33 years of numerical instability. I (Q1067373)

The paper gives an extremly interesting and lyrical presentation of the history of an important concept of numerical analysis - the concept of numerical instability. The development of this concept in connection with the numerical treatment of initial value problems for ODE's is traced back by the author, who has great contributions in this field. The period under consideration begins with the pre-computer epoch (1928) and ends at the beginning of the stiff epoch (1963). In the paper the following questions are discussed: the relation between the stability of the solution of the difference equation and its corresponding differential equation; the relation between consistency, stability and convergence; the qualities of the explicit and the implicit one-step and k-step numerical methods; the gap between the mathematician's and numerical analyst's attitude to words like ''bounded'', ''sufficiently small'', ''convergence'' etc., and to this end, the need of transformation of the pure mathematician's terminology.