NAG Library Routine Document

S07AAF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

S07AAF returns the value of the circular tangent,

\tan x

, via the function name.

2 Specification

FUNCTION S07AAF (

X, IFAIL)

REAL (KIND=nag_wp) S07AAF

INTEGER	IFAIL
REAL (KIND=nag_wp)	X

3 Description

S07AAF calculates an approximate value for the circular tangent of its argument,

\tan x

. It is based on the Chebyshev expansion

\tan θ = θ y (t) = θ \underset{r = 0}{\sum^{'}} c_{r} T_{r} (t)

where

- \frac{π}{4} < θ < \frac{π}{4}

and

- 1 < t < + 1, t = 2 {(\frac{4 θ}{π})}^{2} - 1

The reduction to the standard range is accomplished by taking

x = N π / 2 + θ

where

N

is an integer and

- \frac{π}{4} < θ < \frac{π}{4}

i.e.,

θ = x - (\frac{2 x}{π}) \frac{π}{2}

where

N = [\frac{2 x}{π}] = ​ the nearest integer to ​ \frac{2 x}{π}

From the properties of

\tan x

it follows that

\tan x = \{\begin{array}{r} \tan θ, & N even \\ - 1 / \tan θ, & N odd \end{array}\}

4 References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications

5 Parameters

1: X – REAL (KIND=nag_wp)Input: On entry: the argument $x$ of the function.
2: IFAIL – INTEGERInput/Output: On entry: IFAIL must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $- 1 or 1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$ . 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 larger in magnitude than $F$ ; the default result returned is zero. The value of $F$ is given in the Users' Note for your implementation.

$IFAIL = 2$: The routine has been called with an argument that is too close (as determined using the relative tolerance $F$ ) to an odd multiple of $π / 2$ , 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 $F$ is given in the Users' Note for your implementation.

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 Further Comments

None.

9 Example

This example reads values of the argument

x

from a file, evaluates the function at each value of

x

and prints the results.

NAG Library Routine DocumentS07AAF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

S07AAF