In this paper, we present the implementation and results of new methodology based on linear algebra. The theory behind these methods is covered in detail in the Supporting Information, available electronically (Shrager and Hendler), In brief, the methods presented search through all possible forward sequential submodels in order to find candidates that can be used to construct a complete model for the BR-photocycle. The methodology is limited only to forward sequential models. If no such models are compatible with the experimental data, none will be found, The procedures apply objective tests and filters to eliminate possibilities that cannot be correct, thus cutting the total number of candidate sequences to be considered. In the current application, which uses six exponentials, the total sequences were cut from 1950 to 49. The remaining sequences were further screened using known experimental criteria. The approach led to a solution which consists of a pair of sequences, one with 5 exponentials showing BR* --> L-f --> M-f --> N --> O --> BR and the other with three exponentials showing BR* --> L-s --> M-s --> BR. The deduced complete kinetic model for the BR photocycle is thus either a single photocycle branched at the L intermediate or a pair of two parallel photocycles, Reasons for preferring the parallel photocycles are presented. Synthetic data constructed on the basis of the parallel photocycles were indistinguishable from the experimental data in a number of analytical tests that were applied.