# 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, where `K = 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.

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)))
```