cherab.phix.inversion.GCV.eta#
- GCV.eta(beta)Source#
Calculate squared regularization norm: \(\eta = ||Lx_\lambda||^2\)
\(\eta\) can be calculated with SVD components as follows:
\[\begin{split}\eta &= \left\| V W \Sigma^{-1} U^\mathsf{T} b \right\|^2\\ &= \left\| \begin{pmatrix} v_1 & \cdots & v_r \end{pmatrix} W \Sigma^{-1} U^\mathsf{T} b \right\|^2\\ &= \left\| W \Sigma^{-1} U^\mathsf{T} b \right\|^2 \quad( \because V^\mathsf{T}V = I_r, \quad\text{i.e.}\quad v_i\cdot v_j = \delta_{ij} ),\end{split}\]where \(W = \text{diag}(w_1(\lambda), ..., w_r(\lambda))\) and \(w_i(\lambda)\) is the window function.
- Parameters:
beta (float) – regularization parameter
- Returns:
squared regularization norm
- Return type: