NAG Library Function Document
nag_exp_integral (s13aac)
1
Purpose
nag_exp_integral (s13aac) returns the value of the exponential integral .
2
Specification
| #include <nag.h> |
| #include <nags.h> |
| double |
nag_exp_integral (double x,
NagError *fail) |
|
3
Description
nag_exp_integral (s13aac) calculates an approximate value for
using Chebyshev expansions, where
is real. For
, the real part of the principal value of the integral is taken. The value
is infinite, and so, when
,
nag_exp_integral (s13aac) exits with an error and returns the largest representable machine number.
For
,
where
.
For
,
where
.
In both cases, .
For
, the approximation is based on expansions proposed by
Cody and Thatcher Jr. (1969). Precautions are taken to maintain good relative accuracy in the vicinity of
, which corresponds to a simple zero of Ei(
).
nag_exp_integral (s13aac) guards against producing underflows and overflows by using the argument
; see the
Users' Note for your implementation for the value of
. To guard against overflow, if
the function terminates and returns the negative of the largest representable machine number. To guard against underflow, if
the result is set directly to zero.
4
References
Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
Cody W J and Thatcher Jr. H C (1969) Rational Chebyshev approximations for the exponential integral Ei Math. Comp. 23 289–303
5
Arguments
- 1:
– doubleInput
-
On entry: the argument of the function.
Constraint:
or .
- 2:
– NagError *Input/Output
-
The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
- 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.
See
Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
- NE_REAL_ARG_LE
-
On entry, and the function is infinite.
The evaluation has been abandoned due to the likelihood of overflow. The argument .
7
Accuracy
Unless stated otherwise, it is assumed that .
If
and
are the relative errors in argument and result respectively, then in principle,
so the relative error in the argument is amplified in the result by at least a factor
. The equality should hold if
is greater than the
machine precision (
due to data errors etc.) but if
is simply a result of round-off in the machine representation, it is possible that an extra figure may be lost in internal calculation and round-off.
The behaviour of this amplification factor is shown in the following graph:
Figure 1
It should be noted that, for absolutely small , the amplification factor tends to zero and eventually the error in the result will be limited by machine precision.
For absolutely large
,
the absolute error in the argument.
For , empirical tests have shown that the maximum relative error is a loss of approximately decimal place.
8
Parallelism and Performance
nag_exp_integral (s13aac) is not threaded in any implementation.
None.
10
Example
The following program reads values of the argument from a file, evaluates the function at each value of and prints the results.
10.1
Program Text
Program Text (s13aace.c)
10.2
Program Data
Program Data (s13aace.d)
10.3
Program Results
Program Results (s13aace.r)
