nlcpy.linalg.svd
- nlcpy.linalg.svd(a, full_matrices=True, compute_uv=True, hermitian=False)[ソース]
Singular Value Decomposition.
When a is a 2D array, it is factorized as
u @ nlcpy.diag(s) @ vh = (u * s) @ vh
, where u and vh are 2D unitary arrays 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_matricesbool, optional
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, whereK = min(M, N)
.- compute_uvbool, optional
Whether or not to compute u and vh in addition to s. True by default.
- hermitianbool, optional
If True, a is assumed to be Hermitian (symmetric if real-valued), enabling a more efficient method for finding singular values. Defaults to False.
- Returns
- u{(..., M, M), (..., M, K)} ndarray
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) ndarray
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)} ndarray
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.
注釈
The decomposition is performed using LAPACK routine
_gesdd
.SVD is usually described for the factorization of a 2D matrix . The higher-dimensional case will be discussed below. In the 2D case, SVD is written as , where , , and . The 1D array s contains the singular values of a and u and vh are unitary. The rows of vh are the eigenvectors of and the columns of u are the eigenvectors of . 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.Examples
>>> import nlcpy as vp >>> from nlcpy import testing >>> a = vp.random.randn(9, 6) + 1j*vp.random.randn(9, 6)
Reconstruction based on full SVD, 2D case:
>>> u, s, vh = vp.linalg.svd(a, full_matrices=True) >>> u.shape, s.shape, vh.shape ((9, 9), (6,), (6, 6)) >>> vp.testing.assert_allclose(a, vp.dot(u[:, :6] * s, vh)) >>> smat = vp.zeros((9, 6), dtype=complex) >>> smat[:6, :6] = vp.diag(s) >>> vp.testing.assert_allclose(a, vp.dot(u, vp.dot(smat, vh)))
Reconstruction based on reduced SVD, 2D case:
>>> u, s, vh = vp.linalg.svd(a, full_matrices=False) >>> u.shape, s.shape, vh.shape ((9, 6), (6,), (6, 6)) >>> vp.testing.assert_allclose(a, vp.dot(u * s, vh)) >>> smat = vp.diag(s) >>> vp.testing.assert_allclose(a, vp.dot(u, vp.dot(smat, vh)))