hide long namesshow long names
hide short namesshow short names
Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

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

NAG Toolbox: nag_specfun_erf_real (s15ae)

Purpose

nag_specfun_erf_real (s15ae) returns the value of the error function erf(x)erf(x), via the function name.

Syntax

[result, ifail] = s15ae(x)
[result, ifail] = nag_specfun_erf_real(x)

Description

nag_specfun_erf_real (s15ae) calculates an approximate value for the error function
x
erf(x) = 2/(sqrt(π))et2dt = 1erfc(x).
0
erf(x)=2π0xe-t2dt=1-erfc(x).
Let x^ be the root of the equation erfc(x)erf(x) = 0erfc(x)-erf(x)=0 (then 0.46875x^0.46875). For |x||x|x^ the value of erf(x)erf(x) is based on the following rational Chebyshev expansion for erf(x)erf(x):
erf(x)xR,m(x2),
erf(x)xR,m(x2),
where R,mR,m denotes a rational function of degree  in the numerator and mm in the denominator.
For |x| > |x|>x^ the value of erf(x)erf(x) is based on a rational Chebyshev expansion for erfc(x)erfc(x): for < |x|4x^<|x|4 the value is based on the expansion
erfc(x)ex2R,m(x);
erfc(x)ex2R,m(x);
and for |x| > 4|x|>4 it is based on the expansion
erfc(x)(ex2)/x(1/(sqrt(π)) + 1/(x2)R,m(1 / x2)).
erfc(x)ex2x(1π+1x2R,m(1/x2)).
For each expansion, the specific values of  and mm are selected to be minimal such that the maximum relative error in the expansion is of the order 10d10-d, where dd is the maximum number of decimal digits that can be accurately represented for the particular implementation (see nag_machine_decimal_digits (x02be)).
For |x|xhi|x|xhi there is a danger of setting underflow in erfc(x)erfc(x). For xxhixxhi, nag_specfun_erf_real (s15ae) returns erf(x) = 1erf(x)=1; for xxhix-xhi it returns erf(x) = 1erf(x)=-1.

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 xx of the function.

Optional Input Parameters

None.

Input Parameters Omitted from the MATLAB Interface

None.

Output Parameters

1:     result – double scalar
The result of the function.
2:     ifail – int64int32nag_int scalar
ifail = 0ifail=0 unless the function detects an error (see [Error Indicators and Warnings]).

Error Indicators and Warnings

There are no failure exits from nag_specfun_erf_real (s15ae). The parameter ifail has been included for consistency with other functions in this chapter.

Accuracy

See Section [Accuracy] in (s15ad).

Further Comments

None.

Example

function nag_specfun_erf_real_example
x = -6;
[result, ifail] = nag_specfun_erf_real(x)
 

result =

    -1


ifail =

                    0


function s15ae_example
x = -6;
[result, ifail] = s15ae(x)
 

result =

    -1


ifail =

                    0



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

© The Numerical Algorithms Group Ltd, Oxford, UK. 2009–2013