We consider the computation of the Euclidean (or L2) norm of an n-dimensional vector in floating-point arithmetic. We review the classical solutions used to avoid spurious overflow or underflow and/or to obtain very accurate results. We modify a recently published algorithm (that uses double-word arithmetic) to allow for a very accurate solution, free of spurious overflows and underflows. To that purpose, we use a double-word square-root algorithm of which we provide a tight error analysis. The returned L2 norm will be within very slightly more than 0.5 ulp from the exact result, which means that we will almost always provide correct rounding.

中文翻译：

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用双字运算精确计算欧氏范数

我们考虑在浮点运算中计算n维向量的欧几里德（或 L2）范数。我们回顾了用于避免虚假溢出或下溢和/或获得非常准确结果的经典解决方案。我们修改了最近发布的算法（使用双字算法）以提供非常准确的解决方案，没有虚假的上溢和下溢。为此，我们使用双字平方根算法，我们对其进行了严格的错误分析。返回的 L2 范数与精确结果的误差略大于 0.5 ulp，这意味着我们几乎总是会提供正确的舍入。