Theory of an interval algebra and its application to numerical analysis

Sunaga, Teruo

doi:10.1007/bf03186528

Cited by 136 publications

(119 citation statements)

References 3 publications

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“…According to a survey paper by Kearfott (1996), a form of interval arithmetic perhaps first appeared in Burkill (1924). Modern interval arithmetic was originally invented independently in the late 1950s by several researchers; including Warmus (1956), Sunaga (1958) and finally Moore (1959), who set firm foundations for the field in his many publications, including the foundational book Moore (1966). Since then, interval arithmetic is being used to rigorously solve numerical problems.…”

Section: A Formal Overview Of Interval Arithmeticmentioning

confidence: 99%

Computing the noncentral-F distribution and the power of the F-test with guaranteed accuracy

2016

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The computations involving the noncentral-F distribution are notoriously difficult to implement properly in floating-point arithmetic: Catastrophic loss of precision, floating-point underflow and overflow, drastically increasing computation time and program hang-ups, and instability due to numerical cancellation have all been reported. It is therefore recommended that existing statistical packages are crosschecked, and the present paper proposes a numerical algorithm precisely for this purpose. To the best of our knowledge, the proposed method is the first method that can compute the noncentrality parameter of the noncentral-F distribution with guaranteed accuracy over a wide parameter range that spans the range relevant for practical applications. Although the proposed method is limited to cases where the the degree of freedom of the denominator of the F test statistic is even, it does not affect its usefulness significantly: All of those algorithmic failures and inaccuracies that we can still reproduce today could have been prevented by simply cross-checking against the proposed method. Two numerical examples are presented where the intermediate computations went wrong silently, but the final result of the computations seemed nevertheless plausible, and eventually erroneous results were published. Cross-checking against the proposed method would have caught the numerical errors in both cases. The source code of the algorithm is available on GitHub, together with self-contained command-line executables. These executables can read the data to be cross-checked from plain text files, making it easy to cross-check any statistical software in an automated fashion.

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Section: A Formal Overview Of Interval Arithmeticmentioning

confidence: 99%

Computing the noncentral-F distribution and the power of the F-test with guaranteed accuracy

2016

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“…It aims at practical use and it addresses non-empty, bounded closed intervals. Other definitions for interval arithmetic appeared at about the same time: let us cite T. Sunaga (Sunaga 1958) in Japan, L.V. Kantorovitch (Kantorovitch 1962) in former USSR.…”

Section: Standardization Effortmentioning

confidence: 99%

Latest Developments on the IEEE 1788 Effort for the Standardization of Interval Arithmetic

Revol¹

2014

Vulnerability, Uncertainty, and Risk

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“…Interval analysis was initially developed to account for the quantification errors introduced by the floating point representation of real numbers with computers and was extended to validated numerics [26], [9], [27].…”

Section: Solving Sip Via Interval Analysis 1) Interval Analysismentioning

confidence: 99%

Safe motion planning computation for databasing balanced movement of humanoid robots

Lengagne

Ramdani

Fraisse

2009

2009 IEEE International Conference on Robotics and Automation

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Motion planning can be seen as a Semi-Infinite Programming problem (SIP) since it involves a finite number of variables over an infinite set of constraints. Most methods solve the SIP problem by transforming it into a finite programming one using a discretization over a prescribed grid. We show that this approach is risky because it can lead to motions which may violate one or several constraints. Then we introduce our new method for planning safe motions. It uses Interval Analysis techniques in order to achieve a safe discretization of the constraints. We show how to implement this method and use it with state-of-the-art constrained optimization packages. Then, we illustrate its capabilities for planning safe motions dedicated to the HOAP-3 humanoid robot.

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Theory of an interval algebra and its application to numerical analysis

Cited by 136 publications

References 3 publications

Computing the noncentral-F distribution and the power of the F-test with guaranteed accuracy

Computing the noncentral-F distribution and the power of the F-test with guaranteed accuracy

Latest Developments on the IEEE 1788 Effort for the Standardization of Interval Arithmetic

Safe motion planning computation for databasing balanced movement of humanoid robots

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