Internal changes have been made to this routine in some implementations at Mark 27.1.1.
This document reflects the updated routine.
The documentation of the Mark 27.1(.0) implementation is also available here.
The routine may be called by the names s15adf or nagf_specfun_erfc_real.
s15adf calculates an approximate value for the complement of the error function
Unless stated otherwise in the Users' Note, s15adf calls the complementary error function supplied by the compiler used for your implementation; as such, details of the underlying algorithm should be obtained from the documentation supplied by the compiler vendor. The following discussion only applies if the Users' Note for your implementation indicates that the compiler's supplied function was not available.
Let be the root of the equation (then ). For the value of is based on the following rational Chebyshev expansion for :
where denotes a rational function of degree in the numerator and in the denominator.
For the value of is based on a rational Chebyshev expansion for : for the value is based on the expansion
and for it is based on the expansion
For each expansion, the specific values of and are selected to be minimal such that the maximum relative error in the expansion is of the order , where is the maximum number of decimal digits that can be accurately represented for the particular implementation (see x02bef).
For there is a danger of setting underflow in (the value of is given in the Users' Note for your implementation).. For , s15adf returns ; for it returns .
Cody W J (1969) Rational Chebyshev approximations for the error function Math.Comp.23 631–637
1: – Real (Kind=nag_wp)Input
On entry: the argument of the function.
2: – IntegerInput/Output
On entry: ifail must be set to , or to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of means that an error message is printed while a value of means that it is not.
If halting is not appropriate, the value or is recommended. If message printing is undesirable, then the value is recommended. Otherwise, the value is recommended. When the value or is used it is essential to test the value of ifail on exit.
On exit: unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
There are no failure exits from s15adf. The argument ifail has been included for consistency with other routines in this chapter.
Unless stated otherwise in the Users' Note, s15adf calls the complementary error function supplied by the compiler used for your implementation. The following discussion only applies if the Users' Note for your implementation indicates that the compiler's supplied function was not available.
If and are relative errors in the argument and result, respectively, then in principle
That is, the relative error in the argument, , is amplified by a factor in the result.
The behaviour of this factor is shown in Figure 1.
It should be noted that near this factor behaves as and hence the accuracy is largely determined by the machine precision. Also, for large negative , where the factor is , accuracy is mainly limited by machine precision. However, for large positive , the factor becomes and to an extent relative accuracy is necessarily lost. The absolute accuracy is given by
so absolute accuracy is guaranteed for all .
8Parallelism and Performance
s15adf is not threaded in any implementation.
Internal changes have been made to this routine as follows:
At Mark 27.1:
Modified to use a compiler-supplied erfc 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 from a file, evaluates the function at each value of and prints the results.