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 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

nag_cos_integral (s13acc) returns the value of the cosine integral

Ci (x)

2 Specification

#include <nag.h>

#include <nags.h>

double

nag_cos_integral (double x, NagError *fail)

3 Description

nag_cos_integral (s13acc) evaluates

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

where

γ

denotes Euler's constant.

The approximation is based on several Chebyshev expansions.

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_REAL_ARG_LE: On entry, x must not be less than or equal to 0.0: $x = 〈value〉$ .
The function is not defined for this value and the result returned is zero.

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| ≃ |(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.

For large values of

x

Ci (x) \sim (\sin x) / x

therefore

\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, for

x > x_{hi}

, where

x_{hi}

is a machine-dependent value,

Ci (x)

in principle has values less than machine precision, and so is set directly to zero.

8 Further Comments

None.

9 Example

The following program 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)