论文标题
误差函数任意准确的分析近似
Arbitrarily Accurate Analytical Approximations for the Error Function
论文作者
论文摘要
在本文中，基于样条的积分近似被用来提出一系列近似值，以比默认的泰勒级数更快地收敛的误差函数。可以通过利用近似ERF（x）大约等于x >> 1的近似值ERF（x）来改进近似值。两个可能的概括是可能的，第一个是基于将集成间隔划分为均等间隔间隔的。第二，它基于使用较大的固定子间隔，并具有已知积分和较小的子互化，其积分将被近似。两种概括都会显着提高准确性。此外，可以将初始近似值和由第一个概括引起的近似值用作自定义动力学系统的输入，以建立具有更好收敛属性的近似值。指示性的结果包括基于四个子截面的第四阶近似值的结果，这导致正面实际线上的相对误差结合为1.43 x 10-7。在正真实的情况下，各种近似值，即实现了10-4、10-6、10-10和10-16的集合相对误差边界。应用程序首先包括上限和下限的函数的定义，该函数的误差函数是任意准确性的。第二，用于错误函数的新系列。第三，EXP（-x2）的近似值的新序列比泰勒串联近似具有更高的收敛性能。第四，满足约束EC（x）^2 + ERF（x）^2 = 1的互补分界函数EC（x）的定义。第五，对正弦信号的功率和谐波失真的任意准确近似，均为错误函数非线性。第六个，用于由误差函数建模的阶跃信号线性过滤的近似表达式。
In this paper a spline based integral approximation is utilized to propose a sequence of approximations to the error function that converge at a significantly faster manner than the default Taylor series. The approximations can be improved by utilizing the approximation erf(x) approximately equal to one for x>>1. Two generalizations are possible, the first is based on demarcating the integration interval into m equally spaced sub-intervals. The second, it based on utilizing a larger fixed sub-interval, with a known integral, and a smaller sub-interval whose integral is to be approximated. Both generalizations lead to significantly improved accuracy. Further, the initial approximations, and the approximations arising from the first generalization, can be utilized as the inputs to a custom dynamical system to establish approximations with better convergence properties. Indicative results include those of a fourth order approximation, based on four sub-intervals, which leads to a relative error bound of 1.43 x 10-7 over the positive real line. Various approximations, that achieve the set relative error bounds of 10-4, 10-6, 10-10 and 10-16, over the positive real, are specified. Applications include, first, the definition of functions that are upper and lower bounds, of arbitrary accuracy, for the error function. Second, new series for the error function. Third, new sequences of approximations for exp(-x2) which have significantly higher convergence properties that a Taylor series approximation. Fourth, the definition of a complementary demarcation function eC(x) which satisfies the constraint eC(x)^2 + erf(x)^2 = 1. Fifth, arbitrarily accurate approximations for the power and harmonic distortion for a sinusoidal signal subject to a error function nonlinearity. Sixth, approximate expressions for the linear filtering of a step signal that is modelled by the error function.
