naginterfaces.library.specfun.erfcx_​real

naginterfaces.library.specfun.erfcx_real(x)

erfcx_real returns the value of the scaled complementary error function $erfcx\left(x\right)$.

For full information please refer to the NAG Library document for s15ag

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/s/s15agf.html

Parameters

xfloat

The argument $x$ of the function.

Returns

ecxfloat

The value of the scaled complementary error function $erfcx\left(x\right)$.

Warns

NagAlgorithmicWarning

(errno $1$)

On entry, $x=⟨value⟩$ and the constant $x_{hi}=⟨value⟩$.

Constraint: $x<x_{hi}$.

(errno $2$)

On entry, $\left|x\right|$ was in the interval $[⟨value⟩,⟨value⟩)$ where $erfcx\left(x\right)$ is approximately $1/(\sqrt{\pi }\times \left|x\right|)$: $x=⟨value⟩$.

(errno $3$)

On entry, $x=⟨value⟩$ and the constant $x_{neg}=⟨value⟩$.

Constraint: $x\ge x_{neg}$.

Notes

erfcx_real calculates an approximate value for the scaled complementary error function

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

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 $erfcx\left(x\right)$ 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 $erfcx\left(x\right)$ 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).

Asymptotically, $erfcx\left(x\right)\sim 1/\left(\sqrt{\pi }\left|x\right|\right)$. There is a danger of setting underflow in $erfcx\left(x\right)$ whenever $x\ge x_{hi}=min(x_{huge},1/\left(\sqrt{\pi }{x}_{tiny}\right))$, where $x_{huge}$ is the largest positive model number (see machine.real_largest) and $x_{tiny}$ is the smallest positive model number (see machine.real_smallest). In this case erfcx_real exits with $errno$ = 1 and returns $erfcx\left(x\right)=0$. For $x$ in the range $1/\left(2\sqrt{ϵ}\right)\le x<x_{hi}$, where $ϵ$ is the machine precision, the asymptotic value $1/\left(\sqrt{\pi }\left|x\right|\right)$ is returned for $erfcx\left(x\right)$ and erfcx_real exits with $errno$ = 2.

There is a danger of setting overflow in $e^{x^2}$ whenever $x<x_{neg}=-\sqrt{\log \left(x_{huge}/2\right)}$. In this case erfcx_real exits with $errno$ = 3 and returns $erfcx\left(x\right)=x_{huge}$.

References

NIST Digital Library of Mathematical Functions

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