Reducing the influence of tiny normwise relative errors on performance profiles
Reducing the influence of tiny normwise relative errors on performance profiles
复制标题
DOI：
10.1145/2491491.2491494
复制
发表时间：
2013-07
期刊：
影响因子：
--
通讯作者：
N. Dingle;N. Higham
中科院分区：
文献类型：
--
作者：
N. Dingle;N. Higham
一个普遍但很少被注意到的现象是，相同精度的浮点数向量 x 和 y 的规范相对误差 ‖x - y‖/‖x‖（其中 y 是 x 的近似值）可能比单位舍入小许多数量级。我们分析了这种现象并表明，在 ∞ 范数中，当 x 具有幅度变化很大的分量并且 x 的最大幅度的每个分量与 y 的相应分量一致时，就会发生这种现象。性能概况是根据特定性能指标来比较竞争算法的一种流行方法。我们表明，由于零或微小的规范相对误差的影响，基于规范相对误差的性能概况可能会产生误导性的印象。我们提出了一种转换，以受控的方式减少这些极端错误的影响，同时保留基础数据的单调性并使性能概况在其左端点保持不变。人工数据和真实数据的数值示例说明了转型的好处。
It is a widespread but little-noticed phenomenon that the normwise relative error ‖x - y‖/‖x‖ of vectors x and y of floating point numbers of the same precision, where y is an approximation to x, can be many orders of magnitude smaller than the unit roundoff. We analyze this phenomenon and show that in the ∞-norm it happens precisely when x has components of widely varying magnitude and every component of x of largest magnitude agrees with the corresponding component of y. Performance profiles are a popular way to compare competing algorithms according to particular measures of performance. We show that performance profiles based on normwise relative errors can give a misleading impression due to the influence of zero or tiny normwise relative errors. We propose a transformation that reduces the influence of these extreme errors in a controlled manner, while preserving the monotonicity of the underlying data and leaving the performance profile unchanged at its left end-point. Numerical examples with both artificial and genuine data illustrate the benefits of the transformation.