Detecting Potential Instabilities of Numerical Algorithms

It has been the standard teaching of today that backward stability analysis is taught as absolute, just as in Newtonian physics time is taught as absolute time. We will prove that it is not true in general. It depends on algorithms. We will prove that forward and mixed stability analysis are absolutely invalid stability analysis in the sense that they have absolutely wrong reference points for detecting huge element growth of any algorithms (if any), even an ”ideal” or ”desirable” backward stability analysis is not so ”ideal” or ”desirable” in general. Any of forward stable, backward stable and mixed stable algorithms as in Demmel, Kahan and Parlett and others’ papers and text books [6], and [8] may not be really stable at all because they may fail to detect and expose any potential instabilities of the algorithm in corresponding stability analysis. Therefore, it is impossible to prove an algorithm is stable according to the standard teaching of today, just as it is impossible to prove a mathematical equation(s)(Maxwell’s,) is a law of physics according to the standard teaching in Newtonian physics.