Catastrophic cancellation (original) (raw)
Unter Auslöschung (engl. cancellation) versteht man in der Numerik den Verlust an Genauigkeit bei der Subtraktion fast gleich großer Gleitkommazahlen.
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| dbo:abstract | In numerical analysis, catastrophic cancellation is the phenomenon that subtracting good approximations to two nearby numbers may yield a very bad approximation to the difference of the original numbers. For example, if there are two studs, one long and the other long, and they are measured with a ruler that is good only to the centimeter, then the approximations could come out to be and . These may be good approximations, in relative error, to the true lengths: the approximations are in error by less than 2% of the true lengths, . However, if the approximate lengths are subtracted, the difference will be , even though the true difference between the lengths is . The difference of the approximations, , is in error by 100% of the magnitude of the difference of the true values, . Catastrophic cancellation may happen even if the difference is computed exactly, as in the example above—it is not a property of any particular kind of arithmetic like floating-point arithmetic; rather, it is inherent to subtraction, when the inputs are approximations themselves. Indeed, in floating-point arithmetic, when the inputs are close enough, the floating-point difference is computed exactly, by the Sterbenz lemma—there is no rounding error introduced by the floating-point subtraction operation. (en) Unter Auslöschung (engl. cancellation) versteht man in der Numerik den Verlust an Genauigkeit bei der Subtraktion fast gleich großer Gleitkommazahlen. (de) 在数值分析中，灾难性抵消（英語：catastrophic cancellation）是指两个大小相近的数值的近似值相减，得到的差值可能和原始数值相减得到的真实的差值有很大差异，因而近似值的差值不能用作真实值差值的近似值。 例如，如果有两个螺柱，一个长，另一个长，用厘米刻度的尺子测量其长度，得到的近似值为和。在相对误差方面，它们是真实长度的良好的近似值：近似值的误差小于真实长度的2%，即 。 但是，如果用这些近似长度相减，则差值为，而长度之间的真实差值是。用近似值算出的差，和用真实值算出的差相比，偏离了100%。 即使差值计算本身是精确的，灾难性抵消仍然有可能发生，如上例所示——它不是哪种类型的运算（如浮点运算）的属性；当输入值本身是近似值时，进行减法运算就必有灾难性抵消。实际上，根据，浮点运算中，当输入值足够接近时，浮点差可以精确计算——浮点减法运算本身并未引入捨入誤差。 (zh) |
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| rdfs:comment | Unter Auslöschung (engl. cancellation) versteht man in der Numerik den Verlust an Genauigkeit bei der Subtraktion fast gleich großer Gleitkommazahlen. (de) 在数值分析中，灾难性抵消（英語：catastrophic cancellation）是指两个大小相近的数值的近似值相减，得到的差值可能和原始数值相减得到的真实的差值有很大差异，因而近似值的差值不能用作真实值差值的近似值。 例如，如果有两个螺柱，一个长，另一个长，用厘米刻度的尺子测量其长度，得到的近似值为和。在相对误差方面，它们是真实长度的良好的近似值：近似值的误差小于真实长度的2%，即 。 但是，如果用这些近似长度相减，则差值为，而长度之间的真实差值是。用近似值算出的差，和用真实值算出的差相比，偏离了100%。 即使差值计算本身是精确的，灾难性抵消仍然有可能发生，如上例所示——它不是哪种类型的运算（如浮点运算）的属性；当输入值本身是近似值时，进行减法运算就必有灾难性抵消。实际上，根据，浮点运算中，当输入值足够接近时，浮点差可以精确计算——浮点减法运算本身并未引入捨入誤差。 (zh) In numerical analysis, catastrophic cancellation is the phenomenon that subtracting good approximations to two nearby numbers may yield a very bad approximation to the difference of the original numbers. For example, if there are two studs, one long and the other long, and they are measured with a ruler that is good only to the centimeter, then the approximations could come out to be and . These may be good approximations, in relative error, to the true lengths: the approximations are in error by less than 2% of the true lengths, . (en) |
| rdfs:label | Auslöschung (numerische Mathematik) (de) Catastrophic cancellation (en) Нищівне скасування (uk) 灾难性抵消 (zh) |
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