compute_svd

cherab.inversion.core.compute_svd(T, H, Q=None, use_gpu=False, dtype=None, sp: Spinner | DummySpinner | None = None) → tuple[Any, Any, Any] | tuple[Any, Any, Any, Any]Source

Compute singular value decomposition (SVD) components of the generalized Tikhonov regularization problem.

This function returns the \(\mathbf{s}\), \(\mathbf{U}\), \(\tilde{\mathbf{V}}\) and \(\mathbf{B}\) from the given matrix \(\mathbf{T}\), \(\mathbf{Q}\), and \(\mathbf{H}\).

Note

The mathmatical notation and calculation procedure is based on the inversion theory.

Parameters:

T(M, N) array_like

Matrix for a linear equation \(\mathbf{T}\in\mathbb{R}^{M\times N}\).

H(N, N) array_like

Regularization matrix \(\mathbf{H} \in \mathbb{R}^{N\times N}\) which must be a symmetric positive semi-definite matrix.

Q(M, M) array_like, optional

Weighted matrix for the residual norm \(\mathbf{Q}\in\mathbb{R}^{M\times M}\), by default None (meaning \(\mathbf{Q} = \mathbf{I}\)). This matrix must be a symmetric positive semi-definite matrix.

use_gpubool, optional

Whether to use GPU or not, by default False. If True, the cupy functionalities is used instead of numpy and scipy ones when calculating the inverse of a sparse matrix, singular value decomposition, inverted solution basis \(\tilde{\mathbf{V}}\), etc. Please ensure cupy is installed before using this option, otherwise an ModuleNotFoundError will be raised.

dtypestr or numpy dtype, optional

Data type of the matrix elements, by default numpy.float64. In case of using GPU, the data type numpy.float32 is a little bit faster and saves memory.

spSpinner or DummySpinner, optional

Spinner object to show the progress of calculation, by default DummySpinner.

Returns:

s{ (r, ), (r-1, ) } array

Vector of singular values like \(\begin{bmatrix}\sigma_1&\cdots&\sigma_r\end{bmatrix}^\mathsf{T}\in\mathbb{R}^r\). If one set \(\mathbf{T}\) as a sparse matrix, \(r\) is reduced by 1 (i.e. \(r \to r-1\)) because of the use of scipy.sparse.linalg.eigsh function to calculate the singular values.

U(M, r) array

Left singular vectors like \(\mathbf{U}\in\mathbb{R}^{M\times r}\).

basis(N, r) array

Inverted solution basis like \(\tilde{\mathbf{V}} \in \mathbb{R}^{N\times r}\).

B(M, M) scipy.sparse.csr_matrix

Matrix \(\mathbf{B}\) coming from \(\mathbf{Q} = \mathbf{B}^\mathsf{T}\mathbf{B}\). Only returned when \(\mathbf{Q}\) is given.

Examples

s, U, basis = compute_svd(T, H)
s, U, basis, B = compute_svd(T, H, Q, dtype=np.float32, use_gpu=True)