Analytical and Numerical Flash-Algorithms for Track Fits

The present Note is intended for track-reconstruction, detector-alignment and detector-evaluation groups. Often regarded as a simple problem, track reconstruction can soon become tangled with prohibitive CPU load and unexpectedly large errors. Detector alignment for instance requires that alignment parameters satisfy proper track reconstruction in all events. Tunning however the alignment parameters reconstructing all tracks in all events for each iteration in parameter space is CPU exhausting, and in practice can be applied only to blocks of i.e. 2000 tracks. If the track fits are also iterative, double-nesting occurs and CPU times reach on the order of 2-4 days for line-fits respectively 20-300 days for helix-fits, in the context of the ATLAS Pixel Detector test-stand. On the other hand if the track fits are semi/analytical, CPU times are 3-4 orders of magnitude smaller and such an approach becomes feasable. The note starts with the basic χ fit, examines the validity of the standard χ approximation in reconstruction contexts specific to ATLAS Pixel Detectors, gives analytical and semi-analytical track reconstruction solutions and presents examples of CPU-clocked processes on a DEC ALPHA machine.