ABSTRACT
Historically Alex Chao wrote a computer code which multiplied matrices that had been manually computed for each type of element. M. Sands and Chao formalisms are characterized by the "existence” of adiabatic emittances. The Chao formalism is closest to the beam envelope formalism to the extent that it only assumes small damping. Chao's formalism of Equations and Sands' integral agree to a high degree when all the necessary conditions are met. The resulting theory, the so-called beam envelope theory, is more accurate than Sands integral formalism and even more accurate than Chao's method. However it can be argued that this improved accuracy, especially over the Chao formalism, is academic. New complex elements are included and the programmer can be certain that all the "synchrotron integral" results will be delivered by the code correctly and self-consistently.
