jax.numpy.linalg.qr#

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

Compute the QR decomposition of an array

JAX implementation of numpy.linalg.qr().

The QR decomposition of a matrix A is given by

\[A = QR\]

Where Q is a unitary matrix (i.e. \(Q^HQ=I\)) and R is an upper-triangular matrix.

Parameters:

  • a (ArrayLike) – array of shape (…, M, N)

  • mode (str) –

    Computational mode. Supported values are:

    • "reduced" (default): return Q of shape (..., M, K) and R of shape (..., K, N), where K = min(M, N).

    • "complete": return Q of shape (..., M, M) and R of shape (..., M, N).

    • "raw": return lapack-internal representations of shape (..., M, N) and (..., K).

    • "r": return R only.

Returns:

A tuple (Q, R) (if mode is not "r") otherwise an array R, where:

  • Q is an orthogonal matrix of shape (..., M, K) (if mode is "reduced") or (..., M, M) (if mode is "complete").

  • R is an upper-triangular matrix of shape (..., M, N) (if mode is "r" or "complete") or (..., K, N) (if mode is "reduced")

with K = min(M, N).

Return type:

Array | QRResult

See also

Examples

Compute the QR decomposition of a matrix:

>>> a = jnp.array([[1., 2., 3., 4.],
...                [5., 4., 2., 1.],
...                [6., 3., 1., 5.]])
>>> Q, R = jnp.linalg.qr(a)
>>> Q  
Array([[-0.12700021, -0.7581426 , -0.6396022 ],
       [-0.63500065, -0.43322435,  0.63960224],
       [-0.7620008 ,  0.48737738, -0.42640156]], dtype=float32)
>>> R  
Array([[-7.8740077, -5.080005 , -2.4130025, -4.953006 ],
       [ 0.       , -1.7870499, -2.6534991, -1.028908 ],
       [ 0.       ,  0.       , -1.0660033, -4.050814 ]], dtype=float32)

Check that Q is orthonormal:

>>> jnp.allclose(Q.T @ Q, jnp.eye(3), atol=1E-5)
Array(True, dtype=bool)

Reconstruct the input:

>>> jnp.allclose(Q @ R, a)
Array(True, dtype=bool)