A new fast track-fit algorithm based on broken lines - Desy

First phase: “Curved line” trajectory <strong>fit</strong>
The case of the additional parameter ∆κ is only slightly more complicated. The linear least squares
expression S (u, ∆κ) is minimized by the solution of the matrix equation:
⎛
Cκ c
⎜
⎝
T
⎞ ⎛ ⎞
∆κ
⎟ ⎜ ⎟
⎟ ⎜ ⎟
c Cu ⎠ ⎝ u ⎠ =
⎛
⎞
Cκ c c c c c c c · · ·
⎜
⎛ ⎞
⎜
c d x x
⎟
⎜
rκ
⎜
c x d x x
⎟
⎜ ⎟
⎜
⎜ ⎟
⎜
c x x d x x
⎟
⎝ru⎠
with matrix ⎜
c x x d x x ⎟
⎜
c x x d x x ⎟
⎜
.
⎝ c
.. ⎟
⎠
i.e. the matrix is a bordered band matrix.
Solution: Cu = LDL T
.
decomposition (6n)
Cuz = c solution for z (5n)
Bκ = � Cκ − c T z �−1 �
∆κ = Bκ rκ − z
variance of curvature (n + 1)
T �
ru curvature (n + 1)
Cu�u = ru solution for �u (5n)
u = �u − z∆κ smoothed coordinates (n)
V. Blobel – University of Hamburg A <strong>new</strong> <strong>fast</strong> <strong>track</strong>-<strong>fit</strong> <strong>algorithm</strong> <strong>based</strong> on broken lines page 14