode='r')
      >>> np.allclose(r, r2)
      True

      >>> q3, r3 = linalg.qr(a, mode='economic')
      >>> q3.shape, r3.shape
      ((9, 6), (6, 6))

      >>> q4, r4, p4 = linalg.qr(a, pivoting=True)
      >>> d = np.abs(np.diag(r4))
      >>> np.all(d[1:] <= d[:-1])
      True
      >>> np.allclose(a[:, p4], np.dot(q4, r4))
      True
      >>> P = np.eye(p4.size)[p4]
      >>> np.allclose(a, np.dot(q4, r4) @ P)
      True
      >>> np.allclose(a @ P.T, np.dot(q4, r4))
      True
      >>> q4.shape, r4.shape, p4.shape
      ((9, 9), (9, 6), (6,))

      >>> q5, r5, p5 = linalg.qr(a, mode='economic', pivoting=True)
      >>> q5.shape, r5.shape, p5.shape
      ((9, 6), (6, 6), (6,))
      >>> P = np.eye(6)[:, p5]
      >>> np.allclose(a @ P, np.dot(q5, r5))
      True