Error estimates for the summation of real numbers with application to floating-point summation
Error estimates for the summation of real numbers with application to floating-point summation
复制标题
实数求和的误差估计及其应用于浮点求和
DOI：
10.1007/s10543-017-0658-9
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
S. Rump
中科院分区：
文献类型：
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作者：
M. Lange;S. Rump
浮点算法的标准Wilkinson型误差估计涉及一个表示浮点数系统的相对舍入误差单位的因子。最近，有研究表明，对于许多标准算法，如矩阵乘法、LU-分解或Cholesky分解，可以用k代替，并且可以消除对k的限制。但是，参数大量使用了底层浮点数集和相应算术的特定属性。在本文中，我们得到了真实的数求和的误差估计，其中每个和都受到某种扰动的影响。最近的结果浮点求和如下作为一个推论，特别是四舍五入到最近和定向舍入的误差估计。我们的新估计是尖锐的，并揭示了浮点方案的必要属性，以允许使用省略高阶项的因子对求和进行先验估计。
Standard Wilkinson-type error estimates of floating-point algorithms involve a factorfordenoting the relative rounding error unit of a floating-point number system. Recently, it was shown that, for many standard algorithms such as matrix multiplication, LU- or Cholesky decomposition,can be replaced by, and the restriction onkcan be removed. However, the arguments make heavy use of specific properties of both the underlying set of floating-point numbers and the corresponding arithmetic. In this paper, we derive error estimates for the summation of real numbers where each sum is afflicted with some perturbation. Recent results on floating-point summation follow as a corollary, in particular error estimates for rounding to nearest and for directed rounding. Our new estimates are sharp and unveil the necessary properties of floating-point schemes to allow for a priori estimates of summation with a factor omitting higher order terms.
在 GPU 上使用快速低精度算术的区间矩阵乘法
DOI：
--
发表时间：
2021
期刊：
影响因子：
--
作者：
K. Ozaki;T. Ogita;D. Mukunoki
通讯作者：
D. Mukunoki