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.

中文翻译：

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误差范围和界限：数值分析的发展以及Lothar Collat​​z所做贡献的影响

1933年，洛萨·科拉兹（Lothar Collat​​z）在《时代报》和《机械学》（ZAMM）上发表了他的第一篇文章。）偏微分方程有限差分方法的误差估计。那时，数字意味着要用手工和机械计算器来计算近似值。然后，在接下来的几十年中，随着计算机技术的进步，越来越多的方法被开发出来。然而，在所有情况下，数值逼近的准确性都是有限的，因此至少需要一个粗略的误差范围，或至多需要紧密的外壳，以确保数值方案的可靠性和近似结果的有效性。在这里，我们回顾了微分方程数值分析，数值迭代以及特征值近似的早期发展。