s10ab returns the value of the hyperbolic sine, sinhx.

Syntax

C#
public static double s10ab(
	double x,
	out int ifail
)
Visual Basic
Public Shared Function s10ab ( _
	x As Double, _
	<OutAttribute> ByRef ifail As Integer _
) As Double
Visual C++
public:
static double s10ab(
	double x, 
	[OutAttribute] int% ifail
)
F#
static member s10ab : 
        x : float * 
        ifail : int byref -> float 

Parameters

x
Type: System..::..Double
On entry: the argument x of the function.
ifail
Type: System..::..Int32%
On exit: ifail=0 unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

Return Value

s10ab returns the value of the hyperbolic sine, sinhx.

Description

s10ab calculates an approximate value for the hyperbolic sine of its argument, sinhx.
For x1 it uses the Chebyshev expansion
sinhx=x×yt=xr=0arTrt
where t=2x2-1.
For 1<xE1,  sinhx=12ex-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 method fails owing to the danger of setting overflow in calculating ex. The result returned for such calls is sinhsignxE1, i.e., it returns the result for the nearest valid argument.

References

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

Error Indicators and Warnings

Errors or warnings detected by the method:
ifail=1
The method has been called with an argument too large in absolute magnitude. There is a danger of setting overflow. The result is the value of sinhx at the closest argument for which a valid call could be made. (See [Description] and the Users' Note for your implementation.)
ifail=-9000
An error occured, see message report.

Accuracy

If δ and ε are the relative errors in the argument and result, respectively, then in principle
εxcothx×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 x2 
εxδ=Δ
where Δ is the absolute error in the argument.

Parallelism and Performance

None.

Further Comments

None.

Example

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

Example program (C#): s10abe.cs

Example program data: s10abe.d

Example program results: s10abe.r

See Also