A Remarkable Example of Catastrophic Cancellation Unraveled

Cuyt, Annie; Verdonk, Brigitte; Becuwe, Stefan; Kuterna, Peter

doi:10.1007/s006070170028

Cited by 16 publications

(9 citation statements)

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“…Problems in the more extreme tail areas are easier to detect because the complementation will cause “catastrophic” cancellation error [5], resulting in a tail area of zero, which is a much clearer signal that something is amiss than an answer that may appear similar to what was expected. This creates a somewhat counter-intuitive situation where inaccuracies in more extreme tail areas (larger exponent) are more obvious than those in less extreme tail areas (smaller exponent).…”

Section: Discussionmentioning

confidence: 99%

How accurate are the extremely small -values used in genomic research: An evaluation of numerical libraries

Bangalore¹,

Wang²,

Allison³

2009

Computational Statistics & Data Analysis

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In the fields of genomics and high dimensional biology (HDB), massive multiple testing prompts the use of extremely small significance levels. Because tail areas of statistical distributions are needed for hypothesis testing, the accuracy of these areas is important to confidently make scientific judgments. Previous work on accuracy was primarily focused on evaluating professionally written statistical software, like SAS, on the Statistical Reference Datasets (StRD) provided by National Institute of Standards and Technology (NIST) and on the accuracy of tail areas in statistical distributions. The goal of this paper is to provide guidance to investigators, who are developing their own custom scientific software built upon numerical libraries written by others. In specific, we evaluate the accuracy of small tail areas from cumulative distribution functions (CDF) of the Chisquare and t-distribution by comparing several open-source, free, or commercially licensed numerical libraries in Java, C, and R to widely accepted standards of comparison like ELV and DCDFLIB. In our evaluation, the C libraries and R functions are consistently accurate up to six significant digits. Amongst the evaluated Java libraries, Colt is most accurate. These languages and libraries are popular choices among programmers developing scientific software, so the results herein can be useful to programmers in choosing libraries for CDF accuracy.

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Section: Discussionmentioning

confidence: 99%

How accurate are the extremely small -values used in genomic research: An evaluation of numerical libraries

Bangalore¹,

Wang²,

Allison³

2009

Computational Statistics & Data Analysis

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“…Rounding errors propagate and compound when a result from one floating point operation is used in further computations. For this reason, expressions of the form ( a + b ) ‐ b may evaluate to 0 for a sufficiently small a ≠ 0 and large b ; a phenomena known as catastrophic cancellation (CC) [CVBK01]. We will show how this impacts any pipeline utilizing matrix multiplications or vector additions.…”

Section: Theoretical Motivationmentioning

confidence: 99%

Dynamic Scene Graph: Enabling Scaling, Positioning, and Navigation in the Universe

Axelsson

Costa

Silva

et al. 2017

Computer Graphics Forum

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In this work, we address the challenge of seamlessly visualizing astronomical data exhibiting huge scale differences in distance, size, and resolution. One of the difficulties is accurate, fast, and dynamic positioning and navigation to enable scaling over orders of magnitude, far beyond the precision of floating point arithmetic. To this end we propose a method that utilizes a dynamically assigned frame of reference to provide the highest possible numerical precision for all salient objects in a scene graph. This makes it possible to smoothly navigate and interactively render, for example, surface structures on Mars and the Milky Way simultaneously. Our work is based on an analysis of tracking and quantification of the propagation of precision errors through the computer graphics pipeline using interval arithmetic. Furthermore, we identify sources of precision degradation, leading to incorrect object positions in screen‐space and z‐fighting. Our proposed method operates without near and far planes while maintaining high depth precision through the use of floating point depth buffers. By providing interoperability with order‐independent transparency algorithms, direct volume rendering, and stereoscopy, our approach is well suited for scientific visualization. We provide the mathematical background, a thorough description of the method, and a reference implementation.

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“…This example shows that the arithmetic of large numbers should be performed very cautiously in scientific computing. A detailed numerical analysis of this catastrophic behavior of rounding errors is presented in [3] and [10]. Thus, we should keep in mind that a very small number of subtractions and roundings (see (5) and (8)) may completely destroy the exact solution [17].…”

Section: Rump's Examplementioning

confidence: 99%

Paradoxes in Numerical Calculations

Brandts¹,

Křížek²,

Zhang³

2016

NNW

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Precise wind energy potential assessment is vital for wind energy generation and planning and development of new wind power plants. This work proposes and evaluates a novel two-stage method for location-specific wind energy potential assessment. It combines accurate statistical modelling of annual wind direction distribution in a given location with supervised machine learning of efficient estimators that can approximate energy efficiency coefficients from the parameters of optimized statistical wind direction models. The statistical models are optimized using differential evolution and energy efficiency is approximated by evolutionary fuzzy rules.

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A Remarkable Example of Catastrophic Cancellation Unraveled

Cited by 16 publications

References 0 publications

How accurate are the extremely small -values used in genomic research: An evaluation of numerical libraries

How accurate are the extremely small -values used in genomic research: An evaluation of numerical libraries

Dynamic Scene Graph: Enabling Scaling, Positioning, and Navigation in the Universe

Paradoxes in Numerical Calculations

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