s07aaf calculates an approximate value for the circular tangent of its argument, . It is based on the Chebyshev expansion
where and .
The reduction to the standard range is accomplished by taking
where is an integer and ,
i.e., where .
From the properties of it follows that
Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
1: – Real (Kind=nag_wp)Input
On entry: the argument of the function.
2: – IntegerInput/Output
On entry: ifail must be set to , . If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value is recommended. If the output of error messages is undesirable, then the value is recommended. Otherwise, if you are not familiar with this argument, the recommended value is . When the value 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).
Error Indicators and Warnings
If on entry or , explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
The routine has been called with an argument that is larger in magnitude than ; the default result returned is zero. The value of is given in the Users' Note for your implementation.
The routine has been called with an argument that is too close (as determined using the relative tolerance ) to an odd multiple of , at which the function is infinite; the routine returns a value with the correct sign but a more or less arbitrary but large magnitude (see Section 7). The value of is given in the Users' Note for your implementation.
An unexpected error has been triggered by this routine. Please
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.
If and are the relative errors in the argument and result respectively, then in principle
That is a relative error in the argument, , is amplified by at least a factor in the result.
Similarly if is the absolute error in the result this is given by
The equalities should hold if is greater than the machine precision ( is a result of data errors etc.) but if is simply the round-off error in the machine it is possible that internal calculation rounding will lose an extra figure.
The graphs below show the behaviour of these amplification factors.
In the principal range it is possible to preserve relative accuracy even near the zero of at but at the other zeros only absolute accuracy is possible. Near the infinities of both the relative and absolute errors become infinite and the routine must fail (error ).
If is odd and the routine could not return better than two figures and in all probability would produce a result that was in error in its most significant figure. Therefore the routine fails and it returns the value
which is the value of the tangent at the nearest argument for which a valid call could be made.
Accuracy is also unavoidably lost if the routine is called with a large argument. If the routine fails (error ) and returns zero. (See the Users' Note for your implementation for specific values of and .)
Parallelism and Performance
s07aaf is not threaded in any implementation.
This example reads values of the argument from a file, evaluates the function at each value of and prints the results.