public static double s15ag(
	double x,
	out int ifail
)

Public Shared Function s15ag ( _
	x As Double, _
	<OutAttribute> ByRef ifail As Integer _
) As Double

public:
static double s15ag(
	double x, 
	[OutAttribute] int% ifail
)

static member s15ag : 
        x : float * 
        ifail : int byref -> float 

s15ag returns the value of the scaled complementary error function $\mathrm{erfcx}\left(x\right)$.

s15ag calculates an approximate value for the scaled complementary error function

Let $\hat{x}$ be the root of the equation $\mathrm{erfc}\left(x\right)-\mathrm{erf}\left(x\right)=0$ (then $\hat{x}\approx 0.46875$). For $\left|x\right|\le \hat{x}$ the value of $\mathrm{erfcx}\left(x\right)$ is based on the following rational Chebyshev expansion for $\mathrm{erf}\left(x\right)$: 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 $\mathrm{erfcx}\left(x\right)$ is based on a rational Chebyshev expansion for $\mathrm{erfc}\left(x\right)$: for $\hat{x}<\left|x\right|\le 4$ the value is based on the expansion and for $\left|x\right|>4$ it is based on the expansion

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 x02be).

Asymptotically, $\mathrm{erfcx}\left(x\right)\sim 1/\left(\sqrt{\pi }\left|x\right|\right)$. There is a danger of setting underflow in $\mathrm{erfcx}\left(x\right)$ whenever $x\ge x_{\mathrm{hi}}=\min \left(x_{\mathrm{huge}},1/\left(\sqrt{\pi }{x}_{\mathrm{tiny}}\right)\right)$, where $x_{\mathrm{huge}}$ is the largest positive model number (see x02al) and $x_{\mathrm{tiny}}$ is the smallest positive model number (see x02ak). In this case s15ag exits with $\mathbf{ifail}=1$ and returns $\mathrm{erfcx}\left(x\right)=0$. For $x$ in the range $1/\left(2\sqrt{\epsilon }\right)\le x<x_{\mathrm{hi}}$, where $\epsilon$ is the machine precision, the asymptotic value $1/\left(\sqrt{\pi }\left|x\right|\right)$ is returned for $\mathrm{erfcx}\left(x\right)$ and s15ag exits with $\mathbf{ifail}=2$.

There is a danger of setting overflow in $e^{x^2}$ whenever $x<x_{\mathrm{neg}}=-\sqrt{\log \left(x_{\mathrm{huge}}/2\right)}$. In this case s15ag exits with $\mathbf{ifail}=3$ and returns $\mathrm{erfcx}\left(x\right)=x_{\mathrm{huge}}$.

The values of $x_{\mathrm{hi}}$, $1/\left(2\sqrt{\epsilon }\right)$ and $x_{\mathrm{neg}}$ are given in the Users' Note for your implementation.

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications

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

Note: s15ag may return useful information for one or more of the following detected errors or warnings.

Errors or warnings detected by the method:

The relative error in computing $\mathrm{erfcx}\left(x\right)$ may be estimated by evaluating where $I^n$ denotes repeated integration. Empirical results suggest that on the interval $\left(\hat{x},2\right)$ the loss in base $b$ significant digits for maximum relative error is around $3.3$, while for root-mean-square relative error on that interval it is $1.2$ (see x02bh for the definition of the model parameter $b$). On the interval $\left(2,20\right)$ the values are around $3.5$ for maximum and $0.45$ for root-mean-square relative errors; note that on these two intervals $\mathrm{erfc}\left(x\right)$ is the primary computation. See also [Accuracy] in s15ad.

None.

None.

This example reads values of the argument $x$ from a file, evaluates the function at each value of $x$ and prints the results.