1: $result$ – double scalar: The result of the function.
2: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$: The function has been called with an argument less than or equal to zero for which the function is not defined. The result returned is zero.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

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.

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.

Open in the MATLAB editor: s13ac_example

function s13ac_example


fprintf('s13ac example results\n\n');

x = [0.2:0.2:1];
n = size(x,2);
result = x;

for j=1:n
  [result(j), ifail] = s13ac(x(j));
end

disp('      x          Ci(x)');
fprintf('%12.3e%12.3e\n',[x; result]);

s13ac_plot;



function s13ac_plot
  x = [0.1:0.1:2.9,3:0.5:26];
  for j=1:numel(x)
    [ci(j), ifail] = s13ac(x(j));
  end

  fig1 = figure;
  plot(x,ci);
  xlabel('x');
  ylabel('Ci(x)');
  title('Cosine Integral Ci(x)');
  axis([0 26 -1.5 1]);

s13ac example results

      x          Ci(x)
   2.000e-01  -1.042e+00
   4.000e-01  -3.788e-01
   6.000e-01  -2.227e-02
   8.000e-01   1.983e-01
   1.000e+00   3.374e-01

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox