NAG Library Function Document

nag_cos_integral (s13acc)

+− 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

1 Purpose

nag_cos_integral (s13acc) returns the value of the cosine integral

Ci (x) = γ + \ln x + \int_{0}^{x} \frac{\cos u - 1}{u} d u, x > 0

where

γ

denotes Euler's constant.

2 Specification

#include <nag.h>

#include <nags.h>

double

nag_cos_integral (double x, NagError *fail)

3 Description

nag_cos_integral (s13acc) calculates an approximate value for

Ci (x)

For

0 < x \leq 16

it is based on the Chebyshev expansion

Ci (x) = \ln x + \sum_{r = 0}^{'} a_{r} T_{r} (t), t = 2 {(\frac{x}{16})}^{2} - 1 .

For

16 < x < x_{hi}

where the value of

x_{hi}

is given in the Users' Note for your implementation,

Ci (x) = \frac{f (x) \sin x}{x} - \frac{g (x) \cos x}{x^{2}}

where

f (x) = \underset{r = 0}{\sum^{'}} f_{r} T_{r} (t)

and

g (x) = \underset{r = 0}{\sum^{'}} g_{r} T_{r} (t)

t = 2 {(\frac{16}{x})}^{2} - 1

For

x \geq x_{hi}

Ci (x) = 0

to within the accuracy possible (see Section 7).

4 References

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

5 Arguments

1: x – doubleInput: On entry: the argument $x$ of the function.
Constraint: $x > 0.0$ .
2: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

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.
NE_REAL_ARG_LE: On entry, $x = ⟨value⟩$ .
Constraint: $x > 0.0$ .
The function has been called with an argument less than or equal to zero for which $Ci (x)$ is not defined.

7 Accuracy

E

and

ε

are the absolute and relative errors in the result and

δ

is the relative error in the argument then in principle these are related by

|E| ≃ |δ \cos x| and ​ |ε| ≃ |\frac{δ \cos x}{Ci (x)}| .

That is accuracy will be limited by machine precision near the origin and near the zeros of

\cos x

, but near the zeros of

Ci (x)

only absolute accuracy can be maintained.

The behaviour of this amplification is shown in Figure 1.

Figure 1

For large values of

x

Ci (x) \sim \frac{\sin x}{x}

therefore

ε \sim δ x \cot x

and since

δ

is limited by the finite precision of the machine it becomes impossible to return results which have any relative accuracy. That is, when

x \geq 1 / δ

we have that

|Ci (x)| \leq 1 / x \sim E

and hence is not significantly different from zero.

Hence

x_{hi}

is chosen such that for values of

x \geq x_{hi}

Ci (x)

in principle would have values less than the machine precision and so is essentially zero.

8 Parallelism and Performance

Not applicable.

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.

NAG Library Function Documentnag_cos_integral (s13acc)