A history of Krylov of product methods

Abstract

In about 1976, the author was preparing a renovation of the elementary course on numerical analysis in Delft University. In relation to the problem of solving a single non-linear equation iteratively, he wondered whether the so-called ‘secant method’ could be generalized to systems of N nonlinear equations with N unknowns. Before starting to read everything on the subject, the author normally tries to think about it unbiased, and so he did this time, and started with (probably) re-inventing the wheel. Would he have seen the book by Ortega and Rheinboldt at that time, he wouldn’t have discovered the ‘new wheel’ IDR, and also CGS and BiCGSTAB probably wouldn’t exist today. Serendipity means something like ‘finding the unsought’, and the strange history of the socalled ‘Krylov Product methods’ shows some examples of this phenomenon.