naginterfaces.library.specfun.erf_​real

naginterfaces.library.specfun.erf_real(x)

erf_real returns the value of the error function $erf\left(x\right)$.

For full information please refer to the NAG Library document for s15ae

https://www.nag.com/numeric/nl/nagdoc_29.3/flhtml/s/s15aef.html

Parameters

xfloat

The argument $x$ of the function.

Returns

resfloat

The value of the error function $erf\left(x\right)$.

Notes

erf_real calculates an approximate value for the error function

$$erf\left(x\right)=\frac{2}{\sqrt{\pi }}{\int }_0^x{e}^{-t^2}dt=1-erfc\left(x\right)\text{.}$$

Let $\hat{x}$ be the root of the equation $erfc\left(x\right)-erf\left(x\right)=0$ (then $\hat{x}\approx 0.46875$). For $\left|x\right|\le \hat{x}$ the value of $erf\left(x\right)$ is based on the following rational Chebyshev expansion for $erf\left(x\right)$:

$$erf\left(x\right)\approx x{R}_{\ell ,m}\left(x^2\right)\text{,}$$

where $R_{\ell ,m}$ denotes a rational function of degree $\ell$ in the numerator and $m$ in the denominator.

For $\left|x\right|>\hat{x}$ the value of $erf\left(x\right)$ is based on a rational Chebyshev expansion for $erfc\left(x\right)$: for $\hat{x}<\left|x\right|\le 4$ the value is based on the expansion

$$erfc\left(x\right)\approx e^{x^2}{R}_{\ell ,m}\left(x\right)\text{;}$$

and for $\left|x\right|>4$ it is based on the expansion

$$erfc\left(x\right)\approx \frac{e^{x^2}}{x}(\frac{1}{\sqrt{\pi }}+\frac{1}{x^2}{R}_{\ell ,m}\left(1/x^2\right))\text{.}$$

For each expansion, the specific values of $\ell$ and $m$ are selected to be minimal such that the maximum relative error in the expansion is of the order ${10}^{-d}$, where $d$ is the maximum number of decimal digits that can be accurately represented for the particular implementation (see machine.decimal_digits).

For $\left|x\right|\ge x_{hi}$ there is a danger of setting underflow in $erfc\left(x\right)$. For $x\ge x_{hi}$, erf_real returns $erf\left(x\right)=1$; for $x\le -x_{hi}$ it returns $erf\left(x\right)=-1$.

References

NIST Digital Library of Mathematical Functions

Cody, W J, 1969, Rational Chebyshev approximations for the error function, Math.Comp. (23), 631–637