Tanh-sinh quadrature用于使用浮点运算的单积分和多积分
Tanh-sinh quadrature for single and multiple integration using floating-point arithmetic
摘要：
在所有的科学研究领域，特别是在物理学中，有或没有端点奇点的一维和多维积分的估计问题是普遍存在的。虽然tanh-sinh求积可以很好地处理大多数情况，但它的使用并没有在物理学家中广泛传播。此外，虽然大多数计算受到有限精度浮点运算的限制，但在文献中大多缺乏对tanh-sinh求积的类似考虑，其中通常假设无限精度浮点数。也很少有资料可用于应用双曲正切-正弦正交多重积分。我们已经研究了在单次和多次积分中使用双曲正切-正弦求积时，与有限精度浮点数相关的风险和限制，同时获得出色的收敛速度。此外，本文提供了一个简单的实施使用有限精度浮点数和避免数值不稳定性的建议。
英文摘要：
The problem of estimating single- and multi-dimensional integrals, with or without end-point singularities, is prevalent in all fields of scientific research, and in particular in physics. Although tanh-sinh quadrature is known to handle most of these cases excellently, its use is not widely spread among physicists. Moreover, while most calculations are limited by the use of finite-precision floating-point arithmetic, similar considerations for tanh-sinh quadrature are mostly lacking in literature, where infinite-precision floating-point numbers are often assumed. Also little information is available on the application of tanh-sinh quadrature to multiple integration. We have investigated the risks and limitations associated with limited-precision floating-point numbers when using tanh-sinh quadrature for both single and multiple integration, while obtaining excellent convergence rates. In addition, this paper provides recommendations for a straightforward implementation using limited-precision floating-point numbers and for avoiding numerical instabilities.
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