Beta Correction Algorithm

The information about the horizontal and vertical beta beat $\delta\beta_i$ at the position of the $i$th quadrupole caused by a strength perturbation $\delta k_j$ of the $j$th quadrupole is contained in the (2*174)x174 sensitivity matrix $\bf S$:

$${\bf S}_{ij}=\frac{\delta\beta_i}{\delta k_j}$$

which can be derived from the model. A Singular Value Decomposition (SVD) technique is then used to ``invert'' $\bf S$ and determine the $\delta k_j$ as a function of the $\delta\beta_i$. Feeding $-\delta k_j$ into the ``beta function correctors'' restores the ideal optics within the error of the beta measurement if the quadrupoles are the only source of the optics perturbation.

Figure: Measured average beta functions ( squares) at the location of 174 quadrupoles in comparison to the model of the unperturbed optics (solid lines)