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NAG Toolbox: nag_specfun_airy_bi_deriv (s17ak)
Purpose
nag_specfun_airy_bi_deriv (s17ak) returns a value for the derivative of the Airy function , via the function name.
Syntax
Description
nag_specfun_airy_bi_deriv (s17ak) calculates an approximate value for the derivative of the Airy function . It is based on a number of Chebyshev expansions.
For
,
where
,
and
and
are expansions in the variable
.
For
,
where
and
are expansions in
.
For
,
where
is an expansion in
.
For
,
where
is an expansion in
.
For
,
where
and
is an expansion in
.
For
the square of the
machine precision, the result is set directly to
. 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
, where
is the
machine precision.
For large positive arguments, where 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:
– double scalar
-
The argument of the function.
Optional Input Parameters
None.
Output Parameters
- 1:
– double scalar
The result of the function.
- 2:
– int64int32nag_int scalar
unless the function detects an error (see
Error Indicators and Warnings).
Error Indicators and Warnings
Errors or warnings detected by the function:
-
-
x is too large and positive. On soft failure, the function returns zero.
-
-
x is too large and negative. On soft failure the function returns zero.
-
An unexpected error has been triggered by this routine. Please
contact
NAG.
-
Your licence key may have expired or may not have been installed correctly.
-
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,
, and the relative error
, are related in principle to the relative error in the argument
, by
In practice, approximate equality is the best that can be expected. When
,
or
is of the order of the
machine precision, the errors in the result will be somewhat larger.
For small
, 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 , the error is, like the function, oscillatory. However, the amplitude of the absolute error grows like . Therefore it becomes impossible to calculate the function with any accuracy if .
For large positive , the relative error amplification is considerable: . 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 from a file, evaluates the function at each value of 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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