NAG CL Interface
s15aec (erf_​real)

s15aec returns the value of the error function $\mathrm{erf}\left(x\right)$.

The function may be called by the names: s15aec, nag_specfun_erf_real or nag_erf.

s15aec calculates an approximate value for the 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 $\mathrm{erf}\left(x\right)$ 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 $\mathrm{erf}\left(x\right)$ 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 X02BEC).

For $\left|x\right|\ge x_{\mathrm{hi}}$ there is a danger of setting underflow in $\mathrm{erfc}\left(x\right)$ (the value of $x_{\mathrm{hi}}$ is given in the Users' Note for your implementation). For $x\ge x_{\mathrm{hi}}$, s15aec returns $\mathrm{erf}\left(x\right)=1$; for $x\le -x_{\mathrm{hi}}$ it returns $\mathrm{erf}\left(x\right)=-1$.

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

None.

See Section 7 in s15adc.

s15aec is not threaded in any implementation.

Internal changes have been made to this function as follows:

• At Mark 27.1:
  Modified to use a compiler-supplied erf function, when available.

For details of all known issues which have been reported for the NAG Library please refer to the Known Issues.

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