XMM-Newton Science Analysis System


bkgfit (ebkgmap-2.10.2) [xmmsas_20230412_1735-21.0.0]

Fitting algorithm:

The vector of amplitudes $\mathbf{a}_{\rm {opt}}$ which minimizes $L$ is approximated by a Newtonian method in $N$ dimensions. In this method, the formula

$$\nabla L = \mathbf{C} \, (\mathbf{a}_{i+1} - \mathbf{a}_{i})$$

where $\mathbf{C}$ is an $N \times N$ matrix of curvature terms given by

  $$C_{i,j} = \frac{\partial^2 L}{\partial a_i \partial a_j},$$ (3)

is iteratively inverted until it is judged to have converged.

$\nabla L$ and $\mathbf{C}$ can be expressed in closed form as

$$(\nabla L)_i = \frac{\partial L}{\partial a_i} = 2\sum_{x=1}^{X}\... ...Y} \bigg(b_{x,y,i} \bigg[1 - \frac{I_{x,y}}{B_{x,y}(\mathbf{a})} \bigg]\bigg),$$

$$C_{i,j} = \frac{\partial^2 L}{\partial a_i \partial a_j} = 2\sum_... ...bigg[ \frac{b_{x,y,i} \, b_{x,y,j} \, I_{x,y}}{B_{x,y}^2(\mathbf{a} )} \bigg].$$