NAG FL Interface
s10abf (sinh)

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1 Purpose

s10abf returns the value of the hyperbolic sine, sinhx, via the function name.

2 Specification

Fortran Interface
Function s10abf ( x, ifail)
Real (Kind=nag_wp) :: s10abf
Integer, Intent (Inout) :: ifail
Real (Kind=nag_wp), Intent (In) :: x
C Header Interface
#include <nag.h>
double  s10abf_ (const double *x, Integer *ifail)
The routine may be called by the names s10abf or nagf_specfun_sinh.

3 Description

s10abf calculates an approximate value for the hyperbolic sine of its argument, sinhx.
For |x|1 it uses the Chebyshev expansion
where t=2x2-1.
For 1<|x|E1,  sinhx=12(ex-e-x)
where E1 is a machine-dependent constant , details of which are given in the Users' Note for your implementation.
For |x|>E1, the routine fails owing to the danger of setting overflow in calculating ex. The result returned for such calls is sinh(signxE1), i.e., it returns the result for the nearest valid argument.

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.
Constraint: |x|E1.
The routine has been called with an argument too large in absolute magnitude. There is a danger of overflow. The result returned is the value of sinhx at the closest argument for which a valid call could be made.
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
|ε| |xcothx×δ|.  
That is the relative error in the argument, x, is amplified by a factor, approximately xcothx. The equality should hold if δ is greater than the machine precision (δ is a result of data errors etc.) but, if δ is simply a result of round-off in the machine representation of x, then it is possible that an extra figure may be lost in internal calculation round-off.
The behaviour of the error amplification factor can be seen in the following graph:
Figure 1
Figure 1
It should be noted that for |x|2
where Δ is the absolute error in the argument.

8 Parallelism and Performance

s10abf 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 (s10abfe.f90)

10.2 Program Data

Program Data (s10abfe.d)

10.3 Program Results

Program Results (s10abfe.r)