{"title":"Accurate Horner methods in real and complex floating-point arithmetic","authors":"Thomas R. Cameron, Stef Graillat","doi":"10.1007/s10543-024-01017-w","DOIUrl":null,"url":null,"abstract":"<p>In this article, we derive accurate Horner methods in real and complex floating-point arithmetic. In particular, we show that these methods are as accurate as if computed in <i>k</i>-fold precision and then rounded into the working precision. When <i>k</i> is two, our methods are comparable or faster than the existing compensated Horner routines. When compared to multi-precision software, such as MPFR and MPC, our methods are significantly faster, up to <i>k</i> equal to eight, that is, up to 489 bits in the significand.</p>","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":"47 1","pages":""},"PeriodicalIF":1.7000,"publicationDate":"2024-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"BIT Numerical Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10543-024-01017-w","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, SOFTWARE ENGINEERING","Score":null,"Total":0}
引用次数: 0
Abstract
In this article, we derive accurate Horner methods in real and complex floating-point arithmetic. In particular, we show that these methods are as accurate as if computed in k-fold precision and then rounded into the working precision. When k is two, our methods are comparable or faster than the existing compensated Horner routines. When compared to multi-precision software, such as MPFR and MPC, our methods are significantly faster, up to k equal to eight, that is, up to 489 bits in the significand.
在本文中，我们推导了实数和复数浮点运算中精确的霍纳方法。特别是，我们证明这些方法的精确度与以 k 倍精度计算，然后四舍五入到工作精度的方法相同。当 k 为 2 时，我们的方法与现有的补偿霍纳例程不相上下，甚至更快。与多精度软件（如 MPFR 和 MPC）相比，我们的方法速度明显更快，最高可达 k 等于 8，即有效值可达 489 位。
期刊介绍：
The journal BIT has been published since 1961. BIT publishes original research papers in the rapidly developing field of numerical analysis. The essential areas covered by BIT are development and analysis of numerical methods as well as the design and use of algorithms for scientific computing. Topics emphasized by BIT include numerical methods in approximation, linear algebra, and ordinary and partial differential equations.
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