Error bounds on complex floating-point multiplication
Error bounds on complex floating-point multiplication
复制标题
复数浮点乘法的误差范围
DOI：
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发表时间：
2007
影响因子：
2
通讯作者：
P. Zimmermann
中科院分区：
文献类型：
--
作者：
R. Brent;Colin Percival;P. Zimmermann
Given floating-point arithmetic with $t$-digit base-$eta$ significands in which all arithmetic operations are performed as if calculated to infinite precision and rounded to a nearest representable value, we prove that the product of complex values $z_0$ and $z_1$ can be computed with maximum absolute error $abs{z_0} abs{z_1} frac{1}{2} eta^{1 - t} sqrt{5}$. In particular, this provides relative error bounds of $2^{-24} sqrt{5}$ and $2^{-53} sqrt{5}$ for {IEEE 754} single and double precision arithmetic respectively, provided that overflow, underflow, and denormals do not occur. We also provide the numerical worst cases for {IEEE 754} single and double precision arithmetic.
Given floating-point arithmetic with $t$-digit base-$eta$ significands in which all arithmetic operations are performed as if calculated to infinite precision and rounded to a nearest representable value, we prove that the product of complex values $z_0$ and $z_1$ can be computed with maximum absolute error $abs{z_0} abs{z_1} frac{1}{2} eta^{1 - t} sqrt{5}$. In particular, this provides relative error bounds of $2^{-24} sqrt{5}$ and $2^{-53} sqrt{5}$ for {IEEE 754} single and double precision arithmetic respectively, provided that overflow, underflow, and denormals do not occur. We also provide the numerical worst cases for {IEEE 754} single and double precision arithmetic.