用复区间算法对Schönhage-Strassen整数乘法算法的严格扩展

Q3 Computer Science
R. Sainudiin, T. Steinke
{"title":"用复区间算法对Schönhage-Strassen整数乘法算法的严格扩展","authors":"R. Sainudiin, T. Steinke","doi":"10.4204/EPTCS.24.19","DOIUrl":null,"url":null,"abstract":"Multiplication of n-digit integers by long multiplication requires O(n 2 ) operations and can be time-consuming. A. Schonhage and V. Strassen published an algorithm in 1970 that is capable of performing the task with only O(nlog(n)) arithmetic operations over C; naturally, finite-precision approximations to C are used and rounding errors need to be accounted for. Overall, using variable-precision fixed-point numbers, this results in an O(n(log(n)) 2+\" )-time algorithm. However, to make this algorithm more efficient and practical we need to make use of hardware-based floating- point numbers. How do we deal with rounding errors? and how do we determine the limits of the fixed-precision hardware? Our solution is to use interval arithmetic to guarantee the correctness of results and deter- mine the hardware's limits. We examine the feasibility of this approach and are able to report that 75,000-digit base-256 integers can be han- dled using double-precision containment sets. This demonstrates that our approach has practical potential; however, at this stage, our implemen- tation does not yet compete with commercial ones, but we are able to demonstrate the feasibility of this technique.","PeriodicalId":54499,"journal":{"name":"Reliable Computing","volume":"41 1","pages":"151-159"},"PeriodicalIF":0.0000,"publicationDate":"2010-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"A Rigorous Extension of the Schönhage-Strassen Integer Multiplication Algorithm Using Complex Interval Arithmetic\",\"authors\":\"R. Sainudiin, T. Steinke\",\"doi\":\"10.4204/EPTCS.24.19\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Multiplication of n-digit integers by long multiplication requires O(n 2 ) operations and can be time-consuming. A. Schonhage and V. Strassen published an algorithm in 1970 that is capable of performing the task with only O(nlog(n)) arithmetic operations over C; naturally, finite-precision approximations to C are used and rounding errors need to be accounted for. Overall, using variable-precision fixed-point numbers, this results in an O(n(log(n)) 2+\\\" )-time algorithm. However, to make this algorithm more efficient and practical we need to make use of hardware-based floating- point numbers. How do we deal with rounding errors? and how do we determine the limits of the fixed-precision hardware? Our solution is to use interval arithmetic to guarantee the correctness of results and deter- mine the hardware's limits. We examine the feasibility of this approach and are able to report that 75,000-digit base-256 integers can be han- dled using double-precision containment sets. This demonstrates that our approach has practical potential; however, at this stage, our implemen- tation does not yet compete with commercial ones, but we are able to demonstrate the feasibility of this technique.\",\"PeriodicalId\":54499,\"journal\":{\"name\":\"Reliable Computing\",\"volume\":\"41 1\",\"pages\":\"151-159\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2010-06-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Reliable Computing\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.4204/EPTCS.24.19\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"Computer Science\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Reliable Computing","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.4204/EPTCS.24.19","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Computer Science","Score":null,"Total":0}
引用次数: 1

摘要

n位整数乘以长乘法需要O(n 2)次运算,并且可能很耗时。A. Schonhage和V. Strassen在1970年发表了一种算法,该算法能够在C上只进行O(nlog(n))次算术运算即可完成任务;当然,使用C的有限精度近似值,并且需要考虑舍入误差。总的来说,使用可变精度的定点数,这将导致O(n(log(n)) 2+“)时间算法。然而,为了使该算法更有效和实用,我们需要利用基于硬件的浮点数。如何处理舍入误差?我们如何确定固定精度硬件的极限?我们的解决方案是使用区间算法来保证结果的正确性,并阻止硬件的限制。我们检验了这种方法的可行性,并且能够报告使用双精度包含集可以处理75000位的以256为基数的整数。这表明我们的方法具有实践潜力;然而,在这个阶段,我们的实现还不能与商业的竞争,但我们能够证明这种技术的可行性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
A Rigorous Extension of the Schönhage-Strassen Integer Multiplication Algorithm Using Complex Interval Arithmetic
Multiplication of n-digit integers by long multiplication requires O(n 2 ) operations and can be time-consuming. A. Schonhage and V. Strassen published an algorithm in 1970 that is capable of performing the task with only O(nlog(n)) arithmetic operations over C; naturally, finite-precision approximations to C are used and rounding errors need to be accounted for. Overall, using variable-precision fixed-point numbers, this results in an O(n(log(n)) 2+" )-time algorithm. However, to make this algorithm more efficient and practical we need to make use of hardware-based floating- point numbers. How do we deal with rounding errors? and how do we determine the limits of the fixed-precision hardware? Our solution is to use interval arithmetic to guarantee the correctness of results and deter- mine the hardware's limits. We examine the feasibility of this approach and are able to report that 75,000-digit base-256 integers can be han- dled using double-precision containment sets. This demonstrates that our approach has practical potential; however, at this stage, our implemen- tation does not yet compete with commercial ones, but we are able to demonstrate the feasibility of this technique.
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
Reliable Computing
Reliable Computing Computer Science-Software
CiteScore
0.20
自引率
0.00%
发文量
0
期刊介绍: eliable Computing accepts manuscripts representing original articles, reviews, presentations of new hardware and software tools, book reviews, information on scientific meetings on relevant topics which are scheduled or have recently been held, etc.
×
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