Error bounds and enclosures: The development of numerical analysis and the impact of the contributions by Lothar Collatz

In 1933 Lothar Collatz published his very first article in Zeitschrift für Angewandte Mathematik und Mechanik (ZAMM) on error estimates for finite difference methods for partial differential equations. At that time, numerics meant calculations of approximations by hand and mechanical calculators. Then, in the next decades, parallel to the progress in computer technology, more and more methods were developed. Nevertheless, in all cases the accuracy of numerical approximations is limited, so that at least rough error bounds or, at best, tight enclosures are required for the reliability of the numerical scheme and the validation of the approximate results. Here, we recall the early development of numerical analysis of differential equations, of numerical iterations, and for the approximation of eigenvalues.