Detecting Potential Instabilities of Numerical Algorithms

TL;DR

The paper challenges the traditional view of stability analysis in numerical algorithms, demonstrating that common methods like forward and backward stability can be fundamentally misleading in detecting potential instabilities.

Contribution

It proves that forward, backward, and mixed stability analyses are invalid for reliably detecting algorithmic instabilities, questioning established teaching and practices.

Findings

Backward stability analysis is not universally reliable.

Forward and mixed stability analyses can fail to detect instabilities.

Traditional stability proofs may not guarantee actual algorithm stability.

Abstract

It has been the standard teaching of today that backward stability analysis is taught as absolute, just as in Newtonian physics time is taught absolute time. We will prove it is not true in general. It depends on algorithms. We will prove that forward and mixed stability anlaysis are absolutely invalid stability analysis in the sense that they have absolutely wrong reference points for detecting huge element growth of any algoritms(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 algorihms as in Demmel, Kahan , Parlett and other's papers and text books, see Demmel(6) and Higham(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 teachin of today, just as it is impossible to prove a mathematical equuation(Maxwell's) is a law of physics according to the standard teaching in Newtonian physics.