China
Africa
Explore chapters and articles related to this topic
Errors in Numerical Computations
Published in Santanu Saha Ray, Numerical Analysis with Algorithms and Programming, 2018
In numerical analysis, numerical stability is a generally desirable property of numerical algorithms. In a stable algorithm, the cumulative effect of the round-off errors remain bounded. When we study an algorithm, our interest is the same as for an expression. We want small changes in the input to only produce small changes in the output. An algorithm or numerical process is called stable if this is true, and it is called unstable if large changes in the output are produced. Analyzing an algorithm for stability is more complicated than determining the condition of an expression, even if the algorithm simply evaluates the expression. This is because an algorithm consists of many basic calculations and each one must be analyzed and, due to round-off error, we must consider the possibility of small errors being introduced in every computed value. For example, evaluating y = x/(1 – x) may be accomplished into two steps: t = 1 – x and y = x/t. Also, we may consider both x and t to have small errors. An algorithm is stable if every step is well-conditioned. It is unstable if any step is ill-conditioned.
Introduction
Published in Victor S. Ryaben’kii, Semyon V. Tsynkov, A Theoretical Introduction to Numerical Analysis, 2006
Victor S. Ryaben’kii, Semyon V. Tsynkov
Most typically, numerical instability manifests itself through a strong amplification of the small round-off errors in the course of computations. A key mechanism for the amplification is the loss of significant digits, which is a purely computer-related phenomenon that only occurs because the numbers inside a computer are represented as finite (binary) fractions (see Section 1.3.3). If computers could operate with infinite fractions (no rounding), then this phenomenon would not take place.
Error Analysis
Published in James P. Howard, Computational Methods for Numerical Analysis with R, 2017
This leads to the search for numerical stability, another important goal of numerical analysis. Numerical stability is the property of numerical algorithm that the approximation errors, from whatever source, are not amplified by the algorithm. This property manifests in two important ways.
An Assessment of Coupling Algorithms in HTR Simulator TINTE
Published in Nuclear Science and Engineering, 2018
Han Zhang, Jiong Guo, Jianan Lu, Fu Li, Yunlin Xu, T. J. Downar
In this case, the numerical result shows that a 2-s time step is fine enough to obtain a converged-in-time solution using the original TINTE, which is used as a reference solution in this case. Similarly, a large time step is used to assess the numerical stability of the coupling algorithms. Here, the phrase “numerical instability” is used to describe the phenomenon that truncation errors are magnified, causing the deviation from the exact solution to grow exponentially.32 The variations of the mean relative errors obtained by different coupling algorithms are presented in Fig. 16. Obviously, the OSSI coupling method is unstable when the large time step is employed since the error of TINTE grows rapidly in the form of oscillations. The maximum relative error of TINTE is 12 times larger than those of TINTE-JFNK and TINTE-Picard, as shown in Table VI. Furthermore, the unbounded error leads the physical variables to exceed the reasonable range and cause the simulation to bad stop. A smaller time step has to be taken to satisfy the stability requirement and improve the accuracy. Therefore, the computational efficiency of TINTE-Picard and TINTE-JFNK is 3.04 and 1.20 times higher than that of TINTE, respectively. This numerical test demonstrates that the OSSI algorithm is only conditionally stable, which means that the time step should be smaller than a certain value to ensure stability. However, the fully implicit coupling algorithms, such as Picard and JFNK, are unconditionally stable, and the limitation of the time step is avoided from the perspective of stability.