NAG FL Interface
s07aaf (tan)

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1 Purpose

s07aaf returns the value of the circular tangent, tanx, via the function name.

2 Specification

Fortran Interface
Function s07aaf ( x, ifail)
Real (Kind=nag_wp) :: s07aaf
Integer, Intent (Inout) :: ifail
Real (Kind=nag_wp), Intent (In) :: x
C Header Interface
#include <nag.h>
double  s07aaf_ (const double *x, Integer *ifail)
The routine may be called by the names s07aaf or nagf_specfun_tan.

3 Description

s07aaf calculates an approximate value for the circular tangent of its argument, tanx. It is based on the Chebyshev expansion
where - π4<θ< π4 and -1 <t<+1 ,   t=2 ( 4θ π ) 2-1 .
The reduction to the standard range is accomplished by taking
where N is an integer and -π4<θ<π4 ,
i.e., θ=x-( 2xπ) π2 where N = [ 2x π ] = ​ the nearest integer to ​ 2x π .
From the properties of tanx it follows that
tanx={ tanθ, Neven -1/tanθ, Nodd }  

4 References

NIST Digital Library of Mathematical Functions

5 Arguments

1: x Real (Kind=nag_wp) Input
On entry: the argument x of the function.
2: ifail Integer Input/Output
On entry: ifail must be set to 0, −1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of −1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value −1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value 0 is recommended. When the value -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry ifail=0 or −1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, x=value and the constant F1=value.
Constraint: |x|F1.
On entry, x=value and the constant F2=value.
The routine has been called with an argument that is too close to an odd multiple of π/2, at which the function is infinite; the routine has returned a value with the correct sign but a more or less arbitrary but large magnitude.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

If δ and ε are the relative errors in the argument and result respectively, then in principle
ε2x sin2x δ.  
That is a relative error in the argument, x, is amplified by at least a factor 2x/sin2x in the result.
Similarly if E is the absolute error in the result this is given by
Excos2x δ.  
The equalities should hold if δ is greater than the machine precision (δ is a result of data errors etc.) but if δ 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.
GnuplotProduced by GNUPLOT 4.6 patchlevel 3 100 101 102 0 π/2 π 3π/2 2π 5π/2 3π 7π/2 |ε/δ| x gnuplot_plot_1
Figure 1
GnuplotProduced by GNUPLOT 4.6 patchlevel 3 100 101 102 0 π/2 π 3π/2 2π 5π/2 3π 7π/2 |ε/δ| x gnuplot_plot_1
Figure 2
In the principal range it is possible to preserve relative accuracy even near the zero of tanx at x=0 but at the other zeros only absolute accuracy is possible. Near the infinities of tanx both the relative and absolute errors become infinite and the routine must fail (indicated by ifail=2).
If N is odd and |θ|xF2 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
-signθ (1|xF2|) -signθtan(π2-|xF2|)  
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 |x|>F1 the routine fails (indicated by ifail=1) and returns zero. (See the Users' Note for your implementation for specific values of F1 and F2.)

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
s07aaf is not threaded in any implementation.

9 Further Comments


10 Example

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

10.1 Program Text

Program Text (s07aafe.f90)

10.2 Program Data

Program Data (s07aafe.d)

10.3 Program Results

Program Results (s07aafe.r)