基于牛顿-拉夫森浮点除法和平方根算法的正确性证明大纲

Marius A. Cornea-Hasegan, Roger A. Golliver, Peter W. Markstein
{"title":"基于牛顿-拉夫森浮点除法和平方根算法的正确性证明大纲","authors":"Marius A. Cornea-Hasegan, Roger A. Golliver, Peter W. Markstein","doi":"10.1109/ARITH.1999.762834","DOIUrl":null,"url":null,"abstract":"This paper describes a study of a class of algorithms for the floating-point divide and square root operations, based on the Newton-Raphson iterative method. The two main goals were. (1) Proving the IEEE correctness of these iterative floating-point algorithms, i.e. compliance with the IEEE-754 standard for binary floating-point operations. The focus was on software driven iterative algorithms, instead of the hardware based implementations that dominated until now. (2) Identifying the special cases of operands that require results. Assistance due to possible overflow, or loss of precision of intermediate This study was initiated in an attempt to prove the IEEE for a class of divide and square root based on the Newton-Rapshson iterative methods. As more insight into the inner workings of these algorithms was gained, it became obvious that a formal study and proof were necessary in order to achieve the desired objectives. The result is a complete and rigorous proof of IEEE correctness for floating-point divide and square root algorithms based on the Newton-Raphson iterative method. Even more, the method used in proving the IEEE correctness of the square root algorithm is applicable in principle to any iterative algorithm, not only based on the Newton-Raphson method. Conditions requiring Software Assistance (SWA) were also determined, and were used to identify cases when alternate algorithms are needed to generate correct results. Overall, this is one important step toward flawless implementation of these floating-point operations based on software implementations.","PeriodicalId":434169,"journal":{"name":"Proceedings 14th IEEE Symposium on Computer Arithmetic (Cat. No.99CB36336)","volume":"40 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"1999-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"62","resultStr":"{\"title\":\"Correctness proofs outline for Newton-Raphson based floating-point divide and square root algorithms\",\"authors\":\"Marius A. Cornea-Hasegan, Roger A. Golliver, Peter W. Markstein\",\"doi\":\"10.1109/ARITH.1999.762834\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"This paper describes a study of a class of algorithms for the floating-point divide and square root operations, based on the Newton-Raphson iterative method. The two main goals were. (1) Proving the IEEE correctness of these iterative floating-point algorithms, i.e. compliance with the IEEE-754 standard for binary floating-point operations. The focus was on software driven iterative algorithms, instead of the hardware based implementations that dominated until now. (2) Identifying the special cases of operands that require results. Assistance due to possible overflow, or loss of precision of intermediate This study was initiated in an attempt to prove the IEEE for a class of divide and square root based on the Newton-Rapshson iterative methods. As more insight into the inner workings of these algorithms was gained, it became obvious that a formal study and proof were necessary in order to achieve the desired objectives. The result is a complete and rigorous proof of IEEE correctness for floating-point divide and square root algorithms based on the Newton-Raphson iterative method. Even more, the method used in proving the IEEE correctness of the square root algorithm is applicable in principle to any iterative algorithm, not only based on the Newton-Raphson method. Conditions requiring Software Assistance (SWA) were also determined, and were used to identify cases when alternate algorithms are needed to generate correct results. Overall, this is one important step toward flawless implementation of these floating-point operations based on software implementations.\",\"PeriodicalId\":434169,\"journal\":{\"name\":\"Proceedings 14th IEEE Symposium on Computer Arithmetic (Cat. No.99CB36336)\",\"volume\":\"40 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"1999-04-14\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"62\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings 14th IEEE Symposium on Computer Arithmetic (Cat. No.99CB36336)\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1109/ARITH.1999.762834\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings 14th IEEE Symposium on Computer Arithmetic (Cat. No.99CB36336)","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1109/ARITH.1999.762834","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 62

摘要

本文研究了一类基于Newton-Raphson迭代法的浮点除法和平方根运算算法。两个主要目标是。(1)证明这些迭代浮点算法的IEEE正确性,即符合IEEE-754二进制浮点运算标准。重点是软件驱动的迭代算法,而不是迄今为止占主导地位的基于硬件的实现。(2)识别需要结果的操作数的特殊情况。本研究是为了证明一类基于Newton-Rapshson迭代方法的除法和平方根的IEEE。随着对这些算法内部工作原理的深入了解,很明显,为了实现预期的目标,有必要进行正式的研究和证明。该结果完整而严格地证明了基于牛顿-拉夫森迭代法的浮点除法和平方根算法的IEEE正确性。而且,证明平方根算法的IEEE正确性的方法原则上适用于任何迭代算法,而不仅仅是基于Newton-Raphson方法。还确定了需要软件辅助(SWA)的条件,并用于识别需要替代算法以生成正确结果的情况。总的来说,这是朝着基于软件实现完美地实现这些浮点操作迈出的重要一步。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Correctness proofs outline for Newton-Raphson based floating-point divide and square root algorithms
This paper describes a study of a class of algorithms for the floating-point divide and square root operations, based on the Newton-Raphson iterative method. The two main goals were. (1) Proving the IEEE correctness of these iterative floating-point algorithms, i.e. compliance with the IEEE-754 standard for binary floating-point operations. The focus was on software driven iterative algorithms, instead of the hardware based implementations that dominated until now. (2) Identifying the special cases of operands that require results. Assistance due to possible overflow, or loss of precision of intermediate This study was initiated in an attempt to prove the IEEE for a class of divide and square root based on the Newton-Rapshson iterative methods. As more insight into the inner workings of these algorithms was gained, it became obvious that a formal study and proof were necessary in order to achieve the desired objectives. The result is a complete and rigorous proof of IEEE correctness for floating-point divide and square root algorithms based on the Newton-Raphson iterative method. Even more, the method used in proving the IEEE correctness of the square root algorithm is applicable in principle to any iterative algorithm, not only based on the Newton-Raphson method. Conditions requiring Software Assistance (SWA) were also determined, and were used to identify cases when alternate algorithms are needed to generate correct results. Overall, this is one important step toward flawless implementation of these floating-point operations based on software implementations.
求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
自引率
0.00%
发文量
0
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信