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

- 1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: The routine has been called with an argument that is too large; the default result returned is zero.

$ifail = 2$: 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.

$ifail = - 99$: 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.

$ifail = - 399$: 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.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

δ

and

ε

are the relative errors in the argument and result respectively, then in principle

ε \geq \frac{2 x}{\sin 2 x} δ .

That is a relative error in the argument,

x

, is amplified by at least a factor

2 x / \sin 2 x

in the result.

Similarly if

E

is the absolute error in the result this is given by

E \geq \frac{x}{\cos^{2} x} 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.

Figure 1

Figure 2

In the principal range it is possible to preserve relative accuracy even near the zero of

\tan x

x = 0

but at the other zeros only absolute accuracy is possible. Near the infinities of

\tan x

both the relative and absolute errors become infinite and the routine must fail (error

2

N

is odd and

|θ| \leq 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

- sign θ (\frac{1}{|x F_{2}|}) ≃ - sign θ \tan (\frac{π}{2} - |x F_{2}|)

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| > F_{1}

the routine fails (error

1

) and returns zero. (See the Users' Note for your implementation for specific values of

F_{1}

and

F_{2}

8 Parallelism and Performance

s07aaf is not threaded in any implementation.

9 Further Comments

None.

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.

s07aa: FL CL CPP AD

NAG FL Interfaces07aaf (tan)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
s07aaf (tan)