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Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_specfun_airy_bi_deriv (s17ak)

▸▿ Contents

1 Purpose

2 Syntax

3 Description

4 References

▸▿ 5 Parameters

5.1 Compulsory Input Parameters

5.2 Optional Input Parameters

5.3 Output Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

Purpose

nag_specfun_airy_bi_deriv (s17ak) returns a value for the derivative of the Airy function

Bi (x)

, via the function name.

Syntax

[result, ifail] = s17ak(x)

[result, ifail] = nag_specfun_airy_bi_deriv(x)

Description

nag_specfun_airy_bi_deriv (s17ak) calculates an approximate value for the derivative of the Airy function

Bi (x)

. It is based on a number of Chebyshev expansions.

For

x < - 5

{Bi}^{'} (x) = \sqrt[4]{- x} [- a (t) \sin z + \frac{b (t)}{ζ} \cos z],

where

z = \frac{π}{4} + ζ

ζ = \frac{2}{3} \sqrt{- x^{3}}

and

a (t)

and

b (t)

are expansions in the variable

t = - 2 {(\frac{5}{x})}^{3} - 1

For

- 5 \leq x \leq 0

{Bi}^{'} (x) = \sqrt{3} (x^{2} f (t) + g (t)),

where

f

and

g

are expansions in

t = - 2 {(\frac{x}{5})}^{3} - 1

For

0 < x < 4.5

{Bi}^{'} (x) = e^{3 x / 2} y (t),

where

y (t)

is an expansion in

t = 4 x / 9 - 1

For

4.5 \leq x < 9

{Bi}^{'} (x) = e^{21 x / 8} u (t),

where

u (t)

is an expansion in

t = 4 x / 9 - 3

For

x \geq 9

{Bi}^{'} (x) = \sqrt[4]{x} e^{z} v (t),

where

z = \frac{2}{3} \sqrt{x^{3}}

and

v (t)

is an expansion in

t = 2 (\frac{18}{z}) - 1

For

|x| <

the square of the machine precision, the result is set directly to

{Bi}^{'} (0)

. This saves time and avoids possible underflows in calculation.

For large negative arguments, it becomes impossible to calculate a result for the oscillating function with any accuracy so the function must fail. This occurs for

x < - {(\frac{\sqrt{π}}{ε})}^{4 / 7}

, where

ε

is the machine precision.

For large positive arguments, where

{Bi}^{'}

grows in an essentially exponential manner, there is a danger of overflow so the function must fail.

References

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

Parameters

Compulsory Input Parameters

1: $x$ – double scalar: The argument $x$ of the function.

Optional Input Parameters

None.

Output Parameters

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:

$ifail = 1$: x is too large and positive. On soft failure, the function returns zero.

$ifail = 2$: x is too large and negative. On soft failure the function returns 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

For negative arguments the function is oscillatory and hence absolute error is appropriate. In the positive region the function has essentially exponential behaviour and hence relative error is needed. The absolute error,

E

, and the relative error

ε

, are related in principle to the relative error in the argument

δ

, by

E ≃ |x^{2} Bi (x)| δ ε ≃ |\frac{x^{2} Bi (x)}{{Bi}^{'} (x)}| δ .

In practice, approximate equality is the best that can be expected. When

δ

ε

E

is of the order of the machine precision, the errors in the result will be somewhat larger.

For small

x

, positive or negative, errors are strongly attenuated by the function and hence will effectively be bounded by the machine precision.

For moderate to large negative

x

, the error is, like the function, oscillatory. However, the amplitude of the absolute error grows like

\frac{{|x|}^{7 / 4}}{\sqrt{π}}

. Therefore it becomes impossible to calculate the function with any accuracy if

{|x|}^{7 / 4} > \frac{\sqrt{π}}{δ}

For large positive

x

, the relative error amplification is considerable:

\frac{ε}{δ} \sim \sqrt{x^{3}}

. However, very large arguments are not possible due to the danger of overflow. Thus in practice the actual amplification that occurs is limited.

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: s17ak_example

function s17ak_example


fprintf('s17ak example results\n\n');

x = [-10    -1    0    1    5    10   20];
n = size(x,2);
result = x;

for j=1:n
  [result(j), ifail] = s17ak(x(j));
end

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

s17ak_plot;



function s17ak_plot
  x = [-10:0.1:3];
  for j = 1:numel(x)
    [Bid(j), ifail] = s17ak(x(j));
  end

  fig1 = figure;
  plot(x,Bid,'-r');
  xlabel('x');
  ylabel('Bi''(x)');
  title('Derivative of Airy Function Bi(x)');
  axis([-10 4 -2 10]);

s17ak example results

      x         Bi'(x)
  -1.000e+01   1.194e-01
  -1.000e+00   5.924e-01
   0.000e+00   4.483e-01
   1.000e+00   9.324e-01
   5.000e+00   1.436e+03
   1.000e+01   1.429e+09
   2.000e+01   9.382e+25

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Chapter Contents

Chapter Introduction

NAG Toolbox