Transreal Integral

Reis, Tiago

doi:10.36285/tm.v0i0.13

Cited by 4 publications

(2 citation statements)

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“…An author [14], unrelated to the author of the present paper, reviewed various approaches to division by zero and introduced the algebraic structure of a premeadow which generalises an associative transfield. A stronger transreal integral was introduced in [38] that is defined on the entire set of transreal numbers and integrates all functions which are properly or improperly integrable on real numbers. In [39], a technique called slipstreaming was described and illustrated an imple-mentation of the Fast Fourier Transform (FFT) on a fine-grained architecture that exploits the exception-free properties of transreal arithmetic to implement a statically assigned systolic dataflow.…”

Section: Transreal Arithmeticmentioning

confidence: 99%

Transreal Foundation for Floating-Point Arithmetic

Anderson¹

2023

Transmath

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Software Engineering relies, to a large extent, on formal software standards and logical means for specifying and verifying computer programs. Among these the IEEE 754 standard for floating-point arithmetic is widely used. We criticise the standard from the standpoint of transreal arithmetic. Transreal arithmetic was derived from projective geometry using a double cover to provide signed infinities on the horizon and nullity at the point of projection. These infinities and nullity have some similarities with IEEE 754 floating-point infinities and NaNs but there are important differences. We explore the differences by analysing the standard at three levels: commentary within the standard, abstract datatypes, and bit patterns. We find that all of the differences are coincident with faults in the standard. Obviously a correct standard would better support the specification, development and testing of numerical software. We discuss how the standard can be corrected, in its own terms, or by adopting transreal arithmetic as its theoretical foundation. We also discuss emulation of transreal arithmetic in IEEE 754 processing systems and address accusations that transreal arithmetic plagiarised the standard.

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Section: Transreal Arithmeticmentioning

confidence: 99%

Transreal Foundation for Floating-Point Arithmetic

Anderson¹

2023

Transmath

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“…Given the ubiquitous practice with the use of the peripheral number +Inf in computing there has been remarkably little theoretical work on that matter. The main approach in this direction, named transreals, has been pioneered by James Anderson and his colleagues in a sequence of papers of which we mention [2,18] and [19]. For further work in this direction see [14].…”

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confidence: 99%

Prospective, Retrospective, and Formal Division

Bergstra

2022

Transmath

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Division by zero is a controversial theme. Why is division by zero a relevant issue and how can this issue be addressed from different perspectives? Fracterm is used as an abbreviation for fractional expression. Three types of occurrence of the division symbol in a fracterm are distinguished: prospective occurrence, retrospective occurrence, and formal occurrence. Mathematics mostly features retrospective occurrences of division, computer programming gives rise to prospective occurrences, and so does automated proof checking. The use of division in an axiom system may indicate the presence of formal occurrences of division symbols.

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Transcomplex Integral

Reis¹,

Anderson²

2020

Transactions on Engineering Technologies

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Transreal Integral

Cited by 4 publications

References 0 publications

Transreal Foundation for Floating-Point Arithmetic

Transreal Foundation for Floating-Point Arithmetic

Prospective, Retrospective, and Formal Division

Transcomplex Integral

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