Analysis and improvements of a systematic componentwise ultimate-bound computation method

We perform in-depth analysis and provide improvements of a systematic componentwise ultimate-bound computation method recently introduced in the literature. This method was shown to have many advantages over traditional ultimate-bound computation methods based on the use of quadratic Lyapunov functions. The analysis performed enhances our understanding of the componentwise methodology, and simplifies the search for improvements. The improvements provided aim at reducing the conservatism of the componentwise ultimate-bound computation methods even further, hence leading to tighter bounds. These improvements do not alter the systematic nature of the method.