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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_specfun_erfc_real (s15ad)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_specfun_erfc_real (s15ad) returns the value of the complementary error function, erfcx, via the function name.


[result, ifail] = s15ad(x)
[result, ifail] = nag_specfun_erfc_real(x)


nag_specfun_erfc_real (s15ad) calculates an approximate value for the complement of the error function
Let x^ be the root of the equation erfcx-erfx=0 (then x^0.46875). For xx^ the value of erfcx is based on the following rational Chebyshev expansion for erfx:
where R,m denotes a rational function of degree  in the numerator and m in the denominator.
For x>x^ the value of erfcx is based on a rational Chebyshev expansion for erfcx: for x^<x4 the value is based on the expansion
and for x>4 it is based on the expansion
For each expansion, the specific values of  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)).
For xxhi there is a danger of setting underflow in erfcx. For xxhi, nag_specfun_erfc_real (s15ad) returns erfcx=0; for x-xhi it returns erfcx=2.


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


Compulsory Input Parameters

1:     x – double scalar
The argument x of the function.

Optional Input Parameters


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

There are no failure exits from nag_specfun_erfc_real (s15ad). The argument ifail has been included for consistency with other functions in this chapter.


If δ and ε are relative errors in the argument and result, respectively, then in principle
ε 2x e -x2 πerfcx δ .  
That is, the relative error in the argument, x, is amplified by a factor 2xe-x2 πerfcx  in the result.
The behaviour of this factor is shown in Figure 1.
Figure 1
Figure 1
It should be noted that near x=0 this factor behaves as 2xπ  and hence the accuracy is largely determined by the machine precision. Also for large negative x, where the factor is xe-x2π , accuracy is mainly limited by machine precision. However, for large positive x, the factor becomes 2x2 and to an extent relative accuracy is necessarily lost. The absolute accuracy E is given by
so absolute accuracy is guaranteed for all x.

Further Comments



This example reads values of the argument x from a file, evaluates the function at each value of x and prints the results.
function s15ad_example

fprintf('s15ad example results\n\n');

x = [-10   -1     0    1     10];
n = size(x,2);
result = x;

for j=1:n
  [result(j), ifail] = s15ad(x(j));

disp('      x         erfc(x)');
fprintf('%12.3e%12.3e\n',[x; result]);

s15ad example results

      x         erfc(x)
  -1.000e+01   2.000e+00
  -1.000e+00   1.843e+00
   0.000e+00   1.000e+00
   1.000e+00   1.573e-01
   1.000e+01   2.088e-45

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

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