On entry: ifail must be set to $0$, $\mathrm{-1}$ 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 $\mathrm{-1}$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $\mathrm{-1}$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry ${\mathbf{ifail}}=0$ or $\mathrm{-1}$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{x}}=\u27e8\mathit{\text{value}}\u27e9$ and the constant ${F}_{1}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: $\left|{\mathbf{x}}\right|\le {F}_{1}$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{x}}=\u27e8\mathit{\text{value}}\u27e9$ and the constant ${F}_{2}=\u27e8\mathit{\text{value}}\u27e9$.
The routine has been called with an argument that is too close to an odd multiple of $\pi /2$, 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.
${\mathbf{ifail}}=-99$
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.
${\mathbf{ifail}}=-399$
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.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
7Accuracy
If $\delta $ and $\epsilon $ are the relative errors in the argument and result respectively, then in principle
The equalities should hold if $\delta $ is greater than the machine precision ($\delta $ is a result of data errors etc.) but if $\delta $ 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.
Figure 1
Figure 2
In the principal range it is possible to preserve relative accuracy even near the zero of $\mathrm{tan}x$ at $x=0$ but at the other zeros only absolute accuracy is possible. Near the infinities of $\mathrm{tan}x$ both the relative and absolute errors become infinite and the routine must fail (indicated by ${\mathbf{ifail}}={\mathbf{2}}$).
If $N$ is odd and $\left|\theta \right|\le x{F}_{2}$ 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 $\left|x\right|>{F}_{1}$ the routine fails (indicated by ${\mathbf{ifail}}={\mathbf{1}}$) and returns zero.
(See the Users' Note for your implementation for specific values of ${F}_{1}$ and ${F}_{2}$.)
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
s07aaf is not threaded in any implementation.
9Further Comments
None.
10Example
This example reads values of the argument $x$ from a file, evaluates the function at each value of $x$ and prints the results.