is the maximum number of decimal digits that can be accurately represented for the particular implementation (see nag_machine_decimal_digits (x02be)).

Asymptotically,

erfcx (x) \sim 1 / (\sqrt{π} |x|)

. There is a danger of setting underflow in

erfcx (x)

whenever

x \geq x_{hi} = \min (x_{huge}, 1 / (\sqrt{π} x_{tiny}))

, where

x_{huge}

is the largest positive model number (see nag_machine_real_largest (x02al)) and

x_{tiny}

is the smallest positive model number (see nag_machine_real_smallest (x02ak)). In this case nag_specfun_erfcx_real (s15ag) exits with

ifail = 1

and returns

erfcx (x) = 0

. For

x

in the range

1 / (2 \sqrt{ε}) \leq x < x_{hi}

, where

ε

is the machine precision, the asymptotic value

1 / (\sqrt{π} |x|)

is returned for

erfcx (x)

and nag_specfun_erfcx_real (s15ag) exits with

ifail = 2

There is a danger of setting overflow in

e^{x^{2}}

whenever

x < x_{neg} = - \sqrt{\log (x_{huge} / 2)}

. In this case nag_specfun_erfcx_real (s15ag) exits with

ifail = 3

and returns

erfcx (x) = x_{huge}

The values of

x_{hi}

1 / (2 \sqrt{ε})

and

x_{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: $x$ – double scalar: The argument $x$ of the function.

Optional Input Parameters

None.

Output Parameters

1: $result$ – double scalar: The result of the function.
2: $ifail$ – int64int32nag_int scalar: $ifail = 0$ 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 $ifail = 1$: Constraint: $x < x_{hi}$ .

W $ifail = 2$: On entry, $|x|$ was in the interval $[_,_)$ where $erfcx (x)$ is approximately $1 / (\sqrt{π} * |x|)$ : .

W $ifail = 3$: Constraint: $x \geq x_{neg}$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

The relative error in computing

erfcx (x)

may be estimated by evaluating

E = \frac{erfcx (x) - e^{x^{2}} \sum_{n = 1}^{\infty} I^{n} erfc (x)}{erfcx (x)},

where

I^{n}

denotes repeated integration. Empirical results suggest that on the interval

(\hat{x}, 2)

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

(2, 20)

the values are around

3.5

for maximum and

0.45

for root-mean-square relative errors; note that on these two intervals

erfc (x)

is the primary computation. See also Accuracy in nag_specfun_erfc_real (s15ad).

Further Comments

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.

Open in the MATLAB editor: s15ag_example

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

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox