检测数值算法的潜在不稳定性
Detecting Potential Instabilities of Numerical Algorithms
AI总结：
本文挑战传统稳定性分析观念，指出forward、backward和mixed稳定性分析在检测算法潜在不稳定性方面存在根本缺陷，质疑其有效性。
摘要：
今天普遍认为的backward稳定性分析被视为绝对正确，正如牛顿物理学中时间被视为绝对时间一样。本文将证明这种观点并不普遍成立。它取决于算法。我们将证明forward和mixed稳定性分析在严格意义上是无效的稳定性分析，因为它们在检测任何算法中可能存在的巨大元素增长时具有绝对错误的参考点，即使对于
英文摘要：
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.
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