NAG Library Manual, Mark 28.3
Interfaces:  FL   CL   CPP   AD 
NAG CL Interface Introduction
S (Specfun) Chapter Contents
S (Specfun) Chapter Introduction
s11ac:  FL   CL   CPP   AD 

NAG CL Interface
s11acc (arccosh)

NAG Library Manual, Mark 28.3
Interfaces:  FL   CL   CPP   AD 
NAG CL Interface Introduction
S (Specfun) Chapter Contents
S (Specfun) Chapter Introduction
s11ac:  FL   CL   CPP   AD 

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© The Numerical Algorithms Group Ltd. 2022
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1 Purpose

s11acc returns the value of the inverse hyperbolic cosine, arccosh⁡x. The result is in the principal positive branch.

2 Specification

#include <nag.h>
double  s11acc (double x, NagError *fail)
The function may be called by the names: s11acc, nag_specfun_arccosh or nag_arccosh.

3 Description

s11acc calculates an approximate value for the inverse hyperbolic cosine, arccosh⁡x. It is based on the relation
arccosh⁡x=ln(x+x2-1).  
This form is used directly for 1<x<10k, where k=n/2+1, and the machine uses approximately n decimal place arithmetic.
For x≥10k, x2-1 is equal to x to within the accuracy of the machine and hence we can guard against premature overflow and, without loss of accuracy, calculate
arccosh⁡x=ln⁡2+ln⁡x.  

4 References

NIST Digital Library of Mathematical Functions

5 Arguments

1: x – double Input
On entry: the argument x of the function.
Constraint: x≥1.0.
2: fail – NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_REAL_ARG_LT
On entry, x=⟨value⟩.
Constraint: x≥1.0.
The function has been called with an argument less than 1.0, for which arccosh⁡x is not defined.

7 Accuracy

If δ and ε are the relative errors in the argument and result respectively, then in principle
|ε|≃ | x x2-1 arccosh⁡x ×δ| .  
That is the relative error in the argument is amplified by a factor at least xx2-1arccosh⁡x in the result. The equality should apply if δ is greater than the machine precision (δ due to data errors etc.) but if δ is simply a result of round-off in the machine representation it is possible that an extra figure may be lost in internal calculation and round-off. The behaviour of the amplification factor is shown in the following graph:
Figure 1
Figure 1
It should be noted that for x>2 the factor is always less than 1.0. For large x we have the absolute error E in the result, in principle, given by
E∼δ.  
This means that eventually accuracy is limited by machine precision. More significantly for x close to 1, x-1∼δ, the above analysis becomes inapplicable due to the fact that both function and argument are bounded, x≥1, arccosh⁡x≥0. In this region we have
E∼δ.  
That is, there will be approximately half as many decimal places correct in the result as there were correct figures in the argument.

8 Parallelism and Performance

s11acc 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.

10.1 Program Text

Program Text (s11acce.c)

10.2 Program Data

Program Data (s11acce.d)

10.3 Program Results

Program Results (s11acce.r)
NAG Library Manual, Mark 28.3
Interfaces:  FL   CL   CPP   AD 
NAG CL Interface Introduction
S (Specfun) Chapter Contents
S (Specfun) Chapter Introduction
s11ac:  FL   CL   CPP   AD 
© The Numerical Algorithms Group Ltd. 2022