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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_specfun_erfcx_real (s15ag)

## Purpose

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

## Syntax

[result, ifail] = s15ag(x)
[result, ifail] = nag_specfun_erfcx_real(x)

## Description

nag_specfun_erfcx_real (s15ag) calculates an approximate value for the scaled complementary error function
 $erfcxx = e x2 erfcx = 2 π e x2 ∫x∞ e-t2 dt = e x2 1- erfx .$
Let $\stackrel{^}{x}$ be the root of the equation $\mathrm{erfc}\left(x\right)-\mathrm{erf}\left(x\right)=0$ (then $\stackrel{^}{x}\approx 0.46875$). For $\left|x\right|\le \stackrel{^}{x}$ the value of $\mathrm{erfcx}\left(x\right)$ is based on the following rational Chebyshev expansion for $\mathrm{erf}\left(x\right)$:
 $erfx≈xRℓ,mx2,$
where ${R}_{\ell ,m}$ denotes a rational function of degree $\ell$ in the numerator and $m$ in the denominator.
For $\left|x\right|>\stackrel{^}{x}$ the value of $\mathrm{erfcx}\left(x\right)$ is based on a rational Chebyshev expansion for $\mathrm{erfc}\left(x\right)$: for $\stackrel{^}{x}<\left|x\right|\le 4$ the value is based on the expansion
 $erfcx≈ex2Rℓ,mx;$
and for $\left|x\right|>4$ it is based on the expansion
 $erfcx≈ex2x1π+1x2Rℓ,m1/x2.$
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 nag_machine_decimal_digits (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 $x\ge {x}_{\mathrm{hi}}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({x}_{\mathrm{huge}},1/\left(\sqrt{\pi }{x}_{\mathrm{tiny}}\right)\right)$, where ${x}_{\mathrm{huge}}$ is the largest positive model number (see nag_machine_real_largest (x02al)) and ${x}_{\mathrm{tiny}}$ is the smallest positive model number (see nag_machine_real_smallest (x02ak)). In this case nag_specfun_erfcx_real (s15ag) exits with ${\mathbf{ifail}}={\mathbf{1}}$ 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 nag_specfun_erfcx_real (s15ag) exits with ${\mathbf{ifail}}={\mathbf{2}}$.
There is a danger of setting overflow in ${e}^{{x}^{2}}$ whenever $x<{x}_{\mathrm{neg}}=-\sqrt{\mathrm{log}\left({x}_{\mathrm{huge}}/2\right)}$. In this case nag_specfun_erfcx_real (s15ag) exits with ${\mathbf{ifail}}={\mathbf{3}}$ 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.

## References

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

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{x}$ – double scalar
The argument $x$ of the function.

None.

### Output Parameters

1:     $\mathrm{result}$ – double scalar
The result of the function.
2:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Note: nag_specfun_erfcx_real (s15ag) may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

W  ${\mathbf{ifail}}=1$
Constraint: ${\mathbf{x}}<{x}_{\mathrm{hi}}$.
W  ${\mathbf{ifail}}=2$
On entry, $\left|{\mathbf{x}}\right|$ was in the interval $\left[_,_\right)$ where $\mathrm{erfcx}\left({\mathbf{x}}\right)$ is approximately $1/\left(\sqrt{\pi }*\left|{\mathbf{x}}\right|\right)$: .
W  ${\mathbf{ifail}}=3$
Constraint: ${\mathbf{x}}\ge {x}_{\mathrm{neg}}$.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

The relative error in computing $\mathrm{erfcx}\left(x\right)$ may be estimated by evaluating
 $E= erfcxx - ex2 ∑ n=1 ∞ Inerfcx erfcxx ,$
where ${I}^{n}$ denotes repeated integration. Empirical results suggest that on the interval $\left(\stackrel{^}{x},2\right)$ 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 nag_machine_model_base (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 nag_specfun_erfc_real (s15ad).

None.

## Example

This example reads values of the argument $x$ from a file, evaluates the function at each value of $x$ and prints the results.
```function s15ag_example

fprintf('s15ag example results\n\n');

x = [-30.0; -6.0; -4.5; -1.0; 1.0; 4.5; 6.0; 7.0e7];

% Catch non-zero ifails
wstat = warning();
warning('OFF');

result = zeros(8, 1);
ifail  = zeros(8, 1, 'int64');
for i=1:8
[result(i), ifail(i)] = s15ag(x(i));
end
fprintf('       x       erfcx(x)    ifail\n');
for i=1:8
fprintf('%10.2e %13.5e     %d\n', x(i), result(i), ifail(i));
end

warning(wstat);

```
```s15ag example results

x       erfcx(x)    ifail
-3.00e+01  1.79769e+308     3
-6.00e+00   8.62246e+15     0
-4.50e+00   1.24593e+09     0
-1.00e+00   5.00898e+00     0
1.00e+00   4.27584e-01     0
4.50e+00   1.22485e-01     0
6.00e+00   9.27766e-02     0
7.00e+07   8.05985e-09     2
```