xorbits.numpy.linalg.svd#

xorbits.numpy.linalg.svd(a, method='tsqr')[source]#

Singular Value Decomposition.

When a is a 2D array, and full_matrices=False, then it is factorized as u @ np.diag(s) @ vh = (u * s) @ vh, where u and the Hermitian transpose of vh are 2D arrays with orthonormal columns and s is a 1D array of a’s singular values. When a is higher-dimensional, SVD is applied in stacked mode as explained below.

Parameters

  • a ((..., M, N) array_like) – A real or complex array with a.ndim >= 2.

  • full_matrices (bool, optional (Not supported yet)) – If True (default), u and vh have the shapes (..., M, M) and (..., N, N), respectively. Otherwise, the shapes are (..., M, K) and (..., K, N), respectively, where K = min(M, N).

  • compute_uv (bool, optional (Not supported yet)) – Whether or not to compute u and vh in addition to s. True by default.

  • hermitian (bool, optional (Not supported yet)) –

    If True, a is assumed to be Hermitian (symmetric if real-valued), enabling a more efficient method for finding singular values. Defaults to False.

    New in version 1.17.0(numpy.linalg).

Returns

  • When compute_uv is True, the result is a namedtuple with the following

  • attribute names

  • U ({ (…, M, M), (…, M, K) } array) – Unitary array(s). The first a.ndim - 2 dimensions have the same size as those of the input a. The size of the last two dimensions depends on the value of full_matrices. Only returned when compute_uv is True.

  • S ((…, K) array) – Vector(s) with the singular values, within each vector sorted in descending order. The first a.ndim - 2 dimensions have the same size as those of the input a.

  • Vh ({ (…, N, N), (…, K, N) } array) – Unitary array(s). The first a.ndim - 2 dimensions have the same size as those of the input a. The size of the last two dimensions depends on the value of full_matrices. Only returned when compute_uv is True.

Raises

LinAlgError – If SVD computation does not converge.

See also

scipy.linalg.svd

Similar function in SciPy.

scipy.linalg.svdvals

Compute singular values of a matrix.

Notes

Changed in version 1.8.0(numpy.linalg): Broadcasting rules apply, see the numpy.linalg documentation for details.

The decomposition is performed using LAPACK routine _gesdd.

SVD is usually described for the factorization of a 2D matrix \(A\). The higher-dimensional case will be discussed below. In the 2D case, SVD is written as \(A = U S V^H\), where \(A = a\), \(U= u\), \(S= \mathtt{np.diag}(s)\) and \(V^H = vh\). The 1D array s contains the singular values of a and u and vh are unitary. The rows of vh are the eigenvectors of \(A^H A\) and the columns of u are the eigenvectors of \(A A^H\). In both cases the corresponding (possibly non-zero) eigenvalues are given by s**2.

If a has more than two dimensions, then broadcasting rules apply, as explained in Linear algebra on several matrices at once. This means that SVD is working in “stacked” mode: it iterates over all indices of the first a.ndim - 2 dimensions and for each combination SVD is applied to the last two indices. The matrix a can be reconstructed from the decomposition with either (u * s[..., None, :]) @ vh or u @ (s[..., None] * vh). (The @ operator can be replaced by the function np.matmul for python versions below 3.5.)

If a is a matrix object (as opposed to an ndarray), then so are all the return values.

Examples

>>> 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))  
True
>>> smat = np.zeros((9, 6), dtype=complex)  
>>> smat[:6, :6] = np.diag(S)  
>>> np.allclose(a, np.dot(U, np.dot(smat, Vh)))  
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))  
True
>>> smat = np.diag(S)  
>>> np.allclose(a, np.dot(U, np.dot(smat, Vh)))  
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))  
True
>>> np.allclose(b, np.matmul(U[..., :3], S[..., None] * Vh))  
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))  
True
>>> np.allclose(b, np.matmul(U, S[..., None] * Vh))  
True
method: {‘tsqr’}, optional

method to calculate qr factorization, tsqr as default TSQR is presented in:

This docstring was copied from numpy.linalg.