Program verification in the presence of complex numbers, functions with branch cuts etc

James H. Davenport; Russell Bradford; Matthew England; David Wilson

doi:10.1109/SYNASC.2012.68

Program verification in the presence of complex numbers, functions with branch cuts etc

James H. Davenport, Russell Bradford, Matthew England, David Wilson

University of Bath

Research output: Chapter in Book/Report/Conference proceeding › Conference proceeding › peer-review

26 Citations (Scopus)

42 Downloads (Pure)

Abstract

In considering the reliability of numerical programs, it is normal to ''limit our study to the semantics dealing with numerical precision'' (Martel, 2005). On the other hand, there is a great deal of work on the reliability of programs that essentially ignores the numerics. The thesis of this paper is that there is a class of problems that fall between these two, which could be described as ''does the low-level arithmetic implement the high-level mathematics''. Many of these problems arise because mathematics, particularly the mathematics of the complex numbers, is more difficult than expected: for example the complex function log is not continuous, writing down a program to compute an inverse function is more complicated than just solving an equation, and many algebraic simplification rules are not universally valid. The good news is that these problems are theoretically capable of being solved, and are practically close to being solved, but not yet solved, in several real-world examples. However, there is still a long way to go before implementations match the theoretical possibilities.

Original language	English
Title of host publication	Proceedings - 14th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing, SYNASC 2012
Publisher	IEEE
Pages	83-88
Number of pages	6
ISBN (Print)	9780769549347
DOIs	https://doi.org/10.1109/SYNASC.2012.68
Publication status	Published - 2012
Externally published	Yes
Event	14th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing, SYNASC 2012 - Timisoara, Romania Duration: 26 Sept 2012 → 29 Sept 2012

Conference

Conference	14th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing, SYNASC 2012
Country/Territory	Romania
City	Timisoara
Period	26/09/12 → 29/09/12

Bibliographical note

© 2012 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.

Copyright © and Moral Rights are retained by the author(s) and/ or other copyright owners. A copy can be downloaded for personal non-commercial research or study, without prior permission or charge. This item cannot be reproduced or quoted extensively from without first obtaining permission in writing from the copyright holder(s). The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the copyright holders.

Keywords

algebra
branch cut
singularity
Verification

ASJC Scopus subject areas

Applied Mathematics
Numerical Analysis

Access to Document

10.1109/SYNASC.2012.68

Post-PrintAccepted author manuscript, 527 KBLicence: CC BY-NC

Cite this

Program verification in the presence of complex numbers, functions with branch cuts etc. / Davenport, James H.; Bradford, Russell; England, Matthew et al.
Proceedings - 14th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing, SYNASC 2012. IEEE, 2012. p. 83-88 6481015.

Research output: Chapter in Book/Report/Conference proceeding › Conference proceeding › peer-review

Davenport, JH, Bradford, R, England, M & Wilson, D 2012, Program verification in the presence of complex numbers, functions with branch cuts etc. in Proceedings - 14th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing, SYNASC 2012., 6481015, IEEE, pp. 83-88, 14th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing, SYNASC 2012, Timisoara, Romania, 26/09/12. https://doi.org/10.1109/SYNASC.2012.68

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abstract = "In considering the reliability of numerical programs, it is normal to ''limit our study to the semantics dealing with numerical precision'' (Martel, 2005). On the other hand, there is a great deal of work on the reliability of programs that essentially ignores the numerics. The thesis of this paper is that there is a class of problems that fall between these two, which could be described as ''does the low-level arithmetic implement the high-level mathematics''. Many of these problems arise because mathematics, particularly the mathematics of the complex numbers, is more difficult than expected: for example the complex function log is not continuous, writing down a program to compute an inverse function is more complicated than just solving an equation, and many algebraic simplification rules are not universally valid. The good news is that these problems are theoretically capable of being solved, and are practically close to being solved, but not yet solved, in several real-world examples. However, there is still a long way to go before implementations match the theoretical possibilities.",

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Program verification in the presence of complex numbers, functions with branch cuts etc

Abstract

Conference

Bibliographical note

Keywords

ASJC Scopus subject areas

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Cite this