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首页> 外文期刊>Zeitschrift fur Angewandte Mathematik und Mechanik >On numerical stability in large scale linear algebraic computations - Plenary Lecture presented at the 75th Annual GAMM Conference, Dresden/Germany, 22-26 March 2004
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On numerical stability in large scale linear algebraic computations - Plenary Lecture presented at the 75th Annual GAMM Conference, Dresden/Germany, 22-26 March 2004

机译：关于大规模线性代数计算中的数值稳定性-在2004年3月22日至26日在德国德累斯顿举行的第75届GAMM年度会议上发表的演讲

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摘要

Numerical solving of real-world problems typically consists of several stages. After a mathematical description of the problem and its proper reformulation and discretisation, the resulting linear algebraic problem has to be solved. We focus on this last stage, and specifically consider numerical stability of iterative methods in matrix computations. In iterative methods, rounding errors have two main effects: They can delay convergence and they can limit the maximal attainable accuracy. It is important to realize that numerical stability analysis is not about derivation of error bounds or estimates. Rather the goal is to find algorithms and their parts that are safe (numerically stable), and to identify algorithms and their parts that are not. Numerical stability analysis demonstrates this important idea. which also guides this contribution. In our survey we first recall the concept of backward stability and discuss its use in numerical stability analysis of iterative methods. Using the backward error approach we then examine the surprising fact that the accuracy of a (final) computed result may be much higher than the accuracy of intermediate computed quantities. We present some examples of rounding error analysis that are fundamental to justify numerically computed results. Our points are illustrated on the Lanczos method, the conjugate gradient (CG) method and the generalised minimal residual (GMRES) method. © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.
机译：实际问题的数值求解通常包括几个阶段。在对问题进行数学描述并对其进行适当的重新格式化和离散化之后，必须解决所得的线性代数问题。我们专注于最后阶段，并特别考虑矩阵计算中迭代方法的数值稳定性。在迭代方法中，舍入误差有两个主要影响：它们会延迟收敛，并且会限制最大可达到的精度。重要的是要认识到，数值稳定性分析与误差范围或估计的推导无关。相反，目标是找到安全的（数字稳定的）算法及其部分，并找出不安全的算法及其部分。数值稳定性分析证明了这一重要思想。这也指导了这一贡献。在我们的调查中，我们首先回顾了向后稳定性的概念，并讨论了其在迭代方法的数值稳定性分析中的使用。然后，使用后向误差方法，我们检查了一个令人惊讶的事实，即（最终）计算结果的精度可能比中间计算量的精度高得多。我们提供了一些舍入误差分析的示例，这些示例对于证明数值计算结果的合理性至关重要。我们的观点在Lanczos方法，共轭梯度（CG）方法和广义最小残差（GMRES）方法上得到了说明。 ＆复制; 2005 WILEY-VCH Verlag GmbH＆Co. KGaA，魏因海姆。

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