cherab.phix.inversion.gcv.GCV.rho#
- GCV.rho(beta)Source#
Calculate squared residual norm: \(\rho = ||Ax_\lambda - b||^2\).
\(\rho\) can be calculated with SVD components as follows:
\[\begin{split}\rho &= \left\| U (I_r - W) U^\mathsf{T} b \right\|^2\\ &= \left\| \begin{pmatrix} u_1 & \cdots & u_r \end{pmatrix} (I_r - W) U^\mathsf{T} b \right\|^2\\ &= \left\| (I_r - W) U^\mathsf{T} b \right\|^2 \quad( \because U^\mathsf{T}U = I_r, \quad\text{i.e.}\quad u_i\cdot u_j = \delta_{ij} ),\end{split}\]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 residual norm
- Return type: