adelie.bcd.root_function

adelie.bcd.root_function(h: float, *, quad: ndarray, linear: ndarray, l1: float)

Computes the BCD root function.

The BCD root function is given by

\[\begin{align*} \varphi(h) := \sum\limits_{i=1}^p \frac{v_i^2}{(\Sigma_{ii} h + \lambda)^2} - 1 \end{align*}\]

where \(h \geq 0\), \(\Sigma \in \mathbb{R}_+^{p\times p}\) diagonal, \(v \in \mathbb{R}^p\), and \(\lambda \geq 0\).

Parameters:

hfloat

The value at which to evaluate the BCD root function.

quad(p,) ndarray

The quadratic component \(\Sigma\).

linear(p,) ndarray

The linear component \(v\).

l1float

The \(\ell_1\) component \(\lambda\).

Returns:

funcfloat

The BCD root function value.