dpnp.linalg.svd

dpnp.linalg.svd(a, full_matrices=True, compute_uv=True, hermitian=False)[source]

Singular Value Decomposition.

For full documentation refer to numpy.linalg.svd.

Parameters:

  • a ((..., M, N) {dpnp.ndarray, usm_ndarray}) -- Input array with a.ndim >= 2.

  • full_matrices ({bool}, optional) -- If True, it returns u and Vh with full-sized matrices. If False, the matrices are reduced in size. Default: True.

  • compute_uv ({bool}, optional) -- If False, it only returns singular values. Default: True.

  • hermitian ({bool}, optional) -- If True, a is assumed to be Hermitian (symmetric if real-valued), enabling a more efficient method for finding singular values. Default: False.

Returns:

  • u ({ (…, M, M), (…, M, K) } dpnp.ndarray) -- Unitary matrix, where M is the number of rows of the input array a. The shape of the matrix u depends on the value of full_matrices. If full_matrices is True, u has the shape (…, M, M). If full_matrices is False, u has the shape (…, M, K), where K = min(M, N), and N is the number of columns of the input array a. If compute_uv is False, neither u or Vh are computed.

  • s ((…, K) dpnp.ndarray) -- Vector containing the singular values of a, sorted in descending order. The length of s is min(M, N).

  • Vh ({ (…, N, N), (…, K, N) } dpnp.ndarray) -- Unitary matrix, where N is the number of columns of the input array a. The shape of the matrix Vh depends on the value of full_matrices. If full_matrices is True, Vh has the shape (…, N, N). If full_matrices is False, Vh has the shape (…, K, N). If compute_uv is False, neither u or Vh are computed.

Examples

>>> import dpnp as np
>>> a = np.random.randn(9, 6) + 1j*np.random.randn(9, 6)
>>> b = np.random.randn(2, 7, 8, 3) + 1j*np.random.randn(2, 7, 8, 3)

Reconstruction based on full SVD, 2D case:

>>> u, s, vh = np.linalg.svd(a, full_matrices=True)
>>> u.shape, s.shape, vh.shape
((9, 9), (6,), (6, 6))
>>> np.allclose(a, np.dot(u[:, :6] * s, vh))
array([ True])
>>> smat = np.zeros((9, 6), dtype=complex)
>>> smat[:6, :6] = np.diag(s)
>>> np.allclose(a, np.dot(u, np.dot(smat, vh)))
array([ True])

Reconstruction based on reduced SVD, 2D case:

>>> u, s, vh = np.linalg.svd(a, full_matrices=False)
>>> u.shape, s.shape, vh.shape
((9, 6), (6,), (6, 6))
>>> np.allclose(a, np.dot(u * s, vh))
array([ True])
>>> smat = np.diag(s)
>>> np.allclose(a, np.dot(u, np.dot(smat, vh)))
array([ True])

Reconstruction based on full SVD, 4D case:

>>> u, s, vh = np.linalg.svd(b, full_matrices=True)
>>> u.shape, s.shape, vh.shape
((2, 7, 8, 8), (2, 7, 3), (2, 7, 3, 3))
>>> np.allclose(b, np.matmul(u[..., :3] * s[..., None, :], vh))
array([ True])
>>> np.allclose(b, np.matmul(u[..., :3], s[..., None] * vh))
array([ True])

Reconstruction based on reduced SVD, 4D case:

>>> u, s, vh = np.linalg.svd(b, full_matrices=False)
>>> u.shape, s.shape, vh.shape
((2, 7, 8, 3), (2, 7, 3), (2, 7, 3, 3))
>>> np.allclose(b, np.matmul(u * s[..., None, :], vh))
array([ True])
>>> np.allclose(b, np.matmul(u, s[..., None] * vh))
array([ True])